Proof. Confinement and existence. For sufficiently large \(R_0\) the end spheres have \[H_{\{r=R\}}=\operatorname{div}_g (\nabla r/|\nabla r|_g) =\frac{n-1}{R}+O(R^{-1-q})>0.\] Set \(Y=\nabla r/|\nabla r|_g\). For each competitor \(E\), and almost every \(R>R_0\), the BV trace formula and Gauss–Green on \(E\cap\{r>R\}\) give \[ P_g(E\setminus\{r>R\})-P_g(E) \le-\int_{E\cap\{r>R\}}\operatorname{div}_gY\,dV_g\le0. \tag{15}\] Indeed the new slice area is the flux through the inner end of this region; the original exterior-boundary flux is at most its perimeter. At a good slicing radius the two traces agree, so these are exactly the terms in the perimeter difference. The inequality is strict if the removed set has positive volume.

Fix \(R_1>R_0\). For each competitor choose a good radius in \((R_0,R_1)\) and clip there. This confines a minimizing sequence to the same compact region. BV compactness and lower semicontinuity produce a global minimizer; containment of \(O\) and emptiness in the fixed tail are closed constraints. Applying the strict inequality to any minimizer, at a good radius below a fixed \(R_2\) with \(R_0<R_2<R_1\), puts every minimizer inside the smaller truncation. There is no need for a radius that is simultaneously a good slice for every competitor.

Recovering the full smooth-cut infimum. A full cut supplies a member of \(\mathcal A\), so \(\min P_g\le a(g)\). In the other direction choose a smooth collar field pointing strictly out of \(O\) on \(B\) and let \(\Phi_\varepsilon\) be its flow. For small positive \(\varepsilon\), the set \(\Phi_\varepsilon(E)\) contains a neighborhood of \(O\), and its perimeter tends to \(P_g(E)\). Approximate its indicator in finitely many charts by convolution and a partition of unity, keeping it identically one on a smaller neighborhood of \(O\) and zero outside a fixed compact region. The BV approximation and coarea formula give smooth level sets containing that neighborhood, with perimeters tending to \(P_g(\Phi_\varepsilon(E))\). Fill every bounded complementary component of such a level set and retain the complementary component containing the end. Its closure in \(X\) is a connected smooth outer domain; filling deletes boundary components and creates none. Its full-cut area is no larger than the approximating perimeter. Letting the approximation error, then \(\varepsilon\), tend to zero proves \(a(g)\le P_g(E)\). This argument also works when \(E\) has singular frontier.

Local regularity. Choose a smooth field \(Z\) supported near \(B\), with \(|Z|_g\le1\) and \(Z=\nu_O\) on \(B\). For any local replacement \(F\) of a minimizer \(E\), submodularity, constrained minimality, and Gauss–Green imply \[\begin{align*} P_g(E)&\le P_g(F\cup O) \le P_g(F)+P_g(O)-P_g(F\cap O),\\ P_g(O)-P_g(F\cap O)&\le \int_{O\setminus F}\operatorname{div}_gZ\,dV_g \le C\operatorname{Vol}_g(E\triangle F). \end{align*}\] The unchanged terms outside the replacement region cancel. Thus, if \(E\triangle F\Subset U\), \[ P_g(E;U)\le P_g(F;U)+C\operatorname{Vol}_g(E\triangle F). \tag{16}\] Ball insertion and deletion, followed by the relative isoperimetric inequality, give an upper perimeter bound \(Cr^{n-1}\) and positive lower density bounds for both phases at every frontier point (Maggi 2012). The volume error in Equation (16) scales as \(r^n\). The local almost-minimizing-boundary theorem gives \(C^{1,\alpha}\) regularity at reduced-boundary points and minimizing boundary cones as tangent sets. Dimension reduction gives singular dimension at most \((n-1)-7=n-8\); see (Bi and Zhu 2026, Theorem 2.27) and the codimension-one theory in (Simon 2018, chap. 7). Off the obstacle the minimizing property has zero error, so the regular frontier is smooth and minimal.

At a contact point a tangent filled set contains the tangent half-space of \(O\). Its stationary boundary cone lies on one side of the bounding plane. Such a cone is planar: the nonnegative height function on its spherical link satisfies weakly \(\Delta_{\rm link}z=-(n-2)z\). Testing with the constant function makes its height integral zero, hence its support lies in the plane. The weak stationary identity is on the whole link and does not discard its singular set. The two density bounds and the multiplicity-one boundary property select a half-space rather than the full or empty set. Density regularity therefore makes the contact point a graph point.

Equation (16), applied to flows with both signs, bounds the graph’s weak mean curvature. Its scalar minimal-surface operator has smooth coefficients and is uniformly elliptic on the bounded gradient range. Difference quotients yield \(W^{2,2}\) locally. In nondivergence form the leading coefficients are \(C^{0,\alpha}\) and the right side is bounded; localized linear \(W^{2,p}\) estimates then give every finite \(p\). The Sobolev embedding allows every \(\alpha<1\). In dimensions \(3\) and \(4\) we also use the stronger smooth-obstacle theorem: for a \(C^2\) obstacle, zero bulk term, and ambient dimension below eight, its entire minimizing frontier is \(C^{1,1}\) (Huisken and Ilmanen 2001, Regularity Theorem 1.3(iii), pp. 368–369). This gives the stated \(W^{2,\infty}\) regularity.

The frontier is nonempty. Otherwise its BV indicator is constant on the connected filled manifold, contrary to containment of the open obstacle and absence from an end tail. Its perimeter is positive; this follows from the relative isoperimetric inequality, or from a regular frontier chart.

Connected exterior and smooth approximation in low dimensions. Take the closure of the measure-theoretic interior as representative. When \(n=3,4\) its frontier is compact embedded \(C^{1,1}\), with filled and exterior sets on its two local sides. Any complementary component other than the end component is relatively compact. Its boundary is a nonempty union of frontier components. Filling it removes positive full perimeter, contradicting minimality. The exterior is connected. The same argument applies in every dimension whenever the frontier is smooth. It only uses the local two-sided hypersurface charts and compactness.

Choose a smooth field \(V\) uniformly transverse to the continuous outward normal of the compact frontier \(T\). Its flow gives a \(C^1\) collar \(\Psi:T\times(-\delta,\delta)\to\widehat X\) with filled side \(t\le0\). For small \(\varepsilon>0\), the flowed set is \(E\cup\Psi(T\times(0,\varepsilon])\); every crossing of the frontier is outward. It contains \(O\) in its interior, and the displaced frontier lies at positive distance from \(B\). Smooth a \(C^1\) defining function in a smaller collar, retaining its positive derivative along \(V\). The resulting smooth zero set is a small graph on the collar fibers. Extend the graph displacement to an isotopy supported in that collar; it fixes \(O\) and the distant end. Enclosure and connectedness of the exterior are therefore preserved. The resulting full cuts converge in \(C^1\) and area to \(T\).

Detachment from a strict inner barrier. If \(H_B<0\) for the normal into \(X\), a fixed distance collar \(0\le t\le\varepsilon\) has \(Y=\partial_t\) and \(\operatorname{div}_gY\le-\beta<0\). For a good level \(s\), put \(O_s=O\cup\{0<t<s\}\) and \(W_s=O_s\setminus E\). Gauss–Green on \(W_s\), including the flux on \(t=0\), gives \[ P_g(E\cup O_s)-P_g(E) \le\int_{W_s}\operatorname{div}_gY\,dV_g \le-\beta\operatorname{Vol}_g(W_s). \tag{17}\] Minimality makes \(W_s\) null. Choosing, for each minimizer, a good \(s\in(\varepsilon/2,\varepsilon)\) proves that the same smaller collar lies in its filled interior. Thus all contact is removed.

For \(n=3,4\) the detached frontier is now smooth and minimal. More generally, in any dimension where the frontier under consideration is smooth, its connected closed exterior \(D\) contains just the original end. It is complete in its intrinsic length metric: an intrinsic Cauchy sequence is Cauchy in \(X\), has a limit in the closed set \(D\), and smooth interior or boundary half-ball charts give intrinsic convergence to that limit. Finally, every full enclosing cut of \(D\) gives a filled competitor containing \(E\) and hence \(O\). Its full area is at least \(P_g(E)\). This proves outer area-minimization with every component and all coincidence counted. ◻

Proof. The uniform asymptotic hypothesis gives one mean-convex tail, so Equation (15) confines all minimizers uniformly. The same proof applies under the common-sphere hypothesis. For the last alternative, existence can be proved without an outer barrier. Minimize inside successively larger closed coordinate truncations. Each constrained minimum exists by the direct method and is at most \(P_{g_t}(O)\). Local BV compactness yields a limiting set containing \(O\), of finite total perimeter. The minimizing values decrease to the infimum over bounded competitors. The limit is locally minimizing away from \(O\), and constrained-minimizing at \(O\): to compare it with a compact replacement, glue that replacement into the truncated minimizers across a thin annulus. Coarea chooses a slicing boundary on which the \(L^1\) trace discrepancy tends to zero, so the added perimeter tends to zero. The minimizing inequality then passes to the limit. This is the local minimizing-boundary compactness argument in the BV framework of (Ambrosio et al. 2000; Maggi 2012).

The limit has finite volume, so its exterior phase is the empty one once its frontier is confined. To see the finite-volume assertion before using confinement, apply the Euclidean isoperimetric inequality to the filled part on the coordinate end with a fixed inner slicing sphere, capping that slice inside the coordinate ball. Its Euclidean perimeter is bounded by the original perimeter times the uniform metric-comparison constant, plus the fixed sphere area. The resulting volume bound is uniform in the outer truncation; local convergence passes it to the limit.

Testing on \(O\) gives a uniform perimeter bound. If a minimizing frontier passes through an end point of radius \(r\), a coordinate ball of radius \(cr\) about that point misses the obstacle. The rescaled metrics are uniformly elliptic with uniformly bounded first derivatives, and the frontier is locally perimeter-minimizing in that ball. The interior perimeter density estimate gives area at least \(c_1r^{n-1}\), with \(c_1\) uniform. This contradicts the fixed upper bound for large \(r\). Thus the limiting frontier lies in a fixed compact region. The connected end tail is therefore almost everywhere full or empty; finite volume makes it empty. The limiting set is an admissible bounded competitor. Lower semicontinuity bounds its perimeter by the infimum approached by the truncated minima, so it is a global minimizer. The same density argument confines every global minimizer, uniformly in \(t\). Thus their bounded filled sides again lie in one compact set.

BV compactness now applies on that compact region. For fixed-metric minimizers \(E_j\to E\) in \(L^1\), lower semicontinuity gives \[a(g)\le P_g(E)\le\liminf_jP_g(E_j)=a(g).\] Thus the limit is a minimizer and the perimeters converge. Set \(\sigma_j=|D\mathbf1_{E_j}|_g\) and \(\sigma=|D\mathbf1_E|_g\). Weighted lower semicontinuity makes every weak limit of \(\sigma_j\) dominate \(\sigma\). Their total masses agree, so \(\sigma_j\rightharpoonup\sigma\). The normal vector measures \(\nu_j\sigma_j\) converge weakly to \(\nu\sigma\) by Gauss–Green.

To pass from normals to tangent planes, cover the compact support by finitely many charts with nonnegative smooth partition weights \(\rho_\alpha\). Choose a smooth matrix \(A_\alpha(x)\) in each chart such that \(|A_\alpha(x)v|_\delta=|v|_g\). The Euclidean vector measures \[\lambda_{\alpha,j}=\rho_\alpha A_\alpha\nu_j\sigma_j \quad\hbox{have}\quad |\lambda_{\alpha,j}|_\delta=\rho_\alpha\sigma_j.\] They converge weakly and their total variations have convergent masses. The localized Reshetnyak continuity theorem (Reshetnyak 1968), in the polar-direction formulation of (Spector 2011, Theorem 1.3), therefore applies to \[(x,z)\longmapsto \Phi\bigl(x,(A_\alpha(x)^{-1}z)^{\perp_g}\bigr), \qquad |z|_\delta=1.\] Insert a continuous compact cutoff equal to one on the localized support, so the integrand is bounded on the chart, and sum over the partition. This proves weak convergence of \(V_{E_j}\), including contact mass: a chart crossing \(B\) is an interior chart of \(\widehat X\). It is convergence of plane integrals, not yet pointwise convergence of tangent planes. Compact supports and bounded masses make the indicated measure topology metrizable. The subsequence argument proves compactness, and the choice \(\Phi(x,\Pi)=\frac12\operatorname{tr}_\Pi h\) proves continuity.

For varying metrics, uniform convergence on the confining region provides \(\varepsilon_t\to0\) such that \[(1-\varepsilon_t)P_g(F)\le P_{g_t}(F) \le(1+\varepsilon_t)P_g(F)\] for every confined competitor. Testing a fixed \(g\)-minimizer and using \(P_g(E_t)\ge a(g)\) shows \(a(g_t)\to a(g)\) and \(P_g(E_t)\to a(g)\). Hence every BV limit is a \(g\)-minimizer and the preceding localized argument gives its plane measure convergence.

On the compact Grassmann bundle the area density has the uniform Taylor expansion \[ P_{g_t}(F)=P_g(F)+t\,a'_F(h)+O(t^2)P_g(F). \tag{21}\] Testing every fixed \(E\in\mathcal T(g)\) gives \(\limsup_{t\downarrow0}(a(g_t)-a(g))/t\le\min_Ea'_E(h)\). For the other bound choose \(E_t\in\mathcal T(g_t)\). Its \(g\)-perimeter is uniformly bounded and at least \(a(g)\), so \[\frac{a(g_t)-a(g)}t\ge a'_{E_t}(h)-O(t).\] Every sequence tending to zero has a subsequence whose plane measures converge to those of a \(g\)-minimizer. The right side then tends to a value at least \(\min_Ea'_E(h)\). This proves both the derivative and its formula, with no uniqueness or smooth choice of a minimizing cut. ◻

Proof. For minimizing sets \(E,F\), submodularity gives \[P_g(E\cup F)+P_g(E\cap F)\le P_g(E)+P_g(F)=2a(g).\] Each term on the left is at least \(a(g)\), so the union and intersection are themselves minimizers and the submodularity deficit is zero as a measure. Let \(x\in T_F\) lie on a free regular sheet of \(T_E\). Blow up both sets at \(x\) along a common sequence, writing \(H\) for the tangent half-space of \(E\) and \(C\) for a tangent set of \(F\). Their unions and intersections converge in \(L^1\) to \(C\cup H\) and \(C\cap H\). Minimizing-boundary compactness makes these minimizing cones, with empty and full sets allowed. The half-space argument in Proposition 8 gives \[C\cup H\in\{H,\mathbb R^n\},\qquad C\cap H\in\{\varnothing,H\}.\] Thus \(C\) is one of \(H,H^c,\varnothing,\mathbb R^n\); the density bounds for \(F\) rule out the latter two.

If \(C=H\), both union and intersection have density-one planar tangent and are regular near \(x\). Their minimal graphs are ordered against the reference sheet and touch it at \(x\). The strong comparison principle makes them agree with it locally, hence so does \(T_F\). If \(C=H^c\), the union has full tangent and the intersection empty tangent. The uniform two-phase density bounds at frontier points put \(x\) outside the supports of both boundaries. In a neighborhood the union is full and the intersection empty, so \(F\) is the complement of \(E\) there. Their common open sheet contributes twice its positive area to \(P_g(E)+P_g(F)\) and zero to the union and intersection. That contradicts the zero submodularity deficit. This excludes the opposite coorientation and proves local agreement.

We next justify the uniform graphical assertion, including the entire support. Suppose \(x_j\in T_{E_j}\) tends to a free regular point \(x\) of a fixed minimizing frontier. Compactness from Proposition 9 gives a minimizing limit \(E_\infty\). Uniform density bounds put \(x\in T_{E_\infty}\), and the preceding argument identifies its sheet near \(x\) with the fixed regular sheet. In a ball about \(x\) missing \(B\), all these boundaries are locally area-minimizing. Their integral varifolds are stationary in the smooth ambient metric, with multiplicity one and no boundary. The limit is smooth with density one.

Choose a sufficiently small continuity radius for its area measure. The limiting mass ratio is close to one and its height and tilt excesses relative to \(T_xT_{E_\infty}\) are small. Plane-measure convergence transfers these properties to \(E_j\). Moving centers to \(x_j\) changes the required radii by a quantity tending to zero. To apply the small-excess theorem in a Riemannian chart, use its smooth-ambient form, or smoothly isometrically embed a neighborhood: stationarity in the ambient manifold gives Euclidean generalized mean curvature bounded by the ambient second fundamental form. After rescaling by the small radius, the required \(L^p\) curvature norm is small for any fixed \(p>n-1\).

Allard’s graphical theorem therefore represents the entire support in a smaller ball by a single \(C^{1,\alpha}\) graph with uniform bounds (Allard 1972; Maggi 2012); the exact small-mass and \(L^p\) mean-curvature formulation is (Simon 2014, chap. 5, Section 5, Theorem 5.2). It leaves no second small sheet or spike outside a selected graphical component. Uniform graph compactness and the identified varifold limit give \(C^1\) convergence; interior estimates for the smooth minimal-graph equation improve this to smooth convergence on smaller domains. A sequence contradicting uniform cylinders or uniform graph bounds would have precisely such a convergent subsequence, which is impossible. The same argument for \(x_j\to x\) proves continuity of the common plane field.

Two distinct local graphs do not meet by the first part of the proof, so their heights can be ordered. Their nonnegative difference solves a linear uniformly elliptic equation, obtained by integrating the derivative of the scalar minimal-graph operator between the two graphs. The preceding interior bounds give uniform coefficient bounds of every order, including the lower-order terms. The Harnack inequality for a nonnegative solution bounds its size on smaller domains by its value at \(y_0\); interior derivative estimates then give Equation (22). If that value is zero, the same principle makes the two graphs coincide. ◻

Proof. Proposition 8 gives the full-perimeter realization and regularity with obstacle \(S\), without invoking any rigidity assertion. For the paths below, the compact variations vanish in the tail, while the charge prototypes and strict conformal direction have metric perturbation \(O_2(r^{2-n})\). After decreasing the parameter interval, these are uniformly asymptotically flat, uniformly positive, and smoothly controlled on the compact region. Thus Proposition 9 applies, including its localized plane-measure continuity. Proposition 10 gives the graph cylinders and Equation (25). ◻

Proof. Choose \(0<\beta<\min(1,q)\), let \(\varrho\) be a smooth positive extension of the end radius, and set \(\rho_0=\varrho^{-n-\beta}\). The strict conformal direction in Lemma 7 supplies \(\phi=O_6(r^{2-n})\), with finite flux, \[ -\Delta\phi=B\sqrt{|d\phi|^2+\varrho^{-2k}}+\rho_0,\qquad \partial_\nu\phi=-1,\qquad \frac{-\Delta\phi}{\rho_0}\longrightarrow1, \tag{29}\] where \(B\ge|K|\) is smooth and \(B=O_5(r^{-1-q})\). The strict direction is \((2\phi g,\phi K)\). The conformal variation formula gives \[\mathfrak C'=-4\phi(\mu+J(v)) +2k\{-\Delta\phi+K(\nabla\phi,v)\}.\] It is strictly positive on active rays, its quotient by \(\rho_0\) tends to \(2k\), and \(-\theta'_+=k\) on \(S\).

Take the real vector space generated by all compact variations, this strict direction, and cut-off end prototypes of the following forms: \[ h=r^{2-n}\delta,\ p=0;\qquad h=0,\quad p-(\mathop{\mathrm{tr}}_\delta p)\delta=\Pi(a),\quad \Pi(a)_{ij}=r^{-n}(a_ix_j+a_jx_i-(a\cdot x)\delta_{ij}). \tag{30}\] The flat scalar linearization of the first is zero outside the cutoff, and \(\partial^j\Pi(a)_{ij}=0\). Their ADM derivatives are respectively \(E'=(n-2)/2\) and \(P'_i=a_i/k\); thus they span all charge derivatives. In the original background their constraint derivatives, also with ray transport, are \(O(r^{-n-q})\). Their normalized values on the active rays therefore tend to zero.

The coefficient of the strict direction is well defined. Otherwise, outside a compact set we would have \(2a\phi g=b r^{2-n}\delta\) for constants \(a\ne0\) and \(b\). Taking the Euclidean trace gives \[\phi=\frac{bn}{2a}\frac{r^{2-n}}{\mathop{\mathrm{tr}}_\delta g} =\frac b{2a}r^{2-n}+O_6(r^{2-n-q}).\] Since \(\Delta_\delta r^{2-n}=0\), this would imply \(\Delta_g\phi=O(r^{-n-q})\), contradicting \((-\Delta_g\phi)/\rho_0\to1\) and \(\beta<q\). This argument also covers bounded or empty active-ray sets.

Adjoin one infinity point to \(\mathfrak A\), using its one-point compactification when it is unbounded and an isolated point otherwise. On this point assign \(2k\) times the strict-direction coefficient. The linear map \[ (h,p)\longmapsto \left(\frac{\mathfrak C'}{\rho_0},\ -\theta'_+,\ \{c a'_C(h)-2k\omega\mathcal E'(h,p)\}_{C\in\mathfrak T}\right) \tag{31}\] takes values in continuous functions on the disjoint compact union \(\mathfrak A^\infty\sqcup S\sqcup\mathfrak T\). Continuity on the last factor is the strict tangent-plane continuity proved above.

Its image contains no strictly positive function. To check this without a constraint-surjectivity assumption, suppose such a variation has strict-direction coefficient \(a>0\), and write its remaining part as \((h_0,p_0)\). Use the path \[ g_t=e^{2at\phi}(g+th_0),\qquad K_t=e^{at\phi}(K+tp_0),\qquad t\ge0. \tag{32}\] On a compact region, strict positivity on the compact active set, continuity on the unit-ball bundle, and Taylor’s formula make all constraints nonnegative for sufficiently small \(t\). This argument also covers nearby inactive rays; the remaining compact inactive set has a positive original minimum.

On the end the conformal identity leaves a positive multiple of the nonnegative original contraction unaltered in sign. The additional non-strict linear error is \(O(tr^{-n-q})\). Quadratic prototype and conformal errors are \(O(t^2r^{2-2n})=O(t^2\rho_0)\), since \(\beta<1\). The strict conformal term is \(2ka t\rho_0(1+o(1))\), uniformly in the transported unit ball. Choose the radius first, then \(t\); it dominates both errors and proves DEC there. Likewise \(\theta_+(S)<0\) for small positive \(t\). Uniform metric equivalence preserves completeness and positive full-cut area, and the perturbations preserve the stated decay, integrability, and differentiable ADM limits. Future timelikeness of the charges is open. Thus the numerical theorem applies to Equation (32).

The third component of Equation (31), together with Equation (26), would give \(2k\omega\mathcal E'<c a'(0+)\), contradicting \(\mathcal E(g_t,K_t)\ge F(a(g_t))\) and equality at \(t=0\). This proves disjointness from the open positive cone. Hahn–Banach separation of an open convex cone from a linear subspace gives a nonzero continuous functional, nonnegative on nonnegative functions, annihilating the image. No closed range is required. By the Riesz representation theorem it consists of three finite nonnegative measures.

The mass of the measure on \(\mathfrak T\) is positive. Otherwise its value on the strict direction would be positive, because the first two components of that direction have positive minima and some nonzero measure would remain on them. Normalize this mass to one, calling that measure \(\pi\). On the finite part of \(\mathfrak A^\infty\) divide its measure by \(\rho_0\), giving \(\Lambda\), and call the boundary measure \(\zeta\). An infinity atom contributes zero on compact variations and prototypes, exactly the tests asserted in Equation (27). It is therefore absent from that identity, without having been assumed to vanish on the strict test. The normalized identity is Equation (27).

Define the moments by \(u(f)=\int f(x)\,d\Lambda\) and \(X(\alpha)=\int\alpha_x(v)\,d\Lambda\). The unit-ball condition gives \(|X|\le u\), and support on \(\mathfrak A\) gives the complementarity identity in Equation (28). ◻

Proof. All pairings in this proof use the fixed original volume density. Direct differentiation gives \[\begin{align*} (2\mu)'={}&\nabla^i\nabla^jh_{ij}-\Delta\mathop{\mathrm{tr}}h-\langle\mathop{\mathrm{Ric}},h\rangle +2\tau(\mathop{\mathrm{tr}}p-\langle K,h\rangle) -2\langle K,p\rangle+2\langle K^2,h\rangle, \tag{37}\\ J'_i={}&\nabla^jp_{ij}-\nabla_i\mathop{\mathrm{tr}}p +h^{jl}(\nabla_iK_{jl}-\nabla_lK_{ij}) +\tfrac12K^{jl}\nabla_i h_{jl} -K_i{}^a\nabla^j h_{aj} +\tfrac12K_i{}^a\nabla_a\mathop{\mathrm{tr}}h, \tag{38}\\ \mathfrak C'={}&(2\mu)'+2J'(v)-J_i h^i{}_jv^j. \tag{39}\end{align*}\] For Equation (38), vary both inverse metrics and both connection terms in \(\mathop{\mathrm{div}}(K-\tau g)\); terms involving \(h\,d\tau\) and \(\tau\,dh\) cancel. The last term of Equation (39) is the metric-square-root ray transport, and must be retained.

On compact interior tests the objective derivative is zero. Integration by parts of the \(p\) terms in Equation (27) gives \[2\{u(\tau g-K)-\mathop{\mathrm{sym}}\nabla X+(\mathop{\mathrm{div}}X)g\}=0.\] Taking its trace yields \(\mathop{\mathrm{div}}X=-u\tau\), and substitution proves Equation (33). The curvature principal part has adjoint \(\mathop{\mathrm{Hess}}u-(\Delta u)g\). Integrating the metric terms of \(2J'(X)\) gives \[\mathcal L_XK-(\mathop{\mathrm{div}}X)K-\mathop{\mathrm{div}}(K(X,\cdot))g.\] The algebraic scalar terms and ray transport give the other terms of Equation (36). The area derivative is \(\tfrac12\langle d\mathcal T,h\rangle\), proving Equation (34). Its trace is \(-k\Delta u+\mathop{\mathrm{tr}}\mathcal S=-ck\,d\sigma/2\); substitution proves Equation (35), including its sign and coefficient.

For the regularity assertion, differentiate Equation (33) and commute derivatives. One convenient formula, in which the commutator convention is explicit, is \[ \begin{split} \nabla_i\nabla_jX_l={}&-\nabla_i(uK_{jl})-\nabla_j(uK_{il}) +\nabla_l(uK_{ij})\\ &+\tfrac12\big([\nabla_i,\nabla_j]X_l -[\nabla_i,\nabla_l]X_j-[\nabla_j,\nabla_l]X_i\big). \end{split} \tag{40}\] The commutators are smooth curvature coefficients times \(X\). Taking traces of Equation (35) and Equation (40) gives a system with diagonal Laplace principal part and smooth first-order lower terms. Its sole measure source has growth \(O(r^{n-1})\), uniformly on compact subsets, by the perimeter bound in Lemma 11.

To spell out the regularity furnished by this growth, a local order-minus-two parametrix for this smooth-coefficient elliptic system has kernel bounded by \(C|x-y|^{2-n}\) and first derivative bounded by \(C|x-y|^{1-n}\); its localized remainder is smooth. These estimates follow by freezing the principal matrix in a small coordinate ball, using the Newton kernel, and successively correcting the smooth coefficient error; first-order terms have one additional integrable power. Applied to a measure with mass at most \(Cr^{n-1}\), the near part of a first difference is \(O(\delta)\) and the annuli between \(\delta\) and a fixed radius contribute \(O(\delta\log(1/\delta))\). Hence the potential is \(C^{0,\alpha}\) for every \(\alpha<1\). The same local parametrix applies to distributional solutions, leaving only a smooth homogeneous remainder. Thus both \(u\) and \(X\) are \(C^{0,\alpha}\). Equation (40), or its trace in divergence form, and the scalar Schauder estimate with Hölder divergence data then give \(X\in C^{1,\alpha}\).

Now \(\mathcal B\) is locally bounded and continuous. The positive tensor in Equation (35) implies a lower bound on the covariant Hessian of \(u\). One can pass from this distributional statement to semiconvexity without already assuming a bounded gradient. In a smooth chart mollify \(u\). The commutator between \(\Gamma\,du\) and convolution is bounded by \(C\varepsilon^\alpha\): integrate its derivative onto the kernel, subtract the value of \(u\) at the center, and use the smooth coefficient oscillation and the \(C^{0,\alpha}\) bound. Thus the mollified covariant Hessian is bounded below uniformly. Restriction to a unit-speed geodesic gives a uniform lower bound for the ordinary second derivative. Uniform convergence transfers this bound to \(u\). Bounds on \(u\) on slightly longer geodesic segments bound both one-sided slopes on shorter segments; hence \(u\) is locally Lipschitz. Equation (40) now has a bounded right side in every component, so \(\nabla^2X\in L^\infty\) and \(X\in C^{1,1}\). Finally bounded \(du\) converts the covariant Hessian lower bound to ordinary coordinate semiconvexity. This also proves the stated closure on first jets. ◻

Proof. Fix relatively compact open neighborhoods of \(\operatorname{supp}s\) whose closures lie in \(U\). All uniform limits below are on the larger of these fixed neighborhoods. Put \[D_\delta=\frac{J\otimes J}{\mu+\delta}.\] We first realize these tensors as the spatial Einstein components of a smooth Gaussian collar. Write \[\mathbf g_\delta=-dt^2+g_\delta(t),\qquad g_\delta(t)=g+2tK+t^2Q_\delta.\] For each fixed \(\delta\) this is Lorentzian on a sufficiently short collar. Its future unit normal is \(\mathbf n=\partial_t\), and the initial second form is \(K\). At \(t=0\), direct contraction of the Gaussian Christoffel symbols \[\boldsymbol\Gamma^0_{ij}=K_{ij},\qquad \boldsymbol\Gamma^i_{0j}=K^i{}_j,\qquad \boldsymbol\Gamma^i_{jk}=\Gamma^i_{jk}(g)\] gives \[\begin{split} \mathbf{Ric}_{00}&=-\operatorname{tr}_gQ_\delta+|K|^2,\\ \mathbf{Ric}_{0i}&=J_i,\\ \mathbf{Ric}_{ij}&=\operatorname{Ric}_{ij}(g) +(Q_\delta)_{ij}+\tau K_{ij}-2(K^2)_{ij},\\ \mathbf R&=R_g+2\operatorname{tr}_gQ_\delta +\tau^2-3|K|^2. \end{split}\] Consequently \(\mathbf G(\mathbf n,\mathbf n)=\mu\), \(\mathbf G(\mathbf n,\partial_i)=J_i\), and \[\mathbf G_{ij}=(Q_\delta)_{ij} -(\operatorname{tr}_gQ_\delta)g_{ij}+T_{ij}, \quad T=\operatorname{Ric}_g+\tau K-2K^2 -\tfrac12(R_g+\tau^2-3|K|^2)g.\] The map \(Q\mapsto Q-(\operatorname{tr}_gQ)g\) is invertible, with inverse \(H\mapsto H-(\operatorname{tr}_gH)g/(n-1)\). Thus the explicit choice \[Q_\delta=D_\delta-T -\frac{\operatorname{tr}_g(D_\delta-T)}{n-1}g\] has \(\mathbf G_{ij}=D_\delta\). This construction fixes every first spacetime metric jet independently of \(\delta\). Collar widths and higher normal derivatives need not have uniform bounds.

The tensors \(D_\delta\) converge in spatial \(C^1\) to \[D_0=\begin{cases}J\otimes J/\mu,&\mu>0,\\0,&\mu=0. \end{cases}\] Here is the verification at the zero set, where division alone would not justify the assertion. Smoothness and nonnegativity give \(d\mu=0\) on \(\{\mu=0\}\). In any smooth local frame, \(|J_i|\leq C\mu\) implies \(dJ_i=0\) there as well. Differentiating the displayed formula gives, with tensor norms taken using \(g\), \[|D_\delta|\leq\mu,\qquad |\nabla D_\delta| \leq 2|\nabla J|+|\nabla\mu|.\] On a small neighborhood of the compact zero set the right sides are uniformly small. On its complement \(\mu\) has a positive lower bound, so the formulas and their first derivatives converge uniformly. Moreover \(|D_0|\leq\mu\) makes \(D_0\) differentiable, with zero derivative, at each zero of \(\mu\); the derivative bound makes this derivative continuous. These facts prove the asserted \(C^1\) convergence.

Let \(\mathbf T_\delta\) denote the Einstein tensor of the collar, and identify its restriction to \(t=0\) with a bilinear form on \(\mathbb R\mathbf n\oplus TU\). Its time-zero components converge spatially in \(C^1\) to a tensor \(\mathbf T_0\). At a point where \(\mu>0\) it is \[\mathbf T_0=\mu^{-1}\alpha\otimes\alpha, \qquad \alpha(\mathbf n)=\mu,\quad \alpha|_{TU}=J.\] The covector \(\alpha\) is causal in the precise sense that \(\alpha(a\mathbf n+w)\geq0\) when \(a\geq|w|_g\) and \(a\geq0\). At an active ray with \(\mu>0\), the inequalities \[0=\mu+J(v)\geq\mu-|J|\,|v|\geq0\] force \(|J|=\mu\), \(|v|=1\), and \(J=-\mu v^\flat\). Thus \(\ell=\mathbf n+v\) is null and \(\alpha(\ell)=0\). Parallel transport \(\ell\) along any smooth curve in \(\{t=0\}\) through the point, using the collar connection, whose first jets are fixed. The transported vector remains future null. The smooth scalar \(\alpha(\ell)\) is nonnegative and vanishes at the point, so its derivative in every spatial direction vanishes there. Differentiating \(\mu^{-1}\alpha\otimes\alpha\) therefore gives \[(\boldsymbol\nabla_Y\mathbf T_0)(W,\ell)=0 \quad (Y\in TU,\ W\in\mathbb R\mathbf n\oplus TU).\] At \(\mu=0\), all the components of \(\mathbf T_0\) and all their spatial first derivatives vanish. Hence the same identity holds for every active \(|v|\leq1\) there, including the timelike vectors \(\mathbf n+v\).

Contracted Bianchi supplies exactly the normal derivative which is needed, without requiring convergence of third normal metric jets. For a spatial orthonormal frame \(e_1,\ldots,e_n\), evaluated at \(t=0\), \[(\boldsymbol\nabla_{\mathbf n}\mathbf T_\delta) (\mathbf n,\ell) =\sum_{i=1}^n(\boldsymbol\nabla_{e_i}\mathbf T_\delta)(e_i,\ell).\] The right side converges uniformly to zero on the compact active-ray set: spatial \(C^1\) convergence was proved above, and all connection coefficients in this formula are fixed at \(t=0\).

Finally use the graph embeddings \(F_a(x)=(a s(x),x)\) in this collar. Differentiating the induced metric and future second form gives \[h=2sK,\qquad p_\delta=\nabla^2s+sQ_\delta.\] Both variations are smooth and compactly supported. We spell out the argument-vector variation in order to include the terms involving \(ds\). Transport \(v\) on the domain by \(v'_0=-sK^\sharp v\), and write \(\ell_a=\mathbf n_a+(F_a)_*v_a\). Covariant differentiation along the graph variation gives \[\dot{\mathbf n}=\nabla s,\qquad \frac{D}{da}(F_a)_*v_a\bigg|_{a=0}=v(s)\mathbf n, \qquad \dot\ell=\nabla s+v(s)\mathbf n.\] In the middle equality, the spatial connection contribution \(sK^\sharp v\) cancels the prescribed ray transport. Gauss–Codazzi on each graph identifies the differentiated ray constraint with \[\begin{split} \tfrac12\mathfrak C'(h,p_\delta;v) ={}&s(\boldsymbol\nabla_{\mathbf n}\mathbf T_\delta) (\mathbf n,\ell) +\mathbf T_\delta(\nabla s,\ell)\\ &+\mathbf T_\delta(\mathbf n,\nabla s+v(s)\mathbf n). \end{split}\] The first term tends uniformly to zero by Bianchi. The second tends to zero because \(\mathbf T_0(\cdot,\ell)=0\) at every active ray. The final term is exactly \(J(\nabla s)+\mu v(s)\); it is zero at a positive-density active ray by \(J=-\mu v^\flat\), and at a zero-density ray by \(J=0\). This proves the uniform assertion. Local finiteness of \(\Lambda\) proves the final statement of the lemma. Only infinitesimal compact tests have been constructed; no energy condition on the neighboring graphs is needed for this use of the multiplier identity. ◻

Proof. Define the aggregate area measure in the open exterior by \[\beta(B)=\int_{\mathcal T}\operatorname{Area}_g(T\cap B)\,d\varpi(T) \qquad\bigl(B\subset\operatorname{int}\Omega\text{ Borel}\bigr).\] It is positive and has mass at most \(A\). On the union of the free frontiers put \(F(x)=\operatorname{tr}_{\Pi(x)}K(x)\), and set \(F=0\) elsewhere. This is a bounded measurable function on the common compact region. Equation (41) says that the signed Radon measure \(F\beta\) vanishes. Hence \(F=0\) \(\beta\)-almost everywhere, and Tonelli’s theorem gives \[0=\int |F|\,d\beta =\int_{\mathcal T}\int_{T\setminus S} |\operatorname{tr}_{T_xT}K|\,dA_g\,d\varpi(T).\] For almost every \(T\), its tangential trace therefore vanishes at area-almost every free point. Each free part is smooth, so a nonzero value would persist on an open patch of positive area. Thus its trace vanishes everywhere there. Minimality gives \(H_T=0\) on that part as well. The common-plane hypothesis is essential: it makes the trace one function of position before positivity is used.

Fix one of these frontiers and a contact point. Extend the smooth one-sided coefficients of \(g,K\) locally across \(S\) only to write graph equations. In a graph chart let \(f\) describe \(T\) and let the smooth function \(\psi\) describe the obstacle. Their ordered graphs have matching normals at contact. On the coincidence set \(Z=\{f=\psi\}\), their first derivatives agree. Sobolev locality applied to \(\partial_j(f-\psi)\in W^{1,2}\) gives \(D^2f=D^2\psi\) almost everywhere on \(Z\). Since \(S\) is marginal, the graph of \(f\) satisfies \(H+\operatorname{tr}_{\rm tan}K=0\) almost everywhere on \(Z\). It satisfies this equation on the free set by the first part of the proof.

We verify that this almost-everywhere equation yields smoothness through contact. In the chart it has the form \[\partial_i\mathcal F^i(y,f,Df)=\mathcal B(y,f,Df),\] with smooth functions \(\mathcal F,\mathcal B\). The matrix \(\mathcal F^i_{p_j}\) is uniformly elliptic on a smaller chart, since \(Df\) is bounded there. These are the usual graph coefficients for mean curvature, with the smooth tangential trace of \(K\) included in \(\mathcal B\). As \(f\in W^{2,2}\), the divergence on the left is an \(L^2\) function. The almost-everywhere identity is therefore also a weak divergence-form equation; there is no additional contact measure. For \(f_k=\partial_k f\), differentiation in distributions gives \[\partial_i(a^{ij}\partial_j f_k) =\partial_i\bigl(\delta_{ik}\mathcal B -\mathcal F^i_{y_k}-\mathcal F^i_z f_k\bigr), \qquad a^{ij}=\mathcal F^i_{p_j}(y,f,Df).\] All displayed coefficients and the right-hand vector field are \(C^{0,\alpha}\), while \(f_k\in W^{1,2}\). Interior divergence-form Schauder regularity gives \(f_k\in C^{1,\alpha}\) on smaller charts. Thus \(f\in C^{2,\alpha}\), and further differentiation and Schauder estimates give smoothness; see (Gilbarg and Trudinger 2001). Consequently \(T\) is a smooth MOTS across every contact point.

If \(T\) touches \(S_i\), the two ordered smooth graphs satisfy the same zero-expansion equation with the same orientation. Their nonnegative difference \(w\) satisfies a linear uniformly elliptic equation \(a^{ij}w_{ij}+b^i w_i+cw=0\) with bounded coefficients. Replacing \(c\) by \(\min(c,0)\) gives an inequality with left side \(\le0\), since \(w\ge0\). The strong minimum principle at a zero of \(w\) forces local coincidence. Therefore \(T\cap S_i\) is both open and closed in the connected hypersurface \(S_i\), so \(S_i\subset T\). Local coincidence also makes \(S_i\) open in \(T\), and compactness makes it closed there; it is one connected component of \(T\). Every remaining component is disjoint from \(S\) and, by compactness, lies a positive distance inside the open exterior. This proves the component assertion.

Suppose first that (a) holds. If \(T\) touches \(S\), it contains \(S\) as a component. Since \(\operatorname{Area}_g(T)=A= \operatorname{Area}_g(S)\), no further component of positive area is possible, and \(T=S\). Otherwise \(T\) is a wholly interior smooth full enclosing MOTS, excluded by (a).

Under (b), any interior component of \(T\) would give exactly a prohibited partial-coincidence cut: all its other components are interior or entire \(S_i\), and \(\theta_+=0\) everywhere. Hence \(T=\bigsqcup_{i\in I}S_i\) for some \(I\subset\{1,\ldots,r\}\). The area identity \[\sum_{i\in I}\operatorname{Area}_g(S_i) =A=\sum_{i=1}^r\operatorname{Area}_g(S_i)\] and the positive area of every \(S_i\) force \(I=\{1,\ldots,r\}\). Thus \(T=S\) in this case too. All conclusions hold outside the single null family discarded above. Integrating the area derivative of this fixed frontier proves Equation (42). ◻

Proof. Apply Lemma 14 to an arbitrary compact smooth lapse \(s\) in the open exterior. The variations have zero ADM and boundary terms. Their active-constraint derivatives tend uniformly to zero on a fixed compact set, where \(\Lambda\) has finite mass. Equation (27) therefore gives \[ \int_{\mathfrak T}\int_{\partial^*C} s\,\mathop{\mathrm{tr}}_{T\partial^*C}K\,dA\,d\pi(C)=0. \tag{44}\] On a uniform graph cylinder from Lemma 11, all sheets through one point have the same tangent plane. Their tangent planes also vary continuously on the union of the sheets: a contrary sequence would contradict multiplicity-one regular convergence. Consequently \(\mathop{\mathrm{tr}}_{T\partial^*C}K\) defines one continuous function \(f\) on this union, and Equation (44) says \(f\,d\sigma=0\) there. There is no cancellation between differing tangent planes at the same point. Hence \(f=0\) almost everywhere for the positive measure \(\sigma\). Tonelli’s theorem and continuity on each smooth graph give \(\mathop{\mathrm{tr}}_{T\Gamma_C}K=0\) everywhere on each participating free graph for almost every \(C\). Choose countably many smaller cylinders covering all free regular points of the family and discard the union of their null exceptional sets. Minimality already gives \(H=0\) on these patches.

Suppose a remaining frontier touches \(S\). Near a contact point write the frontier and \(S\) as \(W^{2,p}\cap C^{1,\alpha}\) and smooth ordered graphs, respectively. Their difference vanishes on the coincidence set. Sobolev locality says that its first and second weak derivatives vanish almost everywhere on that set; this follows by applying first-derivative locality to the function and then to its first derivatives. Thus the graph’s second jet agrees there almost everywhere with that of \(S\). On the coincidence set it has \(H+\mathop{\mathrm{tr}}_{\rm tan}K=\theta_+(S)=0\), and on its free set it has the same equation by Equation (43). It therefore solves the uniformly elliptic scalar zero-expansion graph equation almost everywhere on the entire patch. Linear \(W^{2,p}\) estimates and Schauder bootstrap make it smooth. The strong comparison principle for two ordered solutions of this graph equation makes it coincide locally with \(S\). Connectedness of \(S\), local continuation of this coincidence, and closedness of the frontier make it contain all of \(S\). Since its total perimeter is \(A=\mathop{\mathrm{Area}}(S)\), there is no remaining perimeter for another sheet. The support is exactly \(S\).

Otherwise its compact support is disjoint from the compact set \(S\), so it lies a positive distance inside the open exterior and satisfies Equation (43) on every regular patch. ◻

Proof. Use Fermi coordinates \((y,s)\) about the smooth reference sheet, with \(g=ds^2+g_s\), \(s=0\) on \(\Sigma\), and \(A_{ab}=\tfrac12\partial_s(g_s)_{ab}\). Let \(a>0\) be the mass of the coincident-graph fiber. Its contribution to Equation (35) is \((ca/2)\,ds\otimes ds\,dA_\Sigma\). In the \(ss\) equation this is the density \((ca/2)\delta_0(ds)\) on almost every normal line; the volume and surface densities coincide at \(s=0\). All other graph parameters have height different from zero on every such line, by noncrossing and contact agreement. They contribute no atom at zero. Semiconvexity and local Lipschitz bounds give bounded one-sided traces of \(\partial_su\) on almost every line, and slicing the equation proves \[ (\partial_su)^+-(\partial_su)^-=ca/2. \tag{45}\]

The tangential equations contain no jump measure on the reference sheet. Contributions from nearby sheets must also be controlled before taking traces. Equation (25) implies that a nearby leaf at distance \(|s|\) has tangential normal components \(O(|s|)\) relative to the reference Fermi frame, on a smaller patch. Indeed Harnack compares its height at the fixed base point to its height at the chosen point, and the derivative estimate controls its tilt. Thus the tangential–tangential components of \(d\mathcal N\) in \(|s|<\varepsilon\) have total variation bounded by \(C\varepsilon^2\sigma(\{|s|<\varepsilon\})\). After division by the slab width this tends to zero; the total local \(\sigma\) mass is bounded. This estimate also covers infinitely many accumulating leaves and their diffuse parameter measure.

Average the tangential equation over \(0<s<\varepsilon\) and over \(-\varepsilon<s<0\), and test in the \(y\) variables. The continuous tensor \(\mathcal B(u,X,\nabla X)\) has identical limiting traces. On each parallel sheet, \[(\mathop{\mathrm{Hess}}_g u)_{ab}=(\mathop{\mathrm{Hess}}_{g_s}u|_{s})_{ab} +A_{ab}(s)\partial_su.\] Uniform continuity of \(u\) makes the two averaged intrinsic Hessians converge to the same distribution, and the bounded normal derivatives converge to their one-sided traces in \(L^1\) on smaller base patches. Taking the difference and using the vanishing measure contribution therefore gives \(A_{ab}(0)[\partial_su]=0\) as distributions. Equation (45) and smoothness of \(A\) prove \(A=0\). ◻

Proof. Fix a base point \(y_0\) in the patch. The fiber of minimizers whose graph agrees there has \(\pi\) measure zero. Every neighborhood of \(C\) in the compact metric space \(\mathfrak T\) has positive \(\pi\) measure. Deleting this fiber and the exceptional family from Lemma 16, choose \(C_j\to C\) with distinct graphs at \(y_0\), all satisfying Equation (43) near the component.

Choose a connected relatively compact smooth exhaustion of \(\Sigma\) by domains containing \(y_0\). For each fixed exhaustion domain, multiplicity-one regular convergence and finitely many overlapping graph charts express \(\Gamma_{C_j}\) as one smooth normal graph \(h_j\) over it, with \(h_j\to0\) smoothly. Noncrossing says this graph either agrees with \(\Sigma\) or is strictly on one side. Agreement at one point propagates through overlapping charts by strong comparison; it would contradict distinction at \(y_0\). A subsequence thus has a fixed strict sign on every exhaustion domain.

Divide \(h_j\) by its signed value at \(y_0\), obtaining positive functions normalized to one there. Each satisfies the linear difference equation for the two minimal graphs. The coefficients converge smoothly to those of \(L_0\). Harnack along a finite chain of balls and interior estimates bound the normalized graphs and all their derivatives on each compact subset. Diagonal extraction gives a smooth positive limit \(\varphi\) on all of \(\Sigma\). The same graph differences solve the two difference equations for \(H\pm\mathop{\mathrm{tr}}_{\rm tan}K=0\); their coefficients converge to \(L_\pm\). Passing to the same limit proves all three equations in Equation (48). This argument requires a sequence of neighboring minimizers, not a differentiable family. ◻

Proof. All variations and integrations in this proof have compact support in the open set. Define \[\mathsf B(B,C)=\langle B,C\rangle_g -(\operatorname{tr}_gB)(\operatorname{tr}_gC).\] Temporarily let \(K\) be independent of the metric. Integration of the momentum divergence, at fixed contravariant shift \(X\), gives \[\begin{split} \int(2u\mu+2J(X))\,dV_g &=\int u\{R_g-\mathsf B(K,K)+2\mathsf B(K,b)\}\,dV_g\\ &=\int u\{R_g+\mathsf B(b,b) -\mathsf B(K-b,K-b)\}\,dV_g. \end{split}\] At \(K=b\) the final square has zero first variation, including for a metric variation with the covariant components of \(K\) held fixed. Thus the metric derivative of the constraint functional at this point equals that of the stationary scalar action \(\int u(R_g+\mathsf B(b,b))\,dV_g\).

For completeness, its relation to the spacetime scalar action follows from the Gaussian contraction formulas with lapse and shift. They give \[\mathbf R=R_g+|b|^2+(\operatorname{tr}b)^2 +2\mathbf n(\operatorname{tr}b)-2u^{-1}\Delta u.\] Stationarity implies \(\mathbf n(\operatorname{tr}b)=-u^{-1}X(\operatorname{tr}b)\) and \(\operatorname{div}X=-u\operatorname{tr}b\), whence \[\mathbf R=R_g+\mathsf B(b,b) -2u^{-1}\operatorname{div} \bigl(\nabla u+(\operatorname{tr}b)X\bigr).\] Since \(\sqrt{|\det\mathbf g|}=u\sqrt{\det g}\), the two scalar actions differ by a spatial divergence per unit \(z\)-length. For a spacetime metric variation \(H\), differentiating the connection formula for scalar curvature gives \[\delta(\mathbf R\,dV_{\mathbf g}) =\left[-\langle\mathbf G,H\rangle_{\mathbf g} +\boldsymbol\nabla_\alpha \bigl(\boldsymbol\nabla_\beta H^{\alpha\beta} -\boldsymbol\nabla^\alpha\operatorname{tr}_{\mathbf g}H \bigr)\right]dV_{\mathbf g}.\] At fixed \(u,X\), \(H=h_{ij}(dx^i+X^i dz)(dx^j+X^j dz)\) annihilates the unit normal in each slot. The derivative of the stationary action therefore pairs \(h\) with \(-u\mathbf G_{\mathrm{spatial}}\).

The metric coefficient of the constraint integral before ray transport is \[\nabla^2u-(\Delta u)g+S_b+(J_b)_{(i}X_{j)} +(u\mu_b+J_b(X))g.\] The last term is the volume derivative, and the preceding added term restores the term removed by isometric ray transport in the definition of \(S_b\). Equating the two coefficients proves the identity. This also displays why the volume derivative disappears only after complementarity has been imposed. ◻

Proof. The operator formula follows by varying \(\Sigma\) with velocity \(f\nu\). The mean-curvature derivative is \(-\Delta_\Sigma f-(|A_\Sigma|^2+\operatorname{Ric}_g(\nu,\nu))f\). For the other term, differentiation of \(\operatorname{tr}_\Sigma K=\operatorname{tr}_gK-K(\nu,\nu)\), using \(\dot\nu=-\nabla_\Sigma f\), gives \(f\operatorname{tr}_\Sigma(\nabla_\nu K) +2K(\nu,\nabla_\Sigma f)\). Adding the two derivatives gives \(L_+\); replacing \(K\) by \(-K\) gives \(L_-\).

Fix a compact patch of \(\Sigma\cap\{u>0\}\). Enlarge it slightly to a coordinate neighborhood on which \(u\geq a>0\). All constants below may depend on this neighborhood, \(a\), the smooth data, and the stated local Lipschitz bounds; the only limiting parameter is the mollifier radius \(\varepsilon\downarrow0\). Choose a nonnegative smooth convolution kernel of integral one. Mollify \(u\) and the components of the covector \(w=X^\flat\), and then put \[u_\varepsilon=u*\rho_\varepsilon,\qquad w_\varepsilon=w*\rho_\varepsilon,\qquad X_\varepsilon=g^{-1}w_\varepsilon, \qquad b_\varepsilon=-u_\varepsilon^{-1} \operatorname{sym}\nabla X_\varepsilon.\] Convolutions are used only on smaller coordinate neighborhoods. In particular \(u_\varepsilon\geq a/2\) for small \(\varepsilon\).

We need convergence of \(b_\varepsilon\) through one derivative. For a smooth coefficient \(a_0\) and a Lipschitz function \(f\), let \[C_\varepsilon(a_0,f)=(a_0f)*\rho_\varepsilon -a_0(f*\rho_\varepsilon).\] The integral formula with integrand \((a_0(y)-a_0(x))f(y)\rho_\varepsilon(x-y)\) gives \(\|C_\varepsilon(a_0,f)\|_\infty\leq C\varepsilon\|f\|_\infty\). Differentiation in distributions gives the useful identity \[\partial_j C_\varepsilon(a_0,f) =C_\varepsilon(\partial_j a_0,f) +C_\varepsilon(a_0,\partial_jf).\] The same integral bound, applied to the bounded weak derivative of \(f\), proves \[\|C_\varepsilon(a_0,f)\|_{C^1} \leq C\varepsilon\|f\|_{W^{1,\infty}}.\] In covector coordinates the shift equation reads \(\tfrac12(\partial_iw_j+\partial_jw_i)-\Gamma^l_{ij}w_l=-uK_{ij}\). Thus \[\operatorname{sym}\nabla X_\varepsilon+u_\varepsilon K =C_\varepsilon(\Gamma,w)-C_\varepsilon(K,u)=O_{C^1}(\varepsilon).\] The positive lower bound for \(u_\varepsilon\) and its bounded first derivatives therefore give \[b_\varepsilon\longrightarrow K\quad\hbox{in }C^1.\] Also \(X_\varepsilon\to X\) in \(C^1\), \(u_\varepsilon\to u\) uniformly, and all their first derivatives are uniformly bounded.

Put \(B=-S+(\operatorname{tr}_gS)g/(n-1)\); it is continuous under the stated regularity. To convolve the tensor measure precisely, write \(dV_g=v_g\,dx\) in the chosen chart and convolve the component measures \(v_g^{-1}\,d\mathcal M_{ij}\). Denote the resulting matrix of smooth functions by \(M_\varepsilon\). This matrix is positive semidefinite at each point, because the convolution kernel and \(v_g^{-1}\) are nonnegative. The Hessian equation and the connection commutator give \[\nabla^2u_\varepsilon=B*\rho_\varepsilon +M_\varepsilon+o_{C^0}(1).\] Indeed the sole extra term is \((\Gamma\,du)*\rho_\varepsilon-\Gamma\,d u_\varepsilon\), which is \(O(\varepsilon)\) since \(du\) is bounded. No derivative of \(du\) is required in this estimate.

Define the stationary smooth spacetime metric from \(u_\varepsilon,X_\varepsilon,g\) as in Lemma 19. Its induced second form is \(b_\varepsilon\); let \(\mu_\varepsilon,J_\varepsilon\) be its constraint components. Since \(b_\varepsilon\to K\) in \(C^1\), \[\mu_\varepsilon\to\mu,\qquad J_\varepsilon\to J,\qquad S_{b_\varepsilon}(u_\varepsilon,X_\varepsilon)\to S\] uniformly. The smooth action identity and the convolved Hessian equation consequently give \[u_\varepsilon\mathbf G^\varepsilon_{\mathrm{spatial}} =P_\varepsilon-J_{(i}X_{j)}+o_{C^0}(1),\qquad P_\varepsilon=(\operatorname{tr}_{g(x)}M_\varepsilon)g(x) -M_\varepsilon.\] Here the regular terms cancel because \(B-(\operatorname{tr}_gB)g+S=0\), and the volume term tends to zero by \(u\mu+J(X)=0\). The matrix \(P_\varepsilon\) is positive semidefinite: in a \(g(x)\)-orthonormal eigenbasis for \(M_\varepsilon\) its eigenvalues are the sums of all but one of the nonnegative eigenvalues of \(M_\varepsilon\). It is essential that this trace reversal be applied after convolution, at \(g(x)\). There is no estimate here commuting a variable trace-reversal coefficient past an unbounded Hessian. For the area measure in the statement, the corresponding unmollified positive spatial measure is exactly \[\frac c2\int(g-\nu_D^\flat\otimes\nu_D^\flat)\,dA_D\,d\pi.\]

Let \(\mathbf n_\varepsilon\) be the future slice normal. For the \(g\)-unit normal \(\nu\) of \(\Sigma\), Gauss–Codazzi and nullness yield \[\begin{split} u_\varepsilon\mathbf{Ric}^\varepsilon (\mathbf n_\varepsilon\pm\nu, \mathbf n_\varepsilon\pm\nu) &=u_\varepsilon\mathbf G^\varepsilon (\mathbf n_\varepsilon\pm\nu, \mathbf n_\varepsilon\pm\nu)\\ &\geq u\mu\pm2uJ(\nu)-J(\nu)g(X,\nu)-o(1)\geq-o(1). \end{split}\] To check the last inequality pointwise, if \(\mu=0\) then \(J=0\). Otherwise complementarity and \(u\geq|X|\) imply \(|J|=\mu\), \(|X|=u\), and \(J=-\mu X^\flat/u\). The expression is therefore \(u\mu(1\pm J(\nu)/\mu)^2\). This verifies the null Ricci lower bound uniformly, without bounding \(M_\varepsilon\) or the full spacetime curvature.

We give the smooth focusing identity used to finish. Write \(\theta_\varepsilon=H_\Sigma+\operatorname{tr}_\Sigma b_\varepsilon\) and \(\ell_\varepsilon=\mathbf n_\varepsilon+\nu\). Launch orthogonal null geodesics from the patch, and intersect their local null hypersurface with the stationary slices. For each fixed \(\varepsilon\) this is a smooth construction for some positive time; no uniform time interval is needed. On each slice normalize the null generator by \(\ell=\mathbf n+\nu\). If \(\chi\) denotes its screen second form, commuting covariant derivatives along an affinely parametrized generator in a parallel screen frame gives \[\nabla_\ell\chi_{ab}=-\chi_{ac}\chi_{cb}-\mathcal R_{ab}, \qquad \sum_a\mathcal R_{aa}=\mathbf{Ric}(\ell,\ell),\] where \(\mathcal R\) is the screen Jacobi curvature matrix. Tracing gives \(\ell(\theta)=-|\chi|^2-\mathbf{Ric}(\ell,\ell)\) for that affine normalization. For the slice normalization, \(\boldsymbol\nabla_\ell\ell=\kappa\ell\), so the trace identity instead has the additional term \(\kappa\theta\). Differentiating \(\mathbf g(\ell,\mathbf n)=-1\) gives \[\kappa=\mathbf g(\ell,\boldsymbol\nabla_\ell\mathbf n) =u^{-1}\nu(u)+b(\nu,\nu),\] because the slice-normal acceleration is \(\nabla\log u\).

The velocity with slice coordinate \(z\) as parameter is \(u\ell\). Subtracting the stationary Killing translation \(\partial_z=u\mathbf n+X\) leaves the spatial velocity \(u\nu-X\), whose normal component is \(Q=u-X\cdot\nu\) and tangential component is \(-X^T\). At the initial slice \(\chi=A_\Sigma+b|_{T\Sigma}\). Thus the two descriptions of the expansion derivative give exactly \[\begin{split} L_{+,\varepsilon}Q_{+,\varepsilon} -X_\varepsilon^T(\theta_\varepsilon) ={}&-u_\varepsilon\bigl( |A_\Sigma+(b_\varepsilon)_{\mathrm{tan}}|^2 +\mathbf{Ric}^\varepsilon(\ell_\varepsilon, \ell_\varepsilon)\bigr)\\ &+\bigl(\nu(u_\varepsilon) +u_\varepsilon b_\varepsilon(\nu,\nu)\bigr) \theta_\varepsilon. \end{split}\] The notation \(L_{+,\varepsilon}\) means the displayed spatial linearization with \(K\) replaced by \(b_\varepsilon\), without assuming that \(\theta_\varepsilon\) is zero.

Since \(H_\Sigma=\operatorname{tr}_\Sigma K=0\) and \(b_\varepsilon\to K\) in \(C^1\), one has \(\theta_\varepsilon\to0\) in \(C^1\) on the patch. Both the tangential transport term and the final normalization term therefore tend uniformly to zero; their coefficients involve only bounded first jets. The null Ricci lower bound gives \[L_{+,\varepsilon}Q_{+,\varepsilon} \leq-u_\varepsilon |A_\Sigma+(b_\varepsilon)_{\mathrm{tan}}|^2+o(1).\] The principal part of all these operators is the fixed \(-\Delta_\Sigma\). Their first-order and zeroth-order coefficients converge uniformly, while \(Q_{+,\varepsilon}\to Q_+\) uniformly with uniform local Lipschitz bounds. Pairing with a compactly supported smooth test function and integrating the Laplacian by parts thus passes to the asserted distributional inequality. For the drift term, the gradients converge weakly against integrable test fields, and the error in its coefficient tends uniformly to zero.

Finally replace stationary time \(z\) by \(-z\) and the future normal by its negative. This replaces \((K,X,J)\) by \((-K,-X,-J)\) while leaving \(u,g,S,B,\mathcal M\) unchanged. Applying the plus calculation gives \(Q_-=u+X\cdot\nu\) and the claimed minus inequality. The argument is local on every compact subset of \(\{u>0\}\), so a partition of unity establishes the statement for all test functions specified in the lemma. ◻

Proof. Here is the capacity calculation and the energy argument needed before using a Sobolev inequality. On a fixed compact region cover its singular set by finitely many ambient balls \(B_{r_i}(p_i)\), with \(r_i<\varepsilon\) and \(\sum_i r_i^{k-2}<\varepsilon\). Such covers exist because \(\dim_{\mathcal H}\mathcal S\leq k-7<k-2\). Let \(\theta_i\) be one on \(B_{r_i}(p_i)\), zero outside \(B_{2r_i}(p_i)\), and satisfy \(|\nabla\theta_i|\leq2/r_i\). The cutoff \(\chi=1-\max_i\theta_i\) satisfies \[\int_\Gamma|\nabla\chi|^2\,d\mu \leq C\Lambda\sum_i r_i^{k-2},\qquad \int_\Gamma(1-\chi)^2\,d\mu \leq C\Lambda\sum_i r_i^k.\] The first estimate follows pointwise almost everywhere by selecting an index attaining the maximum. Taking shrinking covers gives \(0\leq\chi_j\leq1\), equal to zero near \(\mathcal S\), converging to one at every regular point, with both displayed integrals tending to zero. Local versions use a slightly larger compact region than the support of the test under consideration.

For a fixed smooth ambient cutoff \(\eta\), test the subsolution inequality with \(\eta^2\chi_j^2w\). Sobolev approximation on the smooth regular part justifies this test. Expanding and applying Young’s inequality gives \[\int \eta^2\chi_j^2|\nabla w|^2\,d\mu \leq 4\int w^2|\nabla(\eta\chi_j)|^2\,d\mu +2F\int\eta^2\chi_j^2w\,d\mu.\] The right side is uniformly bounded because \(w\) is bounded, the area is locally finite, and the capacity energies tend to zero. Fatou’s lemma proves local finite energy. Moreover, \(\chi_jw\) converges to \(w\) in local \(W^{1,2}\): the extra gradient \(w\nabla\chi_j\) tends to zero in \(L^2\), and dominated convergence applies to \((1-\chi_j)\nabla w\). For a nonnegative smooth compact test \(\psi\), the tests \(\psi\chi_j^2\) now pass to the limit in the weak inequality. Indeed, the only extra term is bounded by \[2\|\psi\|_\infty \|\nabla w\|_{L^2(\operatorname{supp}\psi)} \|\nabla\chi_j\|_{L^2},\] and tends to zero. This proves removability, without assuming energy regularity in advance.

The required Sobolev input is the Michael–Simon inequality (Michael and Simon 1973), in the following local \(L^2\) form: \[\left(\int|v|^{2k/(k-2)}\,d\mu\right)^{(k-2)/k} \leq C\int\bigl(|\nabla v|^2+C_{\mathrm{amb}}v^2\bigr)\,d\mu.\] Initially \(v\) is compactly supported in the regular part. To obtain this form, apply the \(L^1\) Michael–Simon inequality to \(|v|^{2(k-1)/(k-2)}\) and use Cauchy–Schwarz. A smooth isometric embedding of a slightly larger compact ambient neighborhood gives a bound for its second fundamental form. Since \(\Gamma\) is minimal in the ambient metric, its Euclidean mean curvature in that embedding is bounded by the trace of this ambient second fundamental form. In particular the bound does not involve \(|A_\Gamma|\). The capacity approximation just proved extends the inequality to the tests used here. Dilating this fixed embedding multiplies its second fundamental form by the dilation scale, which also proves the claimed uniformity in blow-ups.

For completeness, put \(v=w+Fr^2\); if \(F=0\), first put \(v=w+\delta\) and later let \(\delta\downarrow0\). On \(B_r(x)\) this is a positive subsolution of \(\Delta v\geq-r^{-2}v\). Testing with \(\eta^2v^{p-1}\), \(p\geq2\), and applying the preceding Sobolev inequality gives \[\left(\int|\eta v^{p/2}|^{2k/(k-2)}\,d\mu\right)^{(k-2)/k} \leq Cp^2\int (|\nabla\eta|^2+r^{-2}\eta^2)v^p\,d\mu,\] after decreasing \(r_0\) so the bounded ambient term is absorbed in \(r^{-2}\). All power tests follow from boundedness of \(w\) and the energy approximation above. Iteration with \(p_j=2(k/(k-2))^j\) on radii \(r/2+r/2^{j+1}\) yields the displayed submean estimate; the factors converge since \(\sum_j(1+j)/p_j<\infty\). The contribution of the added constant is bounded by \(CFr^2\) using \(\mu(B_r)\leq\Lambda r^k\). Finally, every regular point has a connected smooth neighborhood in one regular component. The boundary within \(\Gamma\) of any regular component is contained in \(\mathcal S\). Zero extension to other components therefore adds no regular interface, and the same capacity argument applies. ◻

Proof. Write \(d=k-1\) for the link dimension and \(\mathcal S_L=(\operatorname{sing}C)\cap S^k\). The product description of the cone away from its vertex and the singular-dimension estimate give \(\dim_{\mathcal H}\mathcal S_L\leq k-8=d-7\) when the link is singular. Its area growth is of order \(r^d\): on a fixed annulus the radial product formula and cone area growth imply this bound by integrating over radial intervals of length comparable to \(r\). Thus \(\mathcal S_L\) has zero \(2\)-capacity in the link by the preceding cover construction. A smooth link needs no capacity cutoffs.

Let \(U\) be the totally geodesic regular component. It lies in a great equator \(E\) and is open in \(E\). Every limit point of \(U\) that is regular belongs to \(U\): a connected local regular chart through that point meets \(U\), so belongs to the same component. Consequently \(U\) is relatively closed as well as open in \(E\setminus\mathcal S_L\). This latter set is connected. One elementary justification is to use stereographic coordinates on \(E\). Given two points outside \(\mathcal S_L\), almost every intermediate point can be joined to both by straight segments missing \(\mathcal S_L\): the set of bad intermediate points is contained in the two radial shadows of that set, whose Hausdorff dimension is at most \(\dim_{\mathcal H}\mathcal S_L+1<d\). Taking countably many bounded pieces handles the point omitted by stereographic projection. Thus \(U=E\setminus\mathcal S_L\), and closedness of the support implies \(E\subset C\cap S^k\).

Any other connected regular link component \(V\) is disjoint from \(E\). Indeed a smooth regular neighborhood of a point of \(E\) already contains the equatorial sheet; as an embedded hypersurface of the same dimension it is that sheet locally. By connectedness \(V\) lies in one strict hemisphere. Its suitably signed equatorial height \(h\) satisfies \[h>0,\qquad |h|\leq1,\qquad |\nabla_Vh|\leq1, \qquad \Delta_Vh=-d h.\] This Laplace identity holds for coordinate functions on a minimal \(d\)-dimensional submanifold of the unit sphere. Test it on \(V\) with the squared link capacity cutoffs \(\chi_j^2\). These tests have compact support in \(V\): any relative boundary of \(V\) in the compact link is singular. They give \[d\int_Vh\chi_j^2\,d\mathcal H^d =2\int_V\chi_j\langle\nabla h,\nabla\chi_j\rangle\,d\mathcal H^d \longrightarrow0.\] Here Cauchy–Schwarz, the finite link area, and the vanishing capacity energy justify the limit. Dominated convergence gives \(\int_Vh=0\), contradicting \(h>0\) on a nonempty open component. Hence no such \(V\) exists. The regular support is contained in \(E\); the singular set has zero link area, and the regular set is dense in the support. Thus the support is precisely \(E\), and multiplicity one makes the cone a hyperplane. ◻

Proof. A nonflat minimizing boundary cone has \(k\geq7\). In logarithmic radius \(s=\log r\) and the regular link \(L\), set \(a=k-2>0\), \(V=|A_L|^2\), and \(Z(s,\omega)=z(e^s\omega)\). The cone metric and the scaling of the second fundamental form give \[Z_{ss}+aZ_s+\Delta_L Z\geq VZ.\] The operator on the left is in divergence form for the measure \(e^{as}\,ds\,d\mathcal H^{k-1}_L\). On any compact \(s\)-strip, test after dropping \(VZ\geq0\) with \(\eta(s)^2\chi_j(\omega)^2 Z\). The energy proof in Lemma 21, with this smooth positive weight, proves finite energy on smaller strips and permits removal of the link capacity cutoffs. In particular, tests constant in the link variable are legitimate. It follows that \[q(s)=\int_LZ(s,\omega)\,d\mathcal H^{k-1}_L(\omega) \quad\hbox{satisfies}\quad q''+a q'\geq0\] on the entire real line in distributions. The energy bound gives \(q\in H^1_{\mathrm{loc}}(\mathbb R)\), so we use its continuous representative. It is bounded and nonnegative.

Such a \(q\) is nondecreasing. To see this, convolve with a smooth nonnegative kernel. For the resulting smooth bounded function, \((e^{as}q')'\geq0\). If \(q'(s_0)<0\), then for every \(s<s_0\) \[q'(s)\leq e^{a(s_0-s)}q'(s_0).\] Integrating backwards forces \(q(s)\to+\infty\) as \(s\to-\infty\), contrary to boundedness. Thus all mollifications are nondecreasing, and so is \(q\).

Suppose \(Z\) is not zero. Positivity on a regular open patch implies \(q(s_0)>0\) for some \(s_0\). Choose \(\rho\in C_c^\infty((1,2))\), \(\rho\geq0\), \(\int\rho=1\), and set \[\rho_T(s)=T^{-1}\rho(s/T),\qquad Z_T(\omega)=\int_{\mathbb R}\rho_T(s)Z(s,\omega)\,ds.\] For \(T>s_0\) one has \(\int_L Z_T\geq q(s_0)>0\), while \(0\leq Z_T\leq\|z\|_\infty\). On every compact regular link patch, integration by parts in \(s\) gives \[\Delta_LZ_T\geq VZ_T-e_T, \qquad |e_T|\leq\|z\|_\infty (\|\rho_T''\|_{L^1}+a\|\rho_T'\|_{L^1}) \leq C/T.\] Local Caccioppoli estimates give uniform \(W^{1,2}\) bounds there. Weak compactness on a countable exhaustion by regular patches and weak-star compactness in \(L^\infty(L)\) provide a common subsequence with limit \(\overline Z\in W^{1,2}_{\mathrm{loc}}(L)\) satisfying \[0\leq\overline Z\leq\|z\|_\infty,\qquad \int_L\overline Z\geq q(s_0),\qquad \Delta_L\overline Z\geq V\overline Z.\] The integral passes to the weak-star limit since the whole link has finite area. There is no assumption here that \(V\) is bounded or integrable at its singular set.

Test the last inequality with \(\chi_j^2\overline Z\), supported away from the singular link. Expanding and using Young’s inequality gives \[\frac12\int_L\chi_j^2|\nabla\overline Z|^2 +\int_LV\chi_j^2\overline Z^2 \leq2\|z\|_\infty^2\int_L|\nabla\chi_j|^2 \longrightarrow0.\] Fatou’s lemma shows that \(\overline Z\) is constant on each regular link component, and that any component with positive constant has \(A_L=0\). There are at most countably many regular components, since the regular link is a second countable manifold. Its positive integral therefore supplies at least one such component. Lemma 22 would make \(C\) a plane, contrary to hypothesis. Hence \(Z\), and therefore \(z\), is zero. ◻

Proof. In exponential coordinates at the fixed point \(p\), let \(g_j\) be the rescaled metrics; these tend smoothly to the Euclidean metric on compact sets. On the rescaled copy of \(\Sigma\), \[\Delta_{\Gamma_j}z_j =(|A_{\Gamma_j}|^2+ \operatorname{Ric}_{g_j}(\nu_j,\nu_j))z_j +2|\nabla z_j|^2/z_j \geq(|A_{\Gamma_j}|^2-Cr_j^2)z_j.\] The zero extension is bounded by \(m_0^{-1}\) and is smooth on every regular component. On a connected regular limit chart, smooth multiplicity-one convergence identifies \(\Gamma_j\) with a single connected graph. The graph lies either in the rescaled \(\Sigma\) or in another component. Passing to a subsequence fixes this alternative. In the latter case \(z_j=0\) there. In the former choose a base point in the chart and normalize \(\varphi_j\) by its value there. The scalar Harnack inequality and interior Schauder estimates (Gilbarg and Trudinger 2001) apply: the graph metrics are smoothly uniformly elliptic and the Jacobi potentials are uniformly bounded with every derivative on a slightly larger regular chart. They bound the normalized positive functions and their reciprocals in every interior \(C^l\) norm. Therefore \(z_j\) has uniform interior \(C^l\) bounds, even if its value at the base point tends to zero; in that case the same normalization proves smooth convergence to zero. If its value has a positive limit, reciprocal convergence proves the claimed differential inequality. It is also valid on zero limit patches. A countable exhaustion by relatively compact regular charts and diagonal extraction produce compatible limits on the entire regular cone. ◻

Proof. If \(\Gamma\) is smooth, its regular components are compact and the first assertion follows from positivity and continuity; the second is vacuous. Otherwise \(k\geq7\), so all preceding lemmas apply. Choose \(C\geq\sup|\operatorname{Ric}_g|\) on the compact ambient neighborhood. Convex truncation of the Jacobi equation shows that \[\begin{split} w_\delta&=(1-\varphi/\delta)_+,\qquad 0\leq w_\delta\leq1,\\ \Delta_\Sigma w_\delta &\geq (|A_\Sigma|^2+\operatorname{Ric}_g(\nu,\nu)) (\varphi/\delta)\,1_{\{\varphi<\delta\}} \geq-C. \end{split}\] The possible measure at the truncation level has the favorable nonnegative sign. Extend these functions by zero to other regular components and use Lemma 21. Fix \(r>0\) small enough that \(C_0Cr^2<1/4\) and cover \(\Gamma\) by finitely many balls of radius \(r/2\) with corresponding \(r\)-balls in the allowed ambient neighborhood. Since \(w_\delta\to0\) at every regular point and the whole perimeter is finite, dominated convergence makes all the finitely many integral terms less than \(1/4\) for small enough \(\delta>0\). Thus \(w_\delta\leq1/2\) everywhere on \(\Sigma\), proving \(\varphi\geq\delta/2\). The radius is fixed before \(\delta\) is sent to zero.

Set \(z=1/\varphi\), extended by zero as before, and write \(M=\|z\|_\infty<\infty\). It satisfies \(\Delta z\geq(|A_\Sigma|^2-C)z\) on \(\Sigma\) and hence \(\Delta z\geq-CM\) on all regular components. Suppose the second assertion fails. After passing to a subsequence, compactness gives a fixed \(p\in\mathcal S\) with \(x_j\to p\) and \(z(x_j)\geq\epsilon>0\). Set \(r_j=4\operatorname{dist}_g(x_j,p)\) and dilate about \(p\). The points \(x_j\) then lie in the rescaled \(B_{1/2}\). A tangent-cone subsequence converges to a nonflat cone, since a plane tangent cone would imply regularity at \(p\). Lemmas 24 and 23 give \(z_j\to0\) smoothly on every compact regular patch of that cone.

This convergence and the uniform bound \(M\) imply \[\int_{\Gamma_j\cap B_1}z_j^2\,d\mu_j\longrightarrow0.\] To verify the assertion at singular points, choose a neighborhood of the cone singular set in \(\overline B_{3/2}\) whose closure has arbitrarily small cone measure. Such neighborhoods exist because the singular set has zero cone measure; they may be chosen with measure-zero boundary. Perimeter-measure convergence bounds the \(\mu_j\)-measure of these neighborhoods by the same arbitrarily small quantity, up to an error tending to zero. On their complement in \(\overline B_1\), support convergence and finitely many regular graph charts give uniform convergence of \(z_j\) to zero. The integral on the omitted neighborhood is bounded by \(M^2\) times its measure. Let first \(j\to\infty\) and then the omitted measure tend to zero. The cone has zero \(k\)-dimensional measure on every sphere of positive radius, so the displayed ball causes no boundary measure issue.

The rescaled inequality is \(\Delta z_j\geq-CMr_j^2\). The uniform scaled form of Lemma 21, on a fixed ball, now implies \(\sup_{\Gamma_j\cap B_{1/2}}z_j\to0\), contradicting \(z(x_j)\geq\epsilon\). This proves the blow-up along every singular approach. In particular, there has been no use of a cone limit for dilations centered at moving singular points, and no uniqueness of the tangent cone is needed. ◻

Proof. The one-dimensional BV representative has finite left and right limits \(v(t-)\) and \(v(t+)\) at every interior point; the set \(J_v=\{t:v(t-)\ne v(t+)\}\) is countable. Since \(f(t+h)-f(t)=\int_t^{t+h}v(s)\,ds\), the one-sided derivatives of \(f\) equal these traces. At a zero of the nonnegative \(f\), \[v(t-)\leq0\leq v(t+).\] Consequently both traces are zero on \(\{f=0\}\setminus J_v\).

We record a direct consequence of the scalar BV chain rule (Ambrosio et al. 2000): for every \(a\in\mathbb R\), \[|D^dv|(\{t:v(t-)=v(t+)=a\})=0.\] It suffices to prove this on a relatively compact interval, where \(|Dv|\) is finite. Choose \(\eta\in C_c^\infty(\mathbb R)\) with \(0\leq\eta\leq1\) and \(\eta(0)=1\), and define \[H_\varepsilon(s)=\int_a^s\eta((r-a)/\varepsilon)\,dr.\] Then \(\|H_\varepsilon\|_\infty\leq C\varepsilon\). The \(C^1\) BV chain rule writes \[D(H_\varepsilon(v)) =\eta((\widetilde v-a)/\varepsilon)D^dv +\sum_{t\in J_v} [H_\varepsilon(v(t+))-H_\varepsilon(v(t-))]\delta_t,\] where \(\widetilde v\) is the common trace off the jump set. The first term converges in total variation to \(1_{\{\widetilde v=a\}}D^dv\) by dominated convergence with respect to \(|D^dv|\). Every summand in the second term tends to zero, and its absolute value is bounded by \(|v(t+)-v(t-)|\); the latter jump sizes have finite sum. Thus the second term tends to zero in total variation. On the other hand \(H_\varepsilon(v)\to0\) uniformly, so its distributional derivative tends to zero. The restricted signed measure \(1_{\{\widetilde v=a\}}D^dv\) is therefore zero, and hence so is its total variation. This proves the asserted level locality. Apply it with \(a=0\) and use that the nonatomic measure \(|D^dv|\) does not charge the countable set \(J_v\). Jump atoms are genuinely excluded from the conclusion: \(f(t)=|t|\) has \(D^2f=2\delta_0\). ◻

Proof. For almost every \(y\) the equation slices to \[D^2u_y=b(y,t)\,dt+\lambda_y, \qquad \lambda_y=\int_Aw(y,a)\bigl(\delta_{h_a(y)}|_I\bigr)\,d\sigma(a), \qquad u_y(t)=u(y,t).\] Here is a precise way to arrange the slice exceptional sets. Test the cylinder equation first with products of a smooth base test and each member of a countable collection of smooth tests dense in \(C^2\) on every compact subinterval of \(I\). Fubini gives the equality outside a single base null set for all tests in that collection. The local Lipschitz bound for \(u_y\) and the locally finite measure bounds on the right extend it to every test by density. They also show that \(u_y'\in BV_{\mathrm{loc}}(I)\). A countable exhaustion by smaller cylinders yields one allowable exceptional set for the entire cylinder. The vanishing condition on \(b\) likewise holds on \(\{u_y=0\}\) for almost every \(y\), by Fubini.

For each fixed \(y\), injectivity of the graph parameter gives \[\lambda_y(\{t\}) =\int_{\{a:h_a(y)=t\}}w(y,a)\,d\sigma(a).\] Every fiber has at most one member. Consequently the contribution of \(\sigma_{\mathrm{na}}\) is precisely the nonatomic part \(\lambda_y^{\mathrm{na}}\), and the contribution of \(\sigma_{\mathrm{at}}\) is atomic. The strictly positive bounded weights preserve this distinction. Lemma 26 applied to \(u_y\) shows that the diffuse part of \(D(u_y')\) does not charge \(Z_y=\{u_y=0\}\). Taking the diffuse parts in the sliced equation and using \(b(y,\cdot)=0\) almost everywhere on \(Z_y\) therefore gives \[\lambda_y^{\mathrm{na}}(Z_y)=0 \quad\hbox{for almost every }y.\] Tonelli’s theorem now yields \[0=\int_A\int_V w(y,a)\,1_{\{h_a(y)\in I,\ u(y,h_a(y))=0\}} \,dy\,d\sigma_{\mathrm{na}}(a).\] The lower bound on \(w\) proves the assertion about almost every graph, and no uncountable intersection of slice exceptional sets has been taken.

To make the final assertion explicit, index the cylinders by \(j\in\mathbb N\). If \(\pi\) is the original measure and \(a_j\) is its graph-height map on the measurable subfamily meeting cylinder \(j\), let \(N_j\) be the null set of nonatomic parameters excluded above. Since \(N_j\) contains no atomic parameters, \(\sigma_j(N_j)=0\), and the pushforward identity gives \(\pi(a_j^{-1}(N_j))=0\). Discard \(\bigcup_{j=1}^\infty a_j^{-1}(N_j)\) together with the previously specified null family. For every remaining minimizing boundary, every nonempty patch of every regular component is represented in the countable cover. Whenever its parameter there is nonatomic, the asserted zero-set exclusion holds. In particular a regular component on which \(u\) vanishes identically can only have atomic parameters in this cover. ◻

Proof. For \(3\leq n\leq7\), the conclusion was proved above using Proposition 15. We therefore assume \(n\geq8\).

One full-measure family of graph patches.

Discard the exceptional family in Lemma 16 and the complement of \(\mathop{\mathrm{supp}}\pi\), which has zero measure because \(\mathfrak T\) is a compact metric space. Use the countable cylinders, height maps \(a_j\), and pushforward measures \(\lambda_j\) fixed in Subsection 4.7; the measures still come from the original \(\pi\), since the discarded family has zero mass.

The \(\partial_t^2u\) component of Equation (35), written in each such cylinder, has exactly the form in Lemma 27. Its positive weight is \(c/(2|d(t-h_a(y))|_g)\). The remaining bounded term is homogeneous and linear in \(u,X,\nabla X,du\). On \(\{u=0\}\), causality implies \(X=0\), and Sobolev locality gives \(du=\nabla X=0\) almost everywhere. The bounded term thus vanishes there almost everywhere. Apply that lemma on each cylinder and remove the countable union of its exceptional families of minimizers. Every remaining non-atomic graph patch has \(u>0\) somewhere; indeed its zero set has zero graph area. This discarding is on the original measure space \(\mathfrak T\) by the pushforward identity, not on an uncountable collection of separately chosen leaves.

A curvature bound on each regular component.

Fix a remaining wholly interior frontier and a connected component \(\Sigma\) of its regular part. If a local graph parameter is an atom, then all the coincident minimizers agree along this entire component. To see this, the coincidence set is open by the regular-sheet comparison principle; a limit point in \(\Sigma\) lies in the other closed frontier and the same principle makes coincidence open there as well. Connectedness makes the set all of \(\Sigma\). Thus the same positive-mass fiber is present in every smaller patch of this component, and Lemma 17 gives \(A_\Sigma=0\) everywhere.

Otherwise take a non-atomic patch of \(\Sigma\). Lemma 18 supplies one positive \(\varphi\) on all of \(\Sigma\) solving Equation (48). It is bounded below and diverges along every singular approach by Lemma 25. The functions \[Q_+=u-g(X,\nu),\qquad Q_-=u+g(X,\nu)\] are nonnegative and bounded on the compact ambient neighborhood. By the diffuse-zero-set exclusion, \(u>0\) somewhere on the chosen patch, so at least one \(Q_\pm\) is positive somewhere. For that sign, \(Q_\pm/\varphi\) attains a positive maximum on \(\Sigma\): it tends to zero at every singular end, and on a component without singular ends the component is compact. All regular frontier components have relative boundary only in the singular set, so these are the only possibilities for escape from compact subsets of \(\Sigma\).

At a point where this ratio is positive one has \(u>0\). Lemma 20 therefore gives \[L_\pm Q_\pm\le -u|A_\Sigma\pm K_{\rm tan}|^2, \qquad L_\pm\varphi=0.\] Dividing by \(\varphi\) removes the zeroth-order term: if \(L_\pm=-\Delta+b_\pm\cdot\nabla+V_\pm\), the operator on the ratio is \(-\Delta+(b_\pm-2\nabla\log\varphi)\cdot\nabla\). Its distributional strong maximum principle applies to the locally Lipschitz ratio. For example, apply the weak Harnack inequality to the nonnegative difference between its maximum and the ratio. The positive maximum set is open in \(\Sigma\) by this principle and closed by continuity; its positivity ensures that the focusing inequality holds throughout every neighborhood used in this argument. Connectedness makes the ratio constant and positive on all of \(\Sigma\). The displayed inequality then forces \(A_\Sigma\pm K_{\rm tan}=0\) there. We conclude in both the atomic and non-atomic cases that \[ |A_\Sigma|\le\sup_{\Gamma_C}|K|. \tag{50}\] The bound is independent of the component. Each regular frontier has at most countably many components; the preceding single full-measure family already makes these conclusions simultaneous.

Removal of singular points.

There can now be no singular point on this frontier. At such a point take a fixed-center tangent cone. Smooth convergence on each regular cone patch and the rescaled bound \(|A_{\rm scaled}|\le r_j\sup|K|\to0\) make every regular cone component totally geodesic. Some regular component exists by the positive density and singular-dimension bound. Its regular link component is totally geodesic, so Lemma 22 makes the cone a multiplicity-one plane. Density regularity then makes the original point regular, a contradiction.

Application of smooth outermostness.

The entire frontier is therefore a compact smooth embedded two-sided hypersurface, with finitely many components by compactness and local regularity. Its filled minimizer has connected exterior containing the end, as proved in Lemma 11. The closure of that exterior in \(\Omega\) is a smooth closed codimension-zero submanifold with this entire frontier as its intrinsic boundary. Since the frontier is wholly interior, it is an enclosing cut of exactly the kind excluded by the outermostness hypothesis. Equation (43) gives \(\theta_+=0\) everywhere with the normal toward the end, a contradiction.

Thus almost every remaining frontier is \(S\). Integrating its fixed area derivative yields Equation (49). The moment regularity and interior equations were obtained before this removal, so they remain valid with zero interior area source. ◻

Proof. The first equation in Equation (53) has the usual finite prolongation. More explicitly, if \(T_{ij}=\nabla_iX_j+\nabla_jX_i=-2uK_{ij}\), the combination \[\nabla_iT_{j\ell}+\nabla_jT_{i\ell}-\nabla_\ell T_{ij}\] equals \(2\nabla_i\nabla_jX_\ell\) plus curvature contractions with \(X\); this follows by commuting the two derivatives in the three terms. Consequently \(\nabla^2X\) is a smooth linear combination of \(u,X,du\). Together with the second equation in Equation (53), this is a closed first-order homogeneous system for \[\mathcal Y=(u,X,du,\nabla X),\qquad \nabla_i\mathcal Y=\mathcal A_i\mathcal Y,\] with smooth coefficients. The interior regularity already obtained for the moments in the preceding section, followed by these equations, gives smoothness. Alternatively their traced equations and smooth elliptic regularity give the first bootstrap directly.

Fix an interior slice of a compact collar of \(S\). Along each collar normal, the displayed system is an ordinary linear differential equation with smooth coefficients on a closed finite interval. Its solution and all derivatives with respect to the footpoint extend to the endpoint. The original solution agrees with this extension by uniqueness along each normal. The tangential equations continue to hold there by continuity. This proves smooth extension without yet discarding measures on the boundary.

Now use metric variations \(h\) whose value and first derivatives vanish on \(S\), with \(p=0\). Their area and expansion variations vanish there. For the smooth interior fields, integration by parts has zero boundary term and zero interior adjoint term. On the other hand the scalar curvature variation at \(S\) can be prescribed freely: in Gaussian coordinates take \(h_{AB}=s^2\psi(y)g_{AB}|_{s=0}\) times a collar cutoff. Its double-divergence minus Laplacian-of-trace at \(s=0\) is \(-2k\psi\), whereas all first-jet and algebraic terms vanish. The multiplier identity therefore annihilates every smooth scalar test against the scalar boundary measure. That measure is zero. The vector boundary measure is dominated by it by Equation (52), and is zero as well. ◻

Proof. All estimates are uniform in the angular variable. Along radial rays, \(|Y(r)|\leq C+\int_{R_0}^r|DY(t)|\,dt\). Integrating Equation (55), taking the supremum on the sphere, and interchanging the two integrals gives \[\sup_{R_0\leq t\leq r,\,\vartheta}|DY(t,\vartheta)| \leq C+C\int_{R_0}^r t^{-1-q} \sup_{R_0\leq s\leq t,\,\vartheta}|DY(s,\vartheta)|\,dt.\] The term containing \(Y\) has the required bound because \[\int_s^r t^{-2-q}\,dt\leq C s^{-1-q}.\] Gronwall’s inequality gives \(DY=O(1)\) and \(Y=O(r)\), hence \(D^2Y=O(r^{-1-q})\). Each Cartesian derivative has a limit along every ray. Comparing two rays on a sphere of radius \(r\), along a spherical path of length at most \(C r\), shows that their derivative values differ by \(O(r^{-q})\). Thus all ray limits form a single constant matrix \(B\), and \(Y(x)=Bx+o(r)\) uniformly on spheres.

The first row of \(B\) is zero: its linear function is nonnegative on the entire unit sphere, and replacing a direction by its negative changes its sign. Dividing \(u\geq|X|_g\) by \(r\) now forces every remaining row of \(B\) to vanish. Terminal integration of \(D^2Y\) gives \(DY=O(r^{-q})\).

If \(q>1\) this already makes \(Y\) bounded. If \(1/2<q<1\), radial integration first gives \(Y=O(r^{1-q})\), and reinsertion gives \(D^2Y=O(r^{-1-2q})\) and, since the derivative limit is zero, \(DY=O(r^{-2q})\). This is integrable because \(2q>1\), so \(Y\) is bounded. If \(q=1\), replace the exponent in this intermediate step by any number strictly between \(1/2\) and \(1\). In all cases choose an intermediate \(\widehat q\) with \(1/2<\widehat q<q\) (or use \(q\) itself away from \(1\)) so that \(DY=O(r^{-a})\) for some \(a>1\). Boundedness of \(Y\) in Equation (55) then gives \(D^2Y=O(r^{-2-q})\), followed by \(DY=O(r^{-1-q})\) and convergence of \(Y\) on each ray with error \(O(r^{-q})\). Angular comparison using this last derivative bound shows that the limits of \(Y\) are the same. Weakening the exponent proves the stated \(O_2\) assertion. ◻

Proof. In the open exterior, where \(u>0\), Equation (53) gives \[b=-u^{-1}\mathop{\mathrm{sym}}\nabla X=K,\qquad \mathop{\mathrm{Hess}}u-(\Delta u)g+\mathcal S=0.\] Apply Lemma 19 with this \(b\). Complementarity eliminates its volume term and yields \[u\mathbf G_{ij}=-J_{(i}X_{j)}.\] Gauss–Codazzi supplies \(\mathbf G(\mathbf n,\mathbf n)=\mu\) and \(\mathbf G(\mathbf n,\partial_i)=J_i\).

If \(\mu>0\), equality in \(0=u\mu+J(X)\geq\mu(u-|X|)\) forces \(|X|=u\) and \(J=-\mu X^\flat/u\). If \(\mu=0\), DEC gives \(J=0\). Since \(\xi^\flat(\mathbf n)=-u\) and its spatial part is \(X^\flat\), these identities give the first part of Equation (59). In particular \(\mu W=0\). The stress tensor has zero spacetime trace and annihilates \(\xi\); the same is therefore true of its Ricci contraction.

For compact variations extending to \(S\), the multiplier identity now reads \[2k\omega\mathcal E'-\frac c2\int_S\mathop{\mathrm{tr}}_S h\,dA =\int_\Omega\{u(2\mu)'+2J'(X)-J_i h^i{}_jX^j\}\,dV_g -\int_S\theta_+'\,d\zeta.\] The free \(p=K'\) boundary term, with integration normal \(-\nu\), is \(2X_\nu\mathop{\mathrm{tr}}_S p-2p(X_{\rm tan},\nu)\). Since \(\theta_+'=\mathop{\mathrm{tr}}_S p\) in this test, arbitrary normal-tangential and tangential components give \(X_{\rm tan}=0\) and \(d\zeta=2X_\nu dA\). For a compact conformal test \((h,p)=(2s g,sK)\), the constraint variation on active rays is \(2k(-\Delta s+K(\nabla s,v))\) and \(\theta_+'=k\partial_\nu s\). Its boundary identity becomes \[-kc\int_Ss\,dA =2k\int_S\{u\partial_\nu s- (\partial_\nu u+K(X,\nu))s\}\,dA -k\int_S\partial_\nu s\,d\zeta.\] The independent value and normal derivative of \(s\) give \(d\zeta=2u\,dA\) and Equation (60).

It remains to justify the dichotomy, since \(u\) might initially vanish at only part of \(S\). One-sided area minimization implies \(H\geq0\): test the area by every nonnegative outward speed. Use Lie derivative variations along a compactly supported vector field \(Y\) equal to \(s\nu\) on \(S\), extended across its collar. Fix an interior active ray \((x,v)\), let \(\phi_t\) be the local flow of \(Y\), and put \((g_t,K_t)=(\phi_t^*g,\phi_t^*K)\). If \(v_t\) is the isometrically identified ray for \(g_t\), then \(w_t=\mathrm d\phi_t(v_t)\) satisfies \(|w_t|_{g,\phi_t x}=|v|_{g,x}\leq1\). Naturality gives the constraint derivative as \(2F'(0)\), where \[F(t)=\mu(\phi_t x)+J_{\phi_t x}(w_t)\geq0.\] Since \(x\) is interior, this smooth function is defined for both signs of sufficiently small \(t\). Activity gives \(F(0)=0\), so \(F'(0)=0\). This argument also applies when \(|v|_g=1\), since it differentiates only along the transported curve in the closed unit-ball bundle. Boundary rays contribute no integral, since Lemma 31 has already eliminated the boundary constraint measure. The area variation is \(\int_S Hs\), and the expansion variation is \(L_+s\), the outward normal linearization of \(\theta_+\). Thus \[ L_+^*u=\frac c2 H\geq0. \tag{61}\] The operator has principal part \(-\Delta_S\) and smooth bounded coefficients. The strong minimum principle for a nonnegative supersolution applies locally: replace its zeroth-order coefficient by a larger nonnegative constant to obtain the usual form, then propagate along overlapping balls. Since \(S\) is connected, \(u>0\) throughout \(S\), or \(u\) is identically zero there.

If \(u>0\) on \(S\), the tangential derivatives of \(W\) vanish, and the normal-normal component of \(\mathop{\mathrm{sym}}\nabla X=-uK\) gives \[ dW|_S=2\kappa X^\flat=2\kappa u\nu^\flat. \tag{62}\] Thus \(W\) is a positive defining function in the exterior collar. If \(u=0\) on \(S\), then \(X=0\), all its tangential derivatives vanish, and its symmetrized equation successively gives \(\nabla_\nu X=0\) in the normal-normal and normal-tangential components. In original exterior distance \(s\), \[ u=s a,\qquad X=s^2Y_*,\qquad a|_S=\kappa, \qquad W=s^2(a^2-s^2|Y_*|_g^2), \tag{63}\] with smooth coefficients. This also proves positivity in a deleted collar. ◻

Proof. First the boundary laws and \(\mathop{\mathrm{sym}}\nabla X=-uK\) give the collars needed below. On a component where \(u>0\), differentiating \(W=u^2-|X|^2\) and using its vanishing tangential derivatives gives \[ dW|_{S_i}=2\kappa_iX^\flat=2\kappa_i u\nu^\flat. \tag{66}\] On a component where \(u=0\), also \(X=0\) and its tangential derivatives vanish. The normal-normal and normal-tangential components of the symmetrized equation give \(\nabla_\nu X=0\). In original exterior distance \(s\), therefore, \[ u=sa,\qquad X=s^2Y_*,\qquad a|_{S_i}=\kappa_i,\qquad W=s^2(a^2-s^2|Y_*|_g^2). \tag{67}\] Both alternatives imply \(W>0\) in a deleted collar. Since \(W\to1\) at the end, the interior zero set is compact and separated from all these collars and the distant end.

We first derive an estimate that permits integration through an arbitrary interior zero set of \(W\). The tensor \(C=g^{-1}-u^{-2}X\otimes X\) is smooth and nonnegative throughout the open exterior, including its zeros of \(W\). Let \(D_{\gamma\eta}=\boldsymbol\nabla_\gamma\xi_\eta\); the Killing equation makes \(D\) skew. In the following calculation \(|D|_{\mathbf g}^2\) denotes the full contraction \(D_{\gamma\eta}D^{\gamma\eta}\), which need not be nonnegative. The Killing wave identity, together with \(\mathop{\mathrm{Ric}}_{\mathbf g}(\xi,\cdot)=0\), gives \(\boldsymbol\square W=-2|D|_{\mathbf g}^2\). The spatial inverse block of \(\mathbf g\) is \(C\), its volume density is \(u\sqrt{\det g}\), and \(W\) is independent of \(z\). Consequently \[ u^{-1}\mathop{\mathrm{div}}_g(uC\,dW)=-2|D|_{\mathbf g}^2. \tag{68}\] This is an identity of smooth functions through all interior zeros.

Fix a relatively compact neighborhood of such a zero, small enough that \(u\) and \(v=|X|_g\) have positive lower bounds there. Write \(e=X/v\), \(T=e^\perp\), and choose a local \(g\)-orthonormal frame \((e,E_1,\ldots,E_{n-1})\). Let \(\mathbf n\) be the future unit normal to the \(z\)-slices, so \(\xi=u\mathbf n+ve\). For a spatial index \(i\) differentiation of \(W=-\mathbf g(\xi,\xi)\) gives \[D_{i\mathbf n}=u^{-1}(-W_i/2-vD_{ie}).\] Putting \(a_a=D_{ae}\) and \(b_{ab}=D_{ab}\), full contraction in this Lorentz orthonormal frame therefore gives the exact expansion \[-2|D|_{\mathbf g}^2 =\frac{|dW|_g^2}{u^2} +\frac{4v}{u^2}\sum_a W_a a_a -\frac{4W}{u^2}\sum_a a_a^2 -2\sum_{a,b}b_{ab}^2.\] On every smaller relatively compact neighborhood, smooth nonnegativity of \(W\) gives \(|dW|_g^2\leq C_0W\). This elementary estimate follows by applying the second-order Taylor bound along short geodesics in the direction \(-\nabla W\). Since \(D\) is bounded, the favorable last two terms imply \[-2|D|_{\mathbf g}^2\leq C_1(W+|d_TW|_g).\] Here \(d_TW=dW-e(W)e^\flat\). The eigenvalues of \(C\) are \(1\) on \(T\) and \(W/u^2\) along \(e\).

Choose a smooth compactly supported cutoff \(\eta\) in this neighborhood. Multiplying Equation (68) by \(\eta^2/(W+\delta)\) and integrating gives \[\begin{align*} E_\delta &:={\int}\frac{u\eta^2|dW|_C^2}{(W+\delta)^2}\,dV_g\\ &=\int\frac{u\eta^2(-2|D|_{\mathbf g}^2)}{W+\delta}\,dV_g +2\int\frac{u\eta\langle d\eta,dW\rangle_C}{W+\delta}\,dV_g. \end{align*}\] The preceding upper bound and Young’s inequality bound the first term by \(E_\delta/4+C_\eta\): use \(W/(W+\delta)\leq1\) and \(|d_TW|_g^2\leq |dW|_C^2\). The same inequality bounds the second term by \(E_\delta/4+C_\eta\), since \(C\leq g^{-1}\). Thus \(E_\delta\leq 2C_\eta\), independently of \(\delta>0\). Fatou’s lemma gives, wherever \(\eta=1\), \[ \int_{\{W>0\}} |d_T\log W|_g^2\,dV_g<\infty. \tag{69}\] The integral is local in the fixed neighborhood, including its zero set as a limiting set. No ellipticity in the direction \(e\) at \(W=0\) was required.

The estimate controls the transverse logarithmic derivative without requiring ellipticity along \(X\). We now derive the twist flux and verify that it contains only this controlled derivative. On \(\mathcal U\) write \(\theta=dz-A_s\), so that \(\mathbf g=-W\theta^2+h_s\) and \[\xi^\flat=-W\theta,\qquad d\xi^\flat=-dW\wedge\theta+WF_s.\] Raising the index of \(\theta\) with \(\mathbf g\) gives no spatial component. Thus the raised spatial block of \(d\xi^\flat\) is \(WC^{ik}C^{jl}F_{s,kl}\). The Killing Maxwell identity \(\boldsymbol\nabla_\gamma(d\xi^\flat)^{\gamma\eta}=0\) and stationarity yield \[ \nabla_iQ^{ij}=0,\qquad Q^{ij}=uW C^{ik}C^{jl}F_{s,kl}, \tag{70}\] where \(\nabla\) is the connection of \(g\). Since \(C\alpha=(W/u^2)X\), multiplication by \(A_s\) and lowering with \(g\) gives \[ \begin{split} g_{i\ell}Q^{\ell j}A_{s,j} &=\frac1u\left[ X^j(d\alpha)_{ij}-\frac{|X|_g^2}{W}W_i +\frac{X(W)}{W}\alpha_i\right]\\ &=\frac1u\left[X^j(d\alpha)_{ij} -|X|_g^2(d_T\log W)_i\right]. \end{split} \tag{71}\] The first line is valid even at zeros of \(X\) in \(\mathcal U\); the second is used only where \(X\ne0\). In particular the flux \(Q^{ij}A_{s,j}\) is exactly perpendicular to \(X\).

To convert the divergence identity into vanishing twist, we must remove three boundaries of integration: the possible interior zero set, the original horizon, and spatial infinity. We check their fluxes separately before taking any limit. The componentwise collars (66)–(67) already separate the compact interior zero set from \(S\) and the distant end. The boundary estimates below are made on each of the finitely many components.

Choose a smooth function \(f\) which vanishes off a distant end neighborhood and equals \(A_{\infty,i}x^i\) sufficiently far out, where \(A_\infty\) is the constant limit of \(A_s\). Put \(\beta=A_s-df\), so \(d\beta=F_s\). Near the zero set and \(S\), \(\beta=A_s\).

For the interior cutoff take a smooth increasing function \(\chi\), zero on \((-\infty,1]\) and one on \([2,\infty)\), and use \(\chi(W/\delta)\) on a fixed neighborhood of the compact zero set. Extend it as one outside a slightly larger such neighborhood. One precise extension is \(1-\rho+\rho\chi(W/\delta)\), with \(\rho=1\) near the zero set; for all sufficiently small \(\delta\), the derivative involving \(d\rho\) vanishes because \(W\) has a positive lower bound on its support. On \(\{\delta<W<2\delta\}\), transversality in Equation (71) gives \[\left|Q^{ij}A_{s,j}\partial_i\chi(W/\delta)\right| \leq C\bigl(|d_T\log W|_g+|d_T\log W|_g^2\bigr) \leq C\bigl(1+|d_T\log W|_g^2\bigr).\] A finite cover by neighborhoods satisfying Equation (69) shows that its integral tends to zero as \(\delta\downarrow0\).

For the boundary cutoff use \(\chi(s/\epsilon)\). If \(u>0\) on \(S_i\), then uniformly in its collar \[W\asymp s,\qquad e=\nabla s+O(s),\qquad d_TW=O(s).\] The flux in Equation (71) is bounded and perpendicular to \(e\), so its contraction with \(ds\) is \(O(s)\). Since the collar volume is \(O(\epsilon)\) and the cutoff derivative is \(O(\epsilon^{-1})\), the boundary error is \(O(\epsilon)\). If \(u=0\) on \(S_i\), use the first line of Equation (71). Here \(d\alpha=O(s)\), \(d\log W=O(s^{-1})\), and \(X=s^2Y\); each term inside its brackets is \(O(s^3)\), whereas \(u\asymp s\). The whole flux is consequently \(O(s^2)\), uniformly also where \(X=0\), and the boundary error is \(O(\epsilon^2)\).

Finally choose an outer cutoff supported in \(\{r<2R\}\) and equal to one on \(\{r<R\}\), with derivative \(O(R^{-1})\). On its annulus the end estimates give \[\begin{gathered} \beta=O(r^{-q_0}),\qquad Q=O(r^{-1-q_0}),\\ \left|\int Q^{ij}\beta_j\partial_i\chi_R\,dV_g\right| \leq CR^{n-2-2q_0}=o(1). \end{gathered}\]

Let \(\zeta\) be the product of these three nonnegative cutoffs. It has compact support in \(\mathcal U\). Testing Equation (70) against \(\zeta\beta\) gives \[\int\frac{\zeta uW}{2} C^{ik}C^{jl}F_{s,ij}F_{s,kl}\,dV_g =-\int Q^{ij}\beta_j\partial_i\zeta\,dV_g.\] The left integrand is nonnegative, and strictly positive where \(F_s\ne0\). Fix any compact subset of \(\mathcal U\) and choose the cutoffs to equal one there. First let \(\delta\downarrow0\), then \(\epsilon\downarrow0\), and finally \(R\to\infty\). The preceding estimates force the twist energy on that compact subset to vanish. Exhausting \(\mathcal U\) proves \(F_s=0\) everywhere on it, without assuming beforehand that the total twist energy is finite. ◻

Proof of Proposition 34. We identify the positive component, prove the distance estimate at its possible interior null frontier, and attach the original boundary.

Step 1: Static equations and the positive component. By Lemma 35, the one-form \(A_s=X^\flat/W\) is closed on \(\{W>0\}\). On a small simply connected set write \(A_s=d\Phi\) and set \(t=z-\Phi\). Then \[\mathbf g=-\lambda^2dt^2+h.\] The vacuum hypothesis applies on this set. The warped metric formulas, obtained from \(\mathbf\Gamma^t_{it}=\lambda^{-1}\lambda_i\) and \(\mathbf\Gamma^i_{tt}=\lambda\lambda^i\), are \[\mathop{\mathrm{Ric}}_{\mathbf g,tt}=\lambda\Delta_h\lambda, \qquad \mathop{\mathrm{Ric}}_{\mathbf g,ij}=\mathop{\mathrm{Ric}}_{h,ij}-\lambda^{-1}\mathop{\mathrm{Hess}}_{h,ij}\lambda.\] They give the equations in Equation (65) on every positive component.

There is a positive component \(\mathcal C\) containing the end, since \(W\to1\). Every other positive component has compact closure in the original exterior: the complement of a coordinate tail is compact. Its continuous function \(\lambda\) is zero on its frontier, including any part of \(S\). A positive maximum would therefore be attained at an interior point, contrary to the strong maximum principle for its harmonic equation. Hence there is only one positive component. If \(\lambda>1\) anywhere in that component, its limit \(1\) at infinity and its zero value on the original frontier force a maximum at a positive interior point of a compact original truncation. The strong maximum principle excludes this. Thus \(\lambda\leq1\); the boundary collar prevents constancy, so \(0<\lambda<1\). This argument uses no regularity of a possible interior frontier. The positive collar of every \(S_i\) belongs to \(\mathcal C\).

Step 2: Distance to an interior zero frontier. An interior zero of \(W\) is infinitely far away for \(h\). Indeed, on a relatively compact neighborhood of such a zero we have \(u,|X|>0\). Put \(e=X/|X|\) and denote its orthogonal complement by \(T\). The exact flux identity Equation (71), with \(F_s=dA_s=0\), gives a uniform bound for \(d_T\log W\) on its positive part. Smooth nonnegativity gives \(|dW|_g^2\leq C W\). Since \(h^{-1}=g^{-1}-u^{-2}X\otimes X\), it follows that \[ |d\log W|_{h^{-1}}^2 =|d_T\log W|_g^2+\frac{(eW)^2}{u^2W}\leq C. \tag{72}\] The integral of \(|d\log W|\) along a finite \(h\)-length curve is bounded; such a curve cannot have \(W\to0\) at an interior point. Also \(h\geq g\). Every finite-length escape curve has a limit in the original complete metric space, with \(S\) included. An interior positive limit is not an escape; an interior zero is excluded by Equation (72). We next describe precisely the remaining limits on \(S\).

Step 3: Attachment at the original boundary. Work on a fixed component \(S_i\). In the positive boundary-lapse case, Equation (66) allows original coordinates \((W,y)\) in the collar. Put \(\alpha=X^\flat\). Its smooth coefficients satisfy \[\alpha=b(W,y)dW+W\beta_A(W,y)dy^A, \qquad b(0,y)=\frac1{2\kappa_i}.\] Indeed its tangential coefficients vanish at \(W=0\), and the normal coefficient follows from \(dW=2\kappa_i\alpha\) there. In coordinates \((\lambda,y)\), \(W=\lambda^2\), the metric is \[\begin{align*} h={}&(4\lambda^2g_{WW}+4b^2)d\lambda^2 +2(2\lambda g_{WA}+2b\lambda\beta_A)d\lambda\,dy^A\\ &+(g_{AB}+\lambda^2\beta_A\beta_B)dy^A dy^B. \end{align*}\] All coefficient functions on the right are evaluated at \((\lambda^2,y)\). Its normal coefficient at zero is \(\kappa_i^{-2}\), its cross coefficients vanish, and its tangential part is \(g|_{TS_i}\). The \(d\lambda^2\) and tangential coefficients are even smooth functions, and the cross coefficients are odd smooth functions. This is a smooth reflected tensor, not merely continuity of its components.

In the zero boundary-lapse case, Equation (67) instead gives directly \[h=g+s^2\frac{Y_*^\flat\otimes Y_*^\flat} {a^2-s^2|Y_*|^2},\qquad \lambda=s\sqrt{a^2-s^2|Y_*|^2}.\] These are smooth one-sided, the first is positive definite at \(S_i\), and \(d\lambda=\kappa_i\,ds\) there. In both cases the static equation extends to \(S\) and gives \(\mathop{\mathrm{Hess}}_h\lambda=0\). Its tangential restriction is \(\kappa_i\) times the second fundamental form, so that form is zero. In Gaussian base distance \(\sigma\) the Taylor expansions are therefore \[ h=d\sigma^2+h_0(y)+\sigma^2h_2(y)+O(\sigma^3),\qquad \lambda=\kappa_i\sigma+a_3(y)\sigma^3+O(\sigma^4). \tag{73}\] Here and below these one-sided remainders have the corresponding differentiated Taylor bounds. The reflected coefficients match through second derivatives; their third derivatives are locally bounded on each side. Thus the even reflection of \(h\) and odd reflection of \(\lambda\) are \(C^{2,\alpha}\) for every \(\alpha<1\). Limits of finite-length curves on \(S\) are limits in this attached collar, as is seen directly from the coordinates \((\lambda,y)\) or \((s,y)\) just constructed.

The finite-length curve criterion proves completeness of the attached base, and gluing two copies along its compact boundary gives a complete metric double. Finally the original end estimates give \[h_{ij}=\delta_{ij}+b_i b_j+O_2(r^{-q_0}),\qquad \lambda=1+O_2(r^{-q_0}).\] A fixed linear change of coordinates takes the constant positive metric \(\delta+b^\flat\otimes b^\flat\) to the Euclidean metric and preserves these decay orders. ◻

Proof. The original compact-complement and completeness assumptions, Equation (57), and Lemma 33 verify every hypothesis of Proposition 34. In particular the geometric density vanishes where \(W>0\), so the stationary metric is vacuum there. That proposition gives the connected positive region \(\mathcal C\), its complete attached static base, and all the asserted local boundary data. Here the boundary constant is the single number \(\kappa\) in Equation (51). We need only show that no interior null frontier remains.

Step 1: A smooth complete Ricci-flat circle manifold. To rule out missing interior sets, form over \(\mathcal C\) the product with a circle of period \(2\pi/\kappa\) and metric \[ \mathfrak h=h+\lambda^2dT^2. \tag{75}\] This product is built from the global static metric and lapse; it requires no global primitive of \(A_s\) or quotient of the stationary spacetime. Collapse the circle over \(S\) by attaching \(S\times D^2\) in the collar. This attachment has the required regularity. To check it explicitly put \(\theta=\kappa T\) and \((z_1,z_2)=(\sigma\cos\theta,\sigma\sin\theta)\). The normal polar part equals \[dz_1^2+dz_2^2+ \frac{(\lambda/(\kappa\sigma))^2-1}{\sigma^2} (z_1dz_2-z_2dz_1)^2.\] By Equation (73), the extra tensor is a smooth quadratic polynomial to leading order, with remainder whose coefficients and derivatives through order two have size \(O(\sigma^{3-j})\). Its second derivatives are Lipschitz away from the axis and extend Hölder continuously with any exponent less than one. The tangential correction is \((z_1^2+z_2^2)h_2+O(\sigma^3)\) and has the same property. Thus \(\mathfrak h\) is a positive \(C^{2,\alpha}\) metric across the disk center. Its Ricci tensor vanishes off the center by the static equations, and on the center by continuity. Harmonic-coordinate elliptic regularity for the Ricci tensor now makes this metric smooth: harmonic coordinates are \(C^{3,\alpha}\), the Ricci equation has principal part \(-\tfrac12\mathfrak h^{ab}\partial_{ab}\mathfrak h_{ij}\), and repeated Schauder estimates apply to its quadratic first-derivative terms (DeTurck and Kazdan 1981).

The circle manifold is complete. Project a finite-length curve to the base, using Equation (75); the preceding length test gives a base limit. Over an interior positive base limit the circle metric is uniformly nondegenerate and the fiber is compact. At \(S\) the disk coordinates give the limit. These observations also prove metric completeness by joining a summably Cauchy subsequence with paths of summable lengths.

Step 2: Excluding an interior frontier. Suppose an interior frontier of \(\mathcal C\) remained. A sequence tending to it in the original metric escapes every compact subset of the circle manifold: the continuous map from the circle attachment to the original exterior maps its compact subsets to compact sets avoiding this frontier. Choose a connected product cross-section \(\{|x|=R\}\times S^1\) sufficiently far out. Its outer side is the asymptotically Euclidean base times a circle, and is noncompact. A connected two-sided cross-section has at most two complement components: each component must meet one of its two connected collar sides. Thus the inner side is a single component and contains the escaping sequence just constructed. The cross-section therefore separates at least two ends of a complete smooth manifold with nonnegative Ricci curvature. Minimizing segments between escaping sequences in the two sides cross the compact separator and have a subsequence converging to a globally minimizing line. The Cheeger–Gromoll splitting theorem (Cheeger and Gromoll 1971) gives an isometric product \(\R\times N\). If \(N\) were noncompact this product would have one end: outside any compact cylinder its two axial sides can be joined through a point of \(N\) outside its compact projection. Hence \(N\) is compact, and its ball volumes grow at most linearly. On the other hand, in the original product end, \(h\) tends after a linear change to the Euclidean metric and \(\lambda\to1\). Uniform metric comparison places a coordinate annulus of radius proportional to \(L\), times the entire circle, inside a ball of radius \(L\) about a fixed point; its volume is at least \(c_0L^n\). This contradicts linear growth. There is no interior frontier. Since \(\Omega\) is connected, \(\mathcal C=\operatorname{int}\Omega\). ◻

Proof. Extend \(\chi w\) by zero across a ball, where \(\chi\) is one far out, and put \(f=\Delta_\delta(\chi w)\). Then \(\int(1+|z|^\eta)|f(z)|\,dz<\infty\). For the fundamental solution \(\Phi(x)=-(d\omega)^{-1}|x|^{-d}\) of \(\Delta_\delta\), the difference \(\chi w-\Phi*f\) is an entire harmonic function tending to zero, and therefore vanishes. In the region \(|z|<|x|/2\), the estimate \[|\Phi(x-z)-\Phi(x)| \le C|x|^{-d-\eta}|z|^\eta\] controls the difference from the monopole. In the remaining region, split off \(|x-z|<|x|/2\). On that ball, the pointwise bound for \(f\) gives \(C|x|^{-d-\epsilon}\) after integration against \(\Phi\); on its complement the finite \(\eta\)-moment gives \(C|x|^{-d-\eta}\). Consequently \(A=-(d\omega)^{-1}\int f\) and the asserted zeroth-order remainder follows. Interior Poisson estimates on annuli rescaled to unit size give every differentiated remainder. This last step avoids differentiating the singular Newton kernel twice under an absolutely convergent integral. ◻

Proof. The conformal Ricci and Laplace formulas, with conformal exponent \(V/d\), give \[ \mathop{\mathrm{Ric}}_{\check h}=\frac{k}{d}\,dV\otimes dV, \qquad \Delta_{\check h}V=0. \tag{78}\] For example \(\Delta_hV=-|dV|_h^2\) and \(\mathop{\mathrm{Ric}}_h=\mathop{\mathrm{Hess}}_hV+dV\otimes dV\); substitution cancels both the Hessian and the scalar multiple of \(h\) in the conformal Ricci formula.

Here are details of the choice of harmonic coordinates. Extend the end coefficients to be Euclidean inside a large radius \(R\), using a cutoff on \(R<r<2R\), and solve for \(y^i=x^i+\psi^i\) satisfying \(\Delta_{\check h}y^i=0\) on the original tail. The weight for \(\psi\) is \(\gamma=1-q_0\in(2-n,1)\setminus\{0\}\). The Euclidean Laplacian has a bounded right inverse from weighted \(C^{0,\alpha}_{\gamma-2}\) to weighted \(C^{2,\alpha}_\gamma\). To see the bound directly when \(\gamma<0\), use its Newton integral; when \(\gamma>0\), replace the kernel on \(|z|>1\) by \(\Phi(x-z)-\Phi(-z)\). The latter subtraction makes the outer integral convergent. Splitting the integral into \(|z|<|x|/2\), comparable radii, and \(|z|>2|x|\) gives growth \((1+|x|)^\gamma\), and rescaled interior estimates give its derivatives and Hölder seminorm. The restrictions on \(\gamma\) are exactly those used in these integrals. Constants in the positive-weight kernel cause no difficulty; this construction fixes one right inverse.

In coordinates \(x=Rz\), the extended coefficients differ from their Euclidean values by \(O(R^{-q_0})\) in the corresponding weighted coefficient norms. Hence \[\psi=T\bigl(-\Delta_{\check h}x -(\Delta_{\check h}-\Delta_\delta)\psi\bigr)\] is solved by a Neumann series for large \(R\). It gives \(\psi=O_2(r^{1-q_0})\), with the scaled Hölder control, and \(Dy=I+O(r^{-q_0})\). Thus \(y\) is a coordinate system sufficiently far out: local invertibility follows from the last bound, and properness and the estimate \(y=x+o(r)\) give injectivity and coverage of a smaller tail. The initial \(O_2\) coefficient bounds also give scaled \(W^{3,p}\) bounds for \(y\) for every finite \(p\) by differentiating its scalar equation once. The metric in \(y\) consequently has controlled \(W^{2,p}\) and \(C^{1,\alpha}\) norms on scaled annuli, for \(p>n\).

In these coordinates the equations have the diagonal principal part \[\check h^{ab}\partial_{ab}\check h_{ij} =Q_{ij}(\check h^{-1},D\check h) -\frac{2k}{d}\,\partial_iV\partial_jV, \qquad \check h^{ab}\partial_{ab}V=0,\] where \(Q\) is quadratic in \(D\check h\). Scalar interior Schauder estimates, first with the controlled \(C^{1,\alpha}\) coefficients and then after differentiation, give \(\check h-\delta,V=O_j(r^{-q_0})\) for every \(j\). It follows that \[\Delta_\delta(\check h_{ij}-\delta_{ij}),\quad \Delta_\delta V=O_j(r^{-2-2q_0}).\] Since \(2+2q_0>n\), Lemma 37 gives \[\check h_{ij}=\delta_{ij}+A_{ij}r^{-d} +O_j(r^{-d-\eta}),\qquad V=A_0r^{-d}+O_j(r^{-d-\eta}).\] The harmonic coordinate identity \(\partial_j(\sqrt{\det\check h}\,\check h^{ij})=0\) has leading term \[d\left(A_{ij}-\tfrac12\mathop{\mathrm{tr}}(A)\delta_{ij}\right)x^j r^{-n}=0.\] All omitted terms are smaller by a positive power of \(r\). Letting \(r\to\infty\) on each ray shows \(A_{ij}-\tfrac12\mathop{\mathrm{tr}}(A)\delta_{ij}=0\). Its trace is \((1-n/2)\mathop{\mathrm{tr}}(A)=0\), so \(A=0\) because \(n\ge3\).

Write \(A_0=-a\). As \(\eta<1\le d\), exponentiation gives \(\lambda=1-a r^{-d}+O_j(r^{-d-\eta})\). Integrating \(\Delta_h\lambda=0\) between \(S\) and a large coordinate sphere, with the inner integration normal \(-\nu_h\), gives \[\lim_{r\to\infty}\int_{S_r}\partial_{\nu_h}\lambda\,dA_h =\kappa\mathop{\mathrm{Area}}_h(S).\] The left side is \(d\omega a\) by the expansion. This proves the coefficient and its strict positivity. ◻

Proof. The scalar conformal formula and \(\Delta_h\lambda=0\) make Equation (79) scalar flat away from the added point, including the seam. At the retained end its exact expression relative to \(\check h\) is \[\gamma_* =\left(\frac{(1+\lambda)^2}{4\lambda}\right)^{2/d}\check h =\left(1+\frac{(1-\lambda)^2}{4\lambda}\right)^{2/d}\check h.\] Proposition 38 therefore gives \(\gamma_* =\delta+O_j(r^{-d-\eta})+O_j(r^{-2d})\). Its ADM flux is zero.

On the second copy the factor relative to \(\check h\) is \[\left(\frac{1-\lambda}{2}\right)^{4/d}\lambda^{-2/d} =\left(\frac a2\right)^{4/d}r^{-4} \bigl(1+O_j(r^{-\eta})\bigr).\] In inverted coordinates \(z=x/|x|^2\), \(\rho=|z|\), the identity \(r^{-4}\delta_{ij}\,dx^i dx^j=\delta_{ij}\,dz^i dz^j\) gives \[ (\gamma_*)_{ij}(z) =\left(\frac a2\right)^{4/d} \bigl(\delta_{ij}+e_{ij}(z)\bigr), \qquad |D^j e(z)|\le C_j\rho^{\eta-j}\quad(j\ge0). \tag{80}\] The orthogonal reflection in the differential of inversion causes no change in these orders. Define \(e(0)=0\). Choose \[ \frac n2<s<\frac{n}{2-\eta}<n. \tag{81}\] Then \(\int_0^1\rho^{n-1+s(\eta-2)}\,d\rho<\infty\). Thus the metric has two weak derivatives in \(L^s\) across the point. For completeness, integrating by parts over \(\rho>\varepsilon\) introduces no missing second-derivative measure: the potentially relevant first-derivative boundary term is \(O(\varepsilon^{n+\eta-2})\to0\); the zeroth-order boundary term also vanishes. Its curvature is consequently the ordinary \(L^s\) tensor computed from these weak derivatives. Indeed, with \[p=\frac{ns}{n-s}>n,\qquad p>2s,\] the first derivatives belong to \(L^p\), so their quadratic products belong to \(L^{p/2}\subset L^s\). Scalar curvature vanishes almost everywhere and therefore distributionally everywhere.

Equation (80) gives a continuous positive definite extension at the added point. The remainder of the compact core has the same property. Every escape from this core along the retained end has infinite length, since its metric is uniformly comparable to the Euclidean metric. This proves completeness of the filled metric. ◻

Proof. We describe the underlying global estimate to fix the solvability class. This manifold has the Euclidean Sobolev inequality \[ \|\varphi\|_{L^{2n/(n-2)}} \le C\|d\varphi\|_{L^2},\qquad \varphi\in C_c^\infty(N). \tag{84}\] To verify the compact part of this assertion, on the end write \[\varphi(r,\theta)=-\int_r^\infty\partial_t\varphi(t,\theta)\,dt.\] Weighted Cauchy–Schwarz gives \[|\varphi(r,\theta)|^2 \le\frac{r^{-d}}d \int_r^\infty|\partial_t\varphi|^2t^{n-1}\,dt.\] Integration over a fixed end annulus bounds its \(L^2\) norm by the global Dirichlet integral. A Poincaré inequality on a connected compact region meeting this annulus gives the same bound on the core. Finite-chart Sobolev inequalities on the core, and the Euclidean Sobolev inequality applied after an end cutoff, now prove Equation (84). Uniform metric comparability suffices in this argument.

Let \(\mathcal D^{1,2}\) be the completion of \(C_c^\infty(N)\) in the Dirichlet norm. Since a compactly supported \(L^s\) function belongs to \(L^{2n/(n+2)}\), Equation (84) and the Lax–Milgram theorem solve \[\int_N\langle dv,d\varphi\rangle_\gamma\,dV_\gamma =-\int_N f\varphi\,dV_\gamma, \qquad v\in\mathcal D^{1,2}.\] They also bound \(\|dv\|_2\) and \(\|v\|_{2n/(n-2)}\) by \(C\|f\|_s\). Local Moser iteration for this divergence equation gives a local \(L^\infty\) bound, because \(s>n/2\). The local \(W^{2,s}\) estimate applies as follows. In coordinates write \(\Delta_\gamma=a^{ij}\partial_{ij}+B^j\partial_j\). Here \(a\) is continuous and uniformly elliptic, and \(B\in L^p\) with \(p=ns/(n-s)>n\). On small balls, freeze \(a\), use the constant-coefficient Calderón–Zygmund estimate, and absorb its oscillation. The drift is absorbed by Hölder and interpolation, \[\|B Dv\|_s\le\|B\|_p\|Dv\|_n, \qquad \|Dv\|_n\le\varepsilon\|D^2v\|_s+C_\varepsilon\|v\|_s,\] where the strict inequality \(p>n\) permits interpolation below the Sobolev exponent for \(Dv\). These scalar interior estimates can first be applied to smooth metric and source approximations; their uniform energy, Moser, and interior bounds pass to the weak solution. This proves the core part of Equation (82); the scalar estimates used here are those of (Gilbarg and Trudinger 2001). The compact bound controls the inner boundary of the end. We next turn it into the decay and flux estimates that will determine the conformal mass change.

For sufficiently large \(r\), \[\Delta_\gamma r^{-b} =b(b-d)r^{-b-2}+O(r^{-b-d-2})<0.\] Compare \(v\) and \(-v\) with \(Cr^{-b}\), choosing \(C\) from the fixed inner sphere bound. The comparison is legitimate on the unbounded end by testing with \((\pm v-Cr^{-b})_+\): this function has zero inner trace and belongs to \(\mathcal D^{1,2}\) since \(2b>d\). It follows that \(|v|\le Cr^{-b}\). Rescaled interior estimates for the smooth homogeneous end equation give all derivative bounds there. A decaying homogeneous solution has neither a positive maximum nor a negative minimum by the weak maximum principle, so the inverse is unique in the stated class.

Finally, \(\Delta_\delta v=O_j(r^{-n-b})\) on the end. Apply Lemma 37 and integrate the weak equation over large coordinate balls to obtain \[-d\omega A_f =\lim_{r\to\infty}\int_{S_r}\partial_{\nu_\gamma}v\,dA_\gamma =\int_N f\,dV_\gamma.\] There is no interior boundary in this distributional identity. All constants above stay bounded under small compact \(W^{2,s}\) metric perturbations: their \(C^0\) norms and ellipticity, Hölder moduli of \(a\), and \(L^p\) norms of \(B\) are uniformly controlled, and the end is fixed. ◻

Proof. Fix a smooth point \(p\). A local Dirichlet Green function, cut off inside its coordinate ball, has distributional Laplacian \(-d\omega\delta_p+f_0\), where \(f_0\) is smooth and supported away from \(p\). Subtract the solution of \(\Delta_\gamma v=f_0\) supplied by Lemma 41. The resulting function \(G\) is harmonic off \(p\), tends to zero at the end, and has the positive pole \(G=r_p^{-d}(1+o(1))\) in geodesic distance from \(p\), with the corresponding differentiated leading term. The maximum principle on the punctured manifold gives \(G>0\). Its distributional equation and the same flux calculation as in Equation (83) give \[\Delta_\gamma G=-d\omega\delta_p, \qquad G=r^{-d}+O_j(r^{-d-\eta_G})\quad\hbox{at infinity}.\] For \(\varepsilon>0\) put \(\gamma_\varepsilon=(1+\varepsilon G)^{4/d}\gamma\) on \(N\setminus\{p\}\). This is scalar flat and complete, with the original end and a second complete end at the pole. Near the pole, the length factor is comparable to \(r_p^{-2}\).

For fixed \(\varepsilon\), sufficiently small spheres about \(p\) have strictly negative mean curvature toward the original end. In fact, writing \(U=1+\varepsilon G\), the conformal mean-curvature law is \[H_{\gamma_\varepsilon} =U^{-2/d}\left(H_\gamma+\frac{2k}{d}\partial_{\nu_\gamma}\log U\right),\] and the expression in parentheses is \(k/r_p-2k/r_p+o(r_p^{-1})<0\). Choose one such sphere \(S_\rho\). Truncate the pole end at a smaller sphere \(S_{\rho_0}\), where \(0<\rho_0<\rho\), retaining the closed ambient exterior \(M_{\rho_0}\). This is a smooth scalar-flat manifold with compact boundary and compact complement to the original Euclidean end. Its ordinary metric completeness follows from its compact smooth core, smooth boundary charts, and complete asymptotically Euclidean tail. This ambient completeness is independent of paths in the singular enclosure to be constructed. We keep \(M_{\rho_0}\) as an ambient neighborhood on both sides of the minimizing frontier that will be constructed.

For the enclosure construction itself, let \(X_\rho\) be the original-end exterior bounded by \(B=S_\rho\), with \(B\) included. Apply the smooth full-enclosure results of (OpenAI 2026a, Lemmas 2.1–2.3) to \((X_\rho,\gamma_\varepsilon)\). These results assume a smooth connected manifold complete with its nonempty compact smooth boundary included, one Euclidean coordinate end with compact complement, and \(g-\delta=O_2(r^{-q})\) for some \(q>0\). A compact smooth filling \(O\) is attached behind \(B\), and perimeter is measured in the resulting boundaryless manifold, counting all free-frontier components and any contact with \(B\). In our application choose the metric on that auxiliary filling to agree with \(\gamma_\varepsilon\) in a collar behind \(S_\rho\), where the original metric is smooth. Its extension farther inside the filling has no role in the scalar-curvature argument on the exterior.

The end condition holds because \(\gamma_\varepsilon-\delta=O_j(r^{-d})\); in particular, large coordinate spheres have mean curvature \(k/r+O(r^{-d-1})>0\). The enclosure lemmas give a compact full-perimeter minimizer with connected end exterior. Since \(H_B<0\) toward \(X_\rho\), their strict-barrier conclusion puts a fixed exterior collar of \(B\) inside the minimizing filled set. Its frontier is therefore detached from \(B\), is outer minimizing, and is locally perimeter minimizing in the smooth metric. Its singular set has Hausdorff dimension at most \(n-8\). These statements use only smooth obstacle geometry and full perimeter, independently of original-data variation or equality.

Consequently this compact frontier is locally perimeter minimizing, outer minimizing, and minimal, with singular set of Hausdorff dimension at most \(n-8\) by minimizing-boundary regularity (Simon 2018). Its area is positive, and the smooth metric is defined on a neighborhood of it. Its original-end exterior has nonnegative scalar curvature, integrable scalar curvature (in fact zero), and the decay \(d>d/2\) required by Theorem 40. To give its enclosing-set ambient realization, use the already constructed \(M_{\rho_0}\): its compact boundary is strictly behind the detached frontier and its metric is smooth through that frontier. Let \(D\) denote the retained closed enclosing-set exterior, including its entire frontier, and let \(d_D\) be the possibly extended path distance defined above, using \(\gamma_\varepsilon\). We verify completeness of every finite-distance class of \(d_D\).

Fix such a class \(C\) and a \(d_D\)-Cauchy sequence \((x_j)\) in \(C\). Choose a subsequence \((x_{j_\ell})\) with \(d_D(x_{j_\ell},x_{j_{\ell+1}})<2^{-\ell-1}\). By the definition of the infimum, these successive points can be joined by rectifiable paths \(c_\ell\subset D\) of length less than \(2^{-\ell}\). Concatenate the paths. Their total length is finite and their tail lengths tend to zero. Ambient distances between points on a tail are bounded by that tail’s length. Ordinary completeness of \(M_{\rho_0}\) therefore gives an ambient endpoint \(x_\infty\), and \(x_\infty\in D\) because \(D\) is closed. Appending this endpoint gives a rectifiable path in \(D\) with length at most the sum of the constituent lengths. The appended limit lies in the same finite-distance class: the completed concatenation joins it to the fixed chain point \(x_{j_1}\) with finite length. Its tails give \[d_D(x_{j_\ell},x_\infty) \le\sum_{r\ge\ell}\operatorname{Length}_{\gamma_\varepsilon}(c_r) \longrightarrow0.\] The Cauchy property and the triangle inequality then give \(d_D(x_j,x_\infty)\to0\) for the whole sequence. Thus \(C\) is complete. No restriction of \(D\) to the finite-distance class of its end has been made; all frontier points and components, and their full perimeter, remain included. The argument asserts neither finite-length access to an arbitrary singular frontier point nor an identification with the length completion of the open exterior. Thus the numerical assertion of Theorem 40 implies \(m_{\rm ADM}(\gamma_\varepsilon)\ge0\). The old-end expansion of \(G\) and the ADM normalization give \[m_{\rm ADM}(\gamma_\varepsilon) =m_{\rm ADM}(\gamma)+2\varepsilon.\] Letting \(\varepsilon\downarrow0\) proves the claim. No property of the discarded pole end other than the displayed barriers is used. ◻

Proof. Put \(c_n=4k/d\). The conformal equation is \[-c_n\Delta_{\gamma'}(1+v)+R_{\gamma'}(1+v)=0.\] Use the inverse \(T_\gamma\) of Lemma 41 to write it as \[ v=T_\gamma\left[ c_n^{-1}R_{\gamma'}(1+v) -(\Delta_{\gamma'}-\Delta_\gamma)v\right]. \tag{86}\] The perturbation is supported in a fixed compact set \(K\) containing the nonsmooth locus. Because \(R_\gamma=0\), both terms in brackets are supported in \(K\). The source norm is \(L^s\) and the solution norm is that in Equation (82). These products are bounded in the indicated spaces: a second-order coefficient perturbation is small in \(L^\infty\), drift perturbations are small in \(L^p\), \(Dv\in L^p\), \(p/2>s\), and \(v\in L^\infty\). Also \(R_{\gamma'}\) is small in \(L^s\) because \(\gamma'\to\gamma\) in \(W^{2,s}\) and \(R_\gamma=0\). Thus Equation (86) is solved by a Neumann series, with \(\|v\|\to0\) as \(\gamma'\to\gamma\). In particular \(U>0\), the corrected metric is complete, and its scalar curvature vanishes distributionally. The coefficient maps \(\gamma\mapsto\gamma^{-1}\), \(\gamma\mapsto\Delta_\gamma\), and \(\gamma\mapsto R_\gamma\) are differentiable between these spaces: their derivatives contain second derivatives linearly and products of two \(L^p\) first derivatives, with \(p/2>s\). Differentiating the convergent operator inverse in Equation (86) therefore proves differentiability in the perturbation parameter.

On the common end \(v\) is harmonic for \(\gamma'\), so it has the monopole expansion of Equation (83). The ADM calculation for \(U=1+A r^{-d}+o_1(r^{-d})\) gives a mass change \(2A\), or equivalently \[m_{\rm ADM}(U^{4/d}\gamma')-m_{\rm ADM}(\gamma') =-\frac{2}{d\omega} \lim_{r\to\infty}\int_{S_r} \partial_{\nu_{\gamma'}}U\,dA_{\gamma'}.\] Integrating \(\Delta_{\gamma'}U=c_n^{-1}R_{\gamma'}U\) proves Equation (85). The flux integral is finite, and its compact-source expression also proves its differentiability.

To prove nonnegativity for the original rough metric, smooth its coefficients in finitely many charts covering its nonsmooth compact set, with a partition of unity, leaving the end fixed. The resulting smooth positive definite metrics \(\gamma_j\) converge in \(W^{2,s}\) on the core and uniformly; their scalar curvatures tend to zero in \(L^s\). The preceding construction provides \(U_j\to1\) and smooth complete scalar-flat metrics \(U_j^{4/d}\gamma_j\). Smoothness of \(U_j\) follows from its smooth scalar elliptic equation. The end expansion is \(\delta+O_\ell(r^{-d})\) at every derivative order \(\ell\), so Lemma 42 applies. Compact smoothing leaves the uncorrected end mass unchanged, and Equation (85) gives \[0\le m_{\rm ADM}(U_j^{4/d}\gamma_j) \longrightarrow m_{\rm ADM}(\gamma).\] This proves nonnegativity at the asserted regularity as well. ◻

Proof. Step 1: Mass variation and weak Ricci flatness. Let \(v_{ij}\) be an arbitrary smooth symmetric tensor compactly supported in a smooth patch of \(\gamma_*\), and put \(\gamma_t=\gamma_*+tv\). For both signs of sufficiently small \(t\), Lemma 43 constructs a positive factor \(U_t\) such that \(\widetilde\gamma_t=U_t^{4/d}\gamma_t\) is scalar flat. The same lemma gives \(m_{\rm ADM}(\widetilde\gamma_t)\ge0\). Since \(U_0=1\) and \(m_{\rm ADM}(\gamma_*)=0\), this differentiable mass has derivative zero at \(t=0\). Equation (85) gives precisely \[\begin{align*} 0 &= -\frac{1}{2k\omega}\int_N R'_{\gamma_*}[v]\,dV_{\gamma_*}\\ &= \frac{1}{2k\omega}\int_N \langle\mathop{\mathrm{Ric}}_{\gamma_*},v\rangle_{\gamma_*}\,dV_{\gamma_*}. \end{align*}\] For the second equality use \(R'[v]=\mathop{\mathrm{div}}\mathop{\mathrm{div}}v-\Delta\mathop{\mathrm{tr}}v-\langle\mathop{\mathrm{Ric}},v\rangle\); the divergence terms integrate to zero because the test is supported in that smooth patch. Hence \(\mathop{\mathrm{Ric}}_{\gamma_*}=0\) on every smooth patch. Its coordinate components belong to \(L^s\), as proved in Lemma 39. The seam and the filled point have zero volume, so \(\mathop{\mathrm{Ric}}_{\gamma_*}=0\) also as a distribution everywhere.

Step 2: Smoothness at the seam and the filled point. We include the regularity argument before applying a smooth comparison theorem. For a locally uniformly elliptic \(W^{2,s}\) metric with \(n/2<s<n\), put \(p=ns/(n-s)>n\). It belongs to \(W^{1,p}\), so local harmonic coordinates exist as \(C^1\) diffeomorphisms of class \(W^{2,p}\) by (Julin et al. 2017, Corollary 2.8) with harmonic exponent two. In original coordinates their equation is \[a^{ij}\partial_{ij}y^\ell+B^j\partial_jy^\ell=0, \qquad a\in W^{2,s},\quad B\in W^{1,s}\cap L^p.\] We justify the extra derivative without assuming strong regularity of the differentiated coordinates. Define the distributional operator on \(W^{1,p}_{\mathrm{loc}}\) functions by \[Lz=\partial_i(a^{ij}\partial_jz) +(B^j-\partial_i a^{ij})\partial_jz .\] Its lower-order product belongs to \(L^{p/2}\subset L^s\) locally; on \(W^{2,s}\) functions it equals \(a^{ij}\partial_{ij}z+B^j\partial_jz\). For \(z=\partial_k y^\ell\in W^{1,p}\), differentiating the scalar coordinate equation by weak product rules therefore gives \[Lz=F,\qquad F=-(\partial_k a^{ij})\partial_{ij}y^\ell -(\partial_kB^j)\partial_jy^\ell\in L^s.\] Indeed \(Dy\) is bounded, \(DB\in L^s\), and \(Da,D^2y\in L^p\) with \(p/2>s\).

Choose a small coordinate ball \(B_r\) and a cutoff \(\chi\in C_c^\infty(B_r)\) equal to one on a smaller ball. Then \(w=\chi z\in W^{1,p}_0(B_r)\) satisfies \(Lw=H\in L^s(B_r)\): the additional terms are \(2a^{ij}(\partial_i\chi)(\partial_jz)\) and \((a^{ij}\partial_{ij}\chi+B^j\partial_j\chi)z\). Construct \(v\in W^{2,s}(B_r)\cap W^{1,s}_0(B_r)\) with \(Lv=H\) by perturbing the constant-coefficient Dirichlet inverse with coefficient matrix \(a(x_0)\) at the ball center. Its second-derivative estimate and the scaled Sobolev inequality give \[\|B Dv\|_{L^s(B_r)} \le C r^{\,1-n/p}\|B\|_{L^p(B_r)} \|D^2v\|_{L^s(B_r)},\qquad 1-\frac np=2-\frac ns>0 .\] Continuity of \(a\) makes the other operator error at most \(C\|a-a(x_0)\|_\infty\|D^2v\|_s\). Both errors are small for sufficiently small \(r\), so a Neumann series constructs \(v\).

Sobolev embedding gives \(v\in W^{1,p}\) with zero trace. Thus \(q=w-v\in W^{1,p}_0(B_r)\subset H^1_0(B_r)\) satisfies \[\partial_i(a^{ij}\partial_jq) +(B^j-\partial_i a^{ij})\partial_jq=0 .\] Testing with \(q\), using ellipticity with constant \(\lambda_0>0\) and the Euclidean Sobolev inequality, gives \[\lambda_0\|Dq\|_2^2 \le C\|B-\operatorname{div}a\|_{L^n(B_r)}\|Dq\|_2^2 .\] The coefficient norm is arbitrarily small on a small ball because \(B,Da\in L^p\) and \(p>n\). Hence \(q=0\). On the smaller ball \(z=v\in W^{2,s}\), proving \(y\in W^{3,s}_{\mathrm{loc}}\).

The inverse coordinates have the same regularity: their third derivatives contain \(D^3y\in L^s\) and quadratic \(D^2y\) terms in \(L^{p/2}\subset L^s\). The metric transformation formula likewise has highest terms \(D^2\gamma_*\), \(D^3(y^{-1})\), and products of two \(L^p\) factors. Consequently the harmonic-coordinate metric remains \(W^{2,s}\). To justify tensoriality, approximate the metric in \(W^{2,s}\) and the coordinate map in \(W^{3,s}\). The latter convergence is also \(C^1\), so the maps and their inverses remain diffeomorphisms on smaller common charts. Their transformed metrics converge in \(W^{2,s}\); the same product bounds imply convergence of their Ricci tensors in \(L^s\). The smooth Ricci transformation law therefore passes to the limit. Composition of the \(L^s\) tensors causes no additional difficulty: approximate the limiting tensor by a smooth tensor and use the uniformly bounded Jacobians of the coordinate maps.

The harmonic-coordinate Ricci identity (DeTurck and Kazdan 1981, Equation (4.3)) now gives the weak system \[(\gamma_*)^{ab}\partial_{ab}(\gamma_*)_{ij} =Q_{ij}(\gamma_*^{-1},D\gamma_*),\] whose right side belongs to \(L^{s_1}\), where \(s_1=ns/(2(n-s))>s\). The principal coefficients are continuous and uniformly elliptic. To obtain the higher regularity without assuming it in advance, localize a component with a cutoff in a small harmonic-coordinate ball. Its equation has right side in \(L^{s_1}\): besides the quadratic term, the cutoff terms contain only the metric and its first derivatives. Solve that Dirichlet equation in \(W^{2,s_1}\) by the constant-coefficient inverse and the small oscillation of the principal coefficients. The difference from the original localized component lies in \(H^1_0\) and solves the homogeneous equation. Writing the equation in divergence form introduces the drift \(-\partial_i(\gamma_*)^{ij}\in L^p\); its \(L^n\) norm is small on this ball. The energy uniqueness argument just used for the coordinate derivatives forces the difference to vanish. Hence each metric component belongs to \(W^{2,s_1}\) on a smaller ball. Repeating strictly improves the exponent until it exceeds \(n\) in finitely many steps. Indeed while \(s_j<n\) the reciprocal recurrence is \(n/s_{j+1}=2n/s_j-2\), starting below two. If an exponent equals \(n\), the embedding into \(W^{1,p'}\) for every finite \(p'\) supplies the next improvement above \(n\). Thus the metric becomes \(C^{1,\alpha}\); the displayed system and successive Schauder estimates make it smooth. The harmonic transition maps are then smooth by their scalar harmonic equations. This establishes a smooth complete Ricci-flat Riemannian structure on the same metric space, including the filled point and the seam.

Step 3: Euclidean volume rigidity. The one asymptotically Euclidean end gives asymptotic volume ratio one. More explicitly, outside each sufficiently large fixed core the metric lies between \((1-\epsilon)\delta\) and \((1+\epsilon)\delta\), while the core contributes only a fixed additive distance and finite volume. Squeezing geodesic balls between the corresponding coordinate balls and then sending \(\epsilon\downarrow0\) gives \[\lim_{R\to\infty} \frac{\mathop{\mathrm{Vol}}_{\gamma_*}(B_{\gamma_*}(o,R))}{(\omega/n)R^n}=1.\] For a complete smooth manifold with nonnegative Ricci curvature, Bishop–Gromov comparison makes this ratio nonincreasing; its small-radius limit is also one. The classical equality case of Bishop–Gromov comparison gives a global Euclidean isometry; this volume-rigidity statement is recalled in (Cavalletti and Manini 2026, 3237, before Theorem 1.1). ◻

Proof. We first derive estimates in the original \(x\) coordinates, without using the ambient spatial projection as a global coordinate system. Since \(\xi\) and \(\mathbf n\) are future timelike, \[u=-\mathbf G(\xi,\mathbf n)>0, \qquad W:=u^2-|X|_g^2=\alpha(s)^2>0.\] Killing translation preserves the induced metric and second form. Its tangential and normal parts therefore give the vacuum Killing initial-data equations \[\begin{align*} \nabla_iX_j+\nabla_jX_i&=-2uK_{ij}, \tag{97}\\ \nabla_i\nabla_ju &=u\bigl(\mathop{\mathrm{Ric}}_{ij}+\tau K_{ij}-2K_i{}^\ell K_{\ell j}\bigr) -(\mathcal L_XK)_{ij}. \tag{98}\end{align*}\] The sign of the first equation follows directly by restricting \(\mathcal L_\xi\mathbf G=0\) to two tangential vectors and using \(K(U,V)=\mathbf G(\boldsymbol\nabla_U\mathbf n,V)\). For the second equation, the normal variation of \(K\) with speed \(u\) in vacuum is \(\nabla^2u-u(\mathop{\mathrm{Ric}}+\tau K-2K^2)\) with this sign convention; adding the tangential variation \(\mathcal L_XK\) gives zero. In particular, no stationary development has to be constructed in this argument: the ambient vacuum metric is already given.

To see the precise differential estimate supplied by these equations, expand \[(\mathcal L_XK)_{ij} =X^\ell\nabla_\ell K_{ij} +K_{\ell j}\nabla_iX^\ell+K_{i\ell}\nabla_jX^\ell.\] Differentiating Equation (97), adding the two permutations with first derivative indices \(i,j\), and subtracting the third permutation expresses \(\nabla_i\nabla_jX_\ell\) as curvature times \(X\) and the three derivatives \(-\nabla_i(uK_{j\ell})-\nabla_j(uK_{i\ell})+ \nabla_\ell(uK_{ij})\). Commuting covariant derivatives accounts for the curvature terms. Thus both Hessians involve only first derivatives of \(u,X\), with coefficients \(K\), and their values, with coefficients among \(\mathop{\mathrm{Ric}}\), the full curvature, \(\nabla K\), and \(K^2\). Passing to coordinate derivatives adds coefficients \(\Gamma(g)\) to first derivatives and \(D\Gamma(g)+\Gamma(g)^2\) to values. The assumed decay consequently gives, for \(Y=(u,X)\), \[ |D^2Y|\le C r^{-1-q}|DY|+C r^{-2-q}|Y|. \tag{99}\]

Applying Lemma 32 to Equation (99) gives \[(u,X)=(u_\infty,X_\infty)+O_2(r^{-q_0}).\] Its hypothesis \(q>1/2\) includes the slow decay permitted when \(n=3\). Equation (94) gives \(W\to1\), so taking limits in \(W=u^2-|X|_g^2\) proves Equation (95).

We next justify the global coordinate assertion on a tail. The spatial projection \(\pi\) of the slice to \(y\) is a local diffeomorphism: a nonzero vector in its differential kernel would be vertical and therefore timelike, whereas every nonzero tangent vector of the slice is spacelike. Choose a target radius \(s_0\) so large that the compact complement of the original coordinate tail, including its inner coordinate sphere, projects inside \(\{|y|<s_0\}\). The map \[\pi:\pi^{-1}(\{|y|>s_0\})\longrightarrow\{|y|>s_0\}\] is proper. Indeed, the preimage of a compact target set cannot escape through the intrinsic end by Equation (94), and cannot approach the excluded compact core by the choice of \(s_0\). Its image is open by the local inverse theorem and closed by properness. It is nonempty and the target is connected, so the image is the whole tail. A proper local diffeomorphism is a covering: its fiber over a point is compact and discrete, hence finite; finitely many local inverse neighborhoods can be shrunk simultaneously, and properness excludes any additional preimages over that neighborhood.

Each component of this covering is one sheet, since \(\R^n\setminus\overline B_{s_0}\) is simply connected for \(n\ge3\). There is only one such component. To check this last point, the complement of the preimage tail is compact by the same escape condition. Each sheet is noncompact, and points on it with \(|y|\to\infty\) leave every intrinsic compact set. Two sheets would therefore give two distinct ends after removing that compact complement. This contradicts the stipulated single end. We have proved that \(y\) is a global coordinate system on this tail, and that the static time \(b\) there is a single-valued function of \(y\).

Let \(\overline g=\pi^*h_0\). Restricting \(\xi^\flat\) to the slice gives \(X_g^\flat=-W\,db\), whence \[ db=-\frac{X_g^\flat}{W},\qquad \overline g=g+\frac{X_g^\flat\otimes X_g^\flat}{W}. \tag{100}\] By Equation (95), \(\overline g\) tends to the positive definite constant matrix \(\delta+X_\infty^\flat\otimes X_\infty^\flat\), with \(O_2(r^{-q_0})\) error. Since \(h_0\) tends to \(\delta\), the identity \(\overline g=(D_xy)^T h_0(y)D_xy\) bounds both \(D_xy\) and its inverse. Integrating \(D_xy\) along intrinsic radial paths yields \(|y|\le Cr\). Conversely, integrate the inverse derivative along target radial paths starting at a fixed target sphere. Its preimage is compact, giving \(r\le C|y|\). Thus the two radii are comparable.

The connection transformation law is now available in genuine coordinate charts: \[ \partial_i\partial_jy^a =\Gamma(\overline g)^\ell_{ij}\partial_\ell y^a -\Gamma(h_0)^a_{bc}(y)\partial_i y^b\partial_j y^c. \tag{101}\] Here \(\Gamma(\overline g)=O_1(r^{-1-q_0})\) and \(\Gamma(h_0)=O_1(s^{-1-p})\). Because \(q_0<p\), bounded first derivatives and comparable radii give \(D_x^2y=O_1(r^{-1-q_0})\). Radial integration, followed by comparison on coordinate spheres, therefore gives a single invertible constant matrix \(B_0\) with \[D_xy=B_0+O_1(r^{-q_0}).\] Equation (100) similarly gives \(db=\beta+O_1(r^{-q_0})\), where \(\beta=-X_\infty^\flat\). Taking limits in the induced metric shows \[ B_0^TB_0-\beta\otimes\beta=I. \tag{102}\] Thus the linear map \(v\mapsto(\beta(v),B_0v)\) is an isometric embedding of Euclidean \(\R^n\) as a spacelike Minkowski hyperplane. A time-orientation-preserving Lorentz transformation takes this hyperplane to \(T=0\). In the resulting coordinates \((T,z)\), \[D_xT=O_1(r^{-q_0}),\qquad D_xz=Q+O_1(r^{-q_0}),\qquad Q^TQ=I.\] Integration gives \(T=o(r)\) and \(z=Qx+o(r)\). Choose a fixed large intrinsic sphere \(r=R_1\) beyond which these coordinates are defined and \(D_xz\) is invertible, and then choose \(Z_0>\max_{r=R_1}|z|\). The map \[z:\{r>R_1,\ |z(x)|>Z_0\}\longrightarrow\{|z|>Z_0\}\] is proper by \(z=Qx+o(r)\) and the exclusion of that inner sphere. The preceding covering argument makes it a single coordinate chart. This use of boosted coordinates is confined to the tail; it requires no extension of the static time to the compact horizon boundary. Rotating \(x\) by \(Q\) and using the chain rule proves Equation (96). ◻

Proof. Expanding Equation (93), the leading perturbation of Minkowski space is \(2M|y|^{-p}(db^2+p^{-1}|dy|^2)\). On \(T=0\) put \[\rho^2=\gamma^2(x^1)^2+\sum_{a=2}^n(x^a)^2,\qquad f=\rho^{-p}, \qquad A=\gamma^2(v^2+p^{-1}),\quad B=p^{-1},\quad C=kp^{-1}\gamma^2v.\] The nonzero relevant leading components of the spacetime perturbation \(\mathsf H\) are \(\mathsf H_{11}=2MAf\), \(\mathsf H_{aa}=2MBf\), and \(\mathsf H_{01}=2MCf\). Moreover \[\partial_1f=-p\gamma^2x^1\rho^{-n},\qquad \partial_af=-px^a\rho^{-n},\qquad \partial_Tf=-p\gamma^2vx^1\rho^{-n}.\] Thus the energy flux vector \(Q_i=\partial_j\mathsf H_{ij} -\partial_i\mathsf H_{jj}\) satisfies \[Q_1=-2MkB\partial_1f=2Mk\gamma^2x^1\rho^{-n},\qquad Q_a=-2M(A+pB)\partial_af=2Mp(A+1)x^a\rho^{-n}.\] Since \(p(A+1)=k\gamma^2\), these expressions give the pointwise identity \[ Q_i=2Mk\gamma^2x_i\rho^{-n}. \tag{105}\]

The chosen sign of the second fundamental form gives, to first order, \[K_{ij}=\tfrac12\bigl(\partial_T\mathsf H_{ij} -\partial_i\mathsf H_{0j} -\partial_j\mathsf H_{0i}\bigr).\] Direct substitution gives \[\begin{align*} K_{11}&=M\gamma^4v(2p+1-pv^2)x^1\rho^{-n},& K_{aa}&=-M\gamma^2vx^1\rho^{-n},\\ K_{1a}&=Mk\gamma^2vx^a\rho^{-n},& K_{ab}&=0\quad(a\ne b). \end{align*}\] Their Euclidean trace is \(M\gamma^4v(p+v^2)x^1\rho^{-n}\). Consequently, for the leading Euclidean trace reversal \(\pi_{ij}=K_{ij}-(\mathop{\mathrm{tr}}_\delta K)\delta_{ij}\), \[ \pi_{1j}=Mk\gamma^2v x_j\rho^{-n},\qquad \pi_{a1}=Mk\gamma^2vx^a\rho^{-n},\qquad \pi_{ab}=-Mk\gamma^4vx^1\rho^{-n}\delta_{ab}. \tag{106}\] Using the exact normal and the exact metric in the trace reversal changes these formulas by \(O_d(r^{-2p-1})\). The energy remainder has first derivatives of this same order. Their sphere integrals are \(O(r^{-p})\), and therefore vanish for every \(n\ge3\).

The ellipsoid \(\{|\operatorname{diag}(\gamma,1,\ldots,1)x|<1\}\) has volume \(\omega/(n\gamma)\). Polar integration therefore gives \[ \int_{S^{n-1}}|\operatorname{diag}(\gamma,1,\ldots,1)\theta|^{-n} \,d\theta=\frac{\omega}{\gamma}. \tag{107}\] Equations (105) and (106), divided by \(2k\omega\) and \(k\omega\), give \(E=M\gamma\) and \(P_1=M\gamma v\). The integrand of a transverse momentum row is a constant times \(x^1x^a\rho^{-n}/r\), whose integral vanishes by reflection. ◻

Proof. Use the coordinates supplied by Lemma 45 and write \(R=|z|\). A constant Lorentz transformation of Equation (93) gives, on every sufficiently small fixed cone \(|T|<\varepsilon R\), \[ \mathbf G_{\alpha\beta} =\eta_{\alpha\beta}+\mathsf H_{\alpha\beta} +O_j(R^{-2p}),\qquad \mathsf H=2M|y(T,z)|^{-p} \left(db^2+\frac1p|dy|^2\right). \tag{109}\] Here \(\eta=-dT^2+|dz|^2\) and the differentials on the right are the constant linear Lorentz differentials. The small cone is chosen so that \(|y(T,z)|\) and \(R\) are comparable. All coordinate derivatives of any fixed order have their corresponding decay, and \(\mathsf H\) is homogeneous of degree \(-p\). Define the leading induced plane tensors by \[ h_{ij}(z)=\mathsf H_{ij}(0,z),\qquad L_{ij}(z)=\frac12\left( \partial_T\mathsf H_{ij} -\partial_i\mathsf H_{0j}-\partial_j\mathsf H_{0i} \right)(0,z). \tag{110}\] The exact reference plane has metric \(\delta+h+O_2(R^{-2p})\) and second form \(L+O_1(R^{-2p-1})\). Its charges were computed in Lemma 46. We will show that precisely the same terms survive in the ADM fluxes of the given hypersurface.

First consider its graph \(T=f(z)\) in these coordinates. Its tangent vectors are \(e_i=\partial_i+f_i\partial_T\), so direct restriction of Equation (109) gives \[g^{\rm gr}_{ij} =\delta_{ij}-f_if_j+\mathsf H_{ij}(f,z) +\mathsf H_{0j}(f,z)f_i+\mathsf H_{0i}(f,z)f_j +\mathsf H_{00}(f,z)f_if_j+O_1(R^{-2p}).\] In particular \[ g^{\rm gr}_{ij}=\delta_{ij}+h_{ij}+e^{\rm gr}_{ij}, \qquad |e^{\rm gr}|+R|De^{\rm gr}| \le C\left( |f|R^{-p-1}+R^{-2q_0}+R^{-p-q_0}+R^{-2p} \right). \tag{111}\] For example, the evaluation difference \(\mathsf H_{ij}(f,z)-\mathsf H_{ij}(0,z)\) has size \(O(|f|R^{-p-1})\); its derivative also has the term \(O(|Df|R^{-p-1})\), included in the displayed bound. Since \(f=o(R)\) and \(2q_0>p\), this proves \(e^{\rm gr}=o_1(R^{-p})\).

For completeness, the normal formula supplies the needed second-form estimate with its sign. In flat spacetime the future normal covector of the graph is \[\mathbf n^\flat =\frac{-dT+df}{\sqrt{1-|Df|^2}}, \qquad K^{\rm flat}_{ij} =\frac{f_{ij}}{\sqrt{1-|Df|^2}}.\] In the perturbed metric, compute instead from \(K_{ij}=-\mathbf G(\mathbf n,\boldsymbol\nabla_{e_i}e_j)\). The coefficient of \(f_{ij}\) changes by \(O(|Df|^2+R^{-p})\). The connection term at \(T=0\), with zero slope, is exactly \(L\) to first order. Evaluating it at \(T=f\) costs \(O(|f|R^{-p-2})\), inserting the graph slopes costs \(O(R^{-p-1}|Df|)\), and quadratic metric corrections cost \(O(R^{-2p-1})\). It follows that \[ \begin{split} K^{\rm gr}_{ij}&=L_{ij}+f_{ij}+e^{K}_{ij},\\ |e^K|&\le C\left( |f|R^{-p-2}+R^{-p-q_0-1} +R^{-3q_0-1}+R^{-2p-1}\right) =o(R^{-p-1}). \end{split} \tag{112}\] The \(R^{-3q_0-1}\) term is the flat nonlinear error \(O(|Df|^2|D^2f|)\). Thus even when \(f_{ij}\) decays more slowly than the plane second form, its only nonnegligible contribution is an ordinary coordinate Hessian.

It remains to evaluate fluxes in the original coordinates. Set \(z=x+\psi(x)\) as in Equation (96) and \(F_0(x)=f(z(x))\). Pulling back the Euclidean metric gives the exact expression \[(D_xz)^TD_xz =I+D\psi+(D\psi)^T+(D\psi)^TD\psi.\] The last term is \(O_1(r^{-2q_0})\). Homogeneity and \(\psi=o(r)\) give \[(z^*h)_{ij}=h_{ij}(x)+o_1(r^{-p}), \qquad (z^*L)_{ij}=L_{ij}(x)+o(r^{-p-1}).\] Indeed, evaluation changes contribute \(\psi\,Dh\) and \(\psi\,DL\); tensor pullback contributes \(D\psi\,h\) and \(D\psi\,L\). Differentiating the metric pullback also produces \(D^2\psi\,h\), of size \(O(r^{-p-q_0-1})\). The chain rule gives a useful exact identity for the Hessian: \[ z^*(D_z^2f)_{ij} =\partial_i\partial_jF_0 -(\partial_af)(z(x))\,\partial_i\partial_j\psi^a. \tag{113}\] Its final term is \(O(r^{-2q_0-1})=o(r^{-p-1})\). Combining Equations (111)– (113) therefore gives \[\begin{align*} g_{ij} &=\delta_{ij}+h_{ij}(x) +\partial_i\psi_j+\partial_j\psi_i+e_{ij}, &e&=o_1(r^{-p}), \tag{114}\\ K_{ij} &=L_{ij}(x)+\partial_i\partial_jF_0+\widetilde e_{ij}, &\widetilde e&=o(r^{-p-1}). \tag{115}\end{align*}\] For the momentum trace reversal, the original metric satisfies \(g-\delta=O(r^{-q_0})\) and \(K=O(r^{-q_0-1})\). Consequently \[(\mathop{\mathrm{tr}}_gK)g_{ij}-(\mathop{\mathrm{tr}}_\delta K)\delta_{ij} =O(r^{-2q_0-1})=o(r^{-p-1}).\] Thus \[ \begin{split} K_{ij}-(\mathop{\mathrm{tr}}_gK)g_{ij} ={}&L_{ij}-(\mathop{\mathrm{tr}}_\delta L)\delta_{ij}\\ &+\partial_i\partial_jF_0-(\Delta F_0)\delta_{ij} +o(r^{-p-1}). \end{split} \tag{116}\]

Both apparent extra fluxes vanish exactly. The energy flux vector of the symmetric-gradient term in Equation (114) is \[Q_i(\psi)=\Delta\psi_i-\partial_i\mathop{\mathrm{div}}\psi, \qquad \partial_iQ_i(\psi)=0.\] Choose a smooth cutoff that is one outside a fixed large sphere and zero on a smaller ball, and use it to extend \(\psi\) smoothly over \(\R^n\). The same formula for the extension is divergence free on the entire ball bounded by every sufficiently large coordinate sphere. The divergence theorem gives \[ \int_{S_r}Q_i(\psi)\,\nu_\delta^i\,dA_\delta=0. \tag{117}\] Similarly, each row of the trace-reversed Hessian obeys \[\partial_j\bigl(\partial_i\partial_jF_0 -(\Delta F_0)\delta_{ij}\bigr)=0.\] Smoothly extending \(F_0\) by the same procedure proves \[ \int_{S_r}\bigl(\partial_i\partial_jF_0 -(\Delta F_0)\delta_{ij}\bigr) \nu_\delta^j\,dA_\delta=0 \quad\hbox{for every }i. \tag{118}\] These are identities for the extended potentials; no decay assumption on an interior extension, and no subtraction of an unknown inner boundary flux, is involved.

Finally, a metric error \(o_1(r^{-p})\) contributes \(o(1)\) to the energy flux because \(|S_r|=O(r^{p+1})\). A second-form error \(o(r^{-p-1})\) contributes \(o(1)\) to momentum. More explicitly the largest product errors above have flux size \(O(r^{p-2q_0})\); the remaining ambient products have size \(O(r^{-q_0})\) or \(O(r^{-p})\), and the evaluation errors have size \(O(|f|/r+|\psi|/r)=o(1)\). All tend to zero. Equations (117) and (118) leave precisely the plane fluxes.

Lemma 46 computes these, with the present second-form convention, as \(E=\gamma M\), \(P_1=\gamma vM\), and \(P_a=0\) for \(a>1\) under \[b=\gamma(T+vz^1),\qquad y^1=\gamma(z^1+vT),\qquad y^a=z^a.\] On that plane \(\xi=\gamma\partial_T-\gamma v\partial_{z^1}\), so its limiting lapse and shift are \((\gamma,-\gamma v,0,\ldots,0)\). Constant spatial rotations give \(E=Mu_\infty\) and \(P=-MX_\infty\) in the original orthonormal end frame. Equation (95) proves the final identity in Equation (108). ◻

Proof. In horizon-regular null coordinates, restriction of the spacetime metric to the future horizon is the degenerate tensor \(r_h^2\sigma\) pulled back from its spherical space of generators. For example, away from the bifurcation sphere the ingoing form is \[\mathbf G=-F(\mathfrak r)\,dv^2 +2\,dv\,d\mathfrak r+\mathfrak r^2\sigma, \qquad F(\mathfrak r)=1-\frac{2M}{\mathfrak r^p},\] and both of its first two terms restrict to zero at \(\mathfrak r=r_h\). Kruskal coordinates give the same restriction at the bifurcation sphere. A tangent vector in the kernel of projection onto the generator sphere would be null, so spacelikeness makes that projection a local diffeomorphism. Fullness makes it bijective, and hence it is a diffeomorphism. Pulling back the displayed degenerate tensor to the section therefore gives a metric isometric to \(r_h^2\sigma\), regardless of the location of the section along individual generators. This proves Equation (119).

The null second fundamental form along the future horizon generator vanishes as well: the tensor \(r_h^2\sigma\) is constant along each generator. For a spacelike hypersurface ending on the section from the exterior, the future outgoing null normal \(\mathbf n+\nu\) is a positive multiple of this generator, where \(\nu\) points into the hypersurface. Scaling the generator multiplies its null second fundamental form by the same positive factor. Its trace is therefore zero. By the chosen convention that trace is \(H+\mathop{\mathrm{tr}}_S K=\theta_+\). The same argument uses a regular future generator in Kruskal coordinates at the bifurcation sphere, where the Killing field itself vanishes. ◻

Proof. Fix \(r_h<r_1<r_2\) and take a smooth function \(\chi\in C_c^\infty((r_1,r_2))\) with a nondegenerate interior critical point. Put \(b=f(\mathfrak r)=\varepsilon\chi(\mathfrak r)\) and choose \(\varepsilon\) so that \[ \sup_{\mathfrak r>r_h}|F(\mathfrak r)f'(\mathfrak r)|<1. \tag{120}\] The zero choice gives the static slice. The induced metric of any of these graphs is \[ g_f=A_f(\mathfrak r)\,d\mathfrak r^2+\mathfrak r^2\sigma, \qquad A_f=F^{-1}-F(f')^2 =F^{-1}\bigl(1-F^2(f')^2\bigr)>0. \tag{121}\] Thus the graph is spacelike. Close to the horizon it is exactly the static slice. To check regularity there without relying on the singular areal coordinate, use static proper distance \(a\ge0\), for which \(d\mathfrak r/da=\sqrt F\). At \(a=0\), \[\mathfrak r(a) =r_h+\frac{F'(r_h)}4a^2+O(a^4),\qquad F'(r_h)=\frac p{r_h}>0,\] and the metric is \(da^2+\mathfrak r(a)^2\sigma\). In Kruskal coordinates the static slice and its future normal extend smoothly to the bifurcation sphere. To see this, let the tortoise coordinate \(r_*\) satisfy \(dr_*/d\mathfrak r=F^{-1}\). The two null coordinates on the slice are opposite constant multiples of \(e^{\kappa r_*}\), where \(\kappa=F'(r_h)/2\) and \(e^{\kappa r_*}=a\) times a smooth positive even function of \(a\). Their derivatives with respect to \(a\) are nonzero at \(a=0\). Our perturbation is supported away from this collar, so its embedding and normal have the same regularity. The boundary is connected, \(K=0\) there, and its mean curvature is \(k\mathfrak r'(0)/r_h=0\).

The graph is unchanged at infinity as well. Radial length to infinity is infinite, and every bounded radial region is compact with the boundary included; hence \((\Omega,g_f)\) is complete as a metric space with that boundary. The induced data have \(g_f-\delta=O_j(s^{-p})\) for every fixed \(j\) in the static isotropic end and \(K=0\) there. They therefore satisfy all the stated finite derivative decay bounds for any \(p/2<q<p\).

The ambient metric is vacuum in every required dimension. This can also be checked directly from its areal form: the only potentially nonzero Ricci components are multiples of \[F''+\frac{k}{\mathfrak r}F',\qquad (k-1)(1-F)-\mathfrak rF',\] and both vanish for \(F=1-2M\mathfrak r^{-p}\) and \(p=k-1\). Gauss–Codazzi therefore gives \(\mu=J=0\) on each graph, so the dominant energy and integrability assumptions hold. Its end charges are \((E,P)=(M,0)\) by Lemma 46 with \(v=0\). To verify non-time-symmetry when \(\varepsilon\ne0\), take the chosen critical point \(\mathfrak r_*\) of \(f\). There \(f'=0\) and the future normal is \(F^{-1/2}\partial_b\). The defining second-form formula then gives \[K_{\mathfrak r\mathfrak r}(\mathfrak r_*) =\sqrt{F(\mathfrak r_*)}\,f''(\mathfrak r_*)\ne0.\]

We prove the required area inequality for every full cut, including disconnected cuts and all portions coinciding with the boundary. Let \(\pi_\sigma:\Omega\to S^k\) denote angular projection and set \[\zeta=r_h^k\pi_\sigma^*(dA_\sigma).\] This is a smooth closed \(k\)-form on the entire exterior with boundary. Its comass in the metric Equation (121), meaning its maximum absolute value on an orthonormal \(k\)-frame, is \((r_h/\mathfrak r)^k\le1\). Indeed it vanishes on the radial direction and has that value on an oriented orthonormal angular frame.

Let \(\Gamma=\partial D\) be a full cut as in Definition 2. Choose \(R\) beyond its entire boundary and far enough that \(D\) contains all \(\mathfrak r\ge R\). The intrinsic compact manifold \(D_R=D\cap\{\mathfrak r\le R\}\) has boundary exactly \(\Gamma\sqcup\{\mathfrak r=R\}\). The second boundary is in the interior of \(D\) before truncation, so this operation introduces no corner. Orient \(\Gamma\) by the normal pointing into \(D\), and hence toward its end. Its orientation as a boundary of \(D_R\) is the opposite one. Stokes’ theorem gives \[ \int_\Gamma\zeta =\int_{\{\mathfrak r=R\}}\zeta =\omega r_h^k. \tag{122}\] The entire intrinsic boundary appears in this formula, even when a part of it lies on \(S\). Applying the comass bound to every component and then summing gives \[\mathop{\mathrm{Area}}_{g_f}(\Gamma) \ge\left|\int_\Gamma\zeta\right| =\omega r_h^k=\mathop{\mathrm{Area}}_{g_f}(S).\] Since \(S\) itself is an admissible full cut, this proves both \(A_{\min}(S)>0\) and \(A_{\min}(S)=\mathop{\mathrm{Area}}_{g_f}(S)\).

It remains to check the trapping exclusion. Write \(d_f=1-F^2(f')^2>0\). The future normal of the graph and the normal to an outward round sphere within it are respectively \[\mathbf n=\frac{F^{-1/2}}{\sqrt{d_f}} \bigl(\partial_b+F^2f'\partial_{\mathfrak r}\bigr), \qquad \nu=\frac{\sqrt F}{\sqrt{d_f}} \bigl(\partial_{\mathfrak r}+f'\partial_b\bigr).\] The round sphere has \(H=k\nu(\mathfrak r)/\mathfrak r\) and \(\mathop{\mathrm{tr}}_{S_{\mathfrak r}}K=k\mathbf n(\mathfrak r)/\mathfrak r\). Consequently \[ \theta_+(S_{\mathfrak r}) =\frac{k\sqrt F}{\mathfrak r\sqrt{d_f}}(1+Ff')>0 \qquad(\mathfrak r>r_h). \tag{123}\] The positivity follows from Equation (120); it does not require a smallness estimate for second derivatives of \(f\).

Suppose a smooth compact embedded two-sided enclosing hypersurface \(\Sigma\) in the interior had \(\theta_+\le0\) everywhere toward the end. At a maximum of \(\mathfrak r\) on \(\Sigma\), say \(\mathfrak r=\mathfrak r_0>r_h\), it is tangent to the round sphere \(S_{\mathfrak r_0}\) and lies on its inner side. The ray with \(\mathfrak r>\mathfrak r_0\) reaches infinity without meeting \(\Sigma\), so the normal toward the end at this point is the positive radial normal, even if \(\Sigma\) is disconnected. At the contact point, tracing the restriction Hessian identity gives \[0\ge\Delta_\Sigma\mathfrak r =\mathop{\mathrm{tr}}_{T\Sigma}\mathop{\mathrm{Hess}}_g\mathfrak r -H_\Sigma\,\nu(\mathfrak r).\] For the tangent round sphere the right-hand side is zero with \(H_\Sigma\) replaced by \(H_{S_{\mathfrak r_0}}\). Since \(\nu(\mathfrak r)>0\), this implies \(H_\Sigma\ge H_{S_{\mathfrak r_0}}\). The tangent spaces coincide, so their tangential traces of \(K\) are equal. Equation (123) now gives \(\theta_+(\Sigma)>0\) at the contact point, a contradiction. All rigidity-subclass hypotheses have been checked. ◻

Proof. Definition 50 supplies a smooth one-ended exterior, complete with compact boundary, and Equation (126) gives the original asymptotic geometry. There is no hypothesis concerning the constraint densities in the geometric propositions of Section 3. The parameter bounds, after shrinking the interval, make \(g_s\) uniformly comparable to \(g\). Together with the scaled first-derivative bounds, this is the last end alternative in Proposition 9; no common mean-convex tail for the whole path has to be assumed.

Explicitly, the bounded-obstacle competitor gives a uniform perimeter bound. The BV exhaustion construction in that proposition first produces a locally minimizing limit of finite volume and bounded perimeter. An end frontier point at radius \(r\) would contribute at least \(cr^2\) area in an obstacle-free coordinate ball of radius comparable to \(r\), by the uniform interior density estimate. Thus the frontier is confined, and finite volume makes its distant tail empty. It is a global minimizing bounded competitor. The same argument confines every minimizing set for every small \(s\).

Proposition 8, with \(n=3\), gives \(C^{1,1}\) regularity at contact and smooth-cut recovery. With the confinement just established, Proposition 9 gives strict perimeter convergence, the localized Reshetnyak–Spector plane-measure convergence and the derivative formula. The no-crossing argument in Proposition 10 gives the common plane. That proposition’s whole-support graphical convergence also shows that this plane is continuous on the union of the free cuts, with its relative topology. Pointwise plane convergence here uses this additional regularity step, not BV convergence alone. ◻

Proof. Adjoin an end point to \(\mathcal A\), collapsing every ray that escapes spatial compact sets to that point; if \(\mathcal A\) is compact, take an additional isolated end point. The bounded fibers make the result a compact space. Assign the value \(8a(H)\) at the added point. If \(\mathcal A\) is noncompact, the active-ray asymptotics prove continuity there. If \(\mathcal A\) is compact or empty, the added point is isolated, so continuity is automatic. Take the disjoint compact union of this space, \(S\), and \(\mathcal T\), and associate to \(H\) the continuous function whose values on the three pieces are \[ \frac{C'_H}{w},\qquad -\theta'_H,\qquad c a'_T(h)-16\pi\mathcal E'(H), \tag{163}\] respectively. Smooth dependence of the boundary expansion and Lemma 59 give the remaining continuity.

We first prove that no image of this linear map is strictly positive at every point of the compact test space. Suppose otherwise, and write \(H=aQ+H_0\). Positivity at the end point gives \(a>0\). Realize its tangent by the actual path \[ g_s=e^{4as\phi}(g+s h_0),\qquad K_s=e^{2as\phi}(K+s p_0). \tag{164}\] The metric remains positive and uniformly comparable to \(g\) for small \(|s|\). The varied data retain the required asymptotic decay with exponent \(\min\{q,1\}>1/2\), and their charge variations have finite limits. The conformal factor has its stated flux. Thus the path has differentiable ADM fluxes.

Here are the estimates which ensure nonlinear feasibility. On the end the tensors in \(H_0\) are \(O_2(r^{-1})\) and \(O_1(r^{-2})\), and their far-end Euclidean linear constraints vanish. Terms in their linear constraint variation containing a background error cost \(O(s r^{-3-q})\), while new quadratic terms cost \(O(s^2r^{-4})\). Changing the orthonormal identification has the same bound: the background momentum is \(O(r^{-2-q})\), and the metric variation is \(O(r^{-1})\). Applying the exact conformal law to the intermediate data \((g+s h_0,K+s p_0)\), and keeping the nonnegative background term with its positive scaling factor, therefore gives, uniformly over the full unit ball of test vectors, \[ e^{4as\phi}C_s =C+s\bigl(8a w+o(w)\bigr)+O(s^2w) \quad\hbox{on the end}. \tag{165}\] The \(o(w)\) is uniform as \(r\to\infty\) at fixed \(H\), and the quadratic bound is uniform for small \(s\). The gradient square in the conformal formula is \(O(r^{-4})=O(w)\) because \(\delta<1\). One may use the composition of the two natural isometries of the unit balls in deriving this estimate; it tests the same DEC. At active rays its derivative agrees with Equation (156), since the difference of two isometric identifications is tangent to the unit sphere and is annihilated by \(J\) at an active ray.

It follows from Equation (165) that the DEC holds on a fixed far end for all sufficiently small positive \(s\). On the remaining compact ball bundle, the derivative is strictly positive on the closed active set. A neighborhood of that set has the same positive derivative with a fixed smaller margin. On its compact complement the original \(C\) has a positive minimum. Taylor’s formula now gives the DEC throughout the compact region as well. The strict inequality \(-\theta'_H>0\) on compact \(S\) gives \(\theta_{+,s}<0\) for small positive \(s\). The new sources are integrable: outside a compact set their changes, after the harmless positive rescaling of the original sources, have the integrable bounds just displayed. The ADM vector stays future timelike. Hence these are data to which Proposition 51 applies.

Finally, strict positivity on the compact cut space gives \[c\min_{T\in\mathcal T}a'_T(h)-16\pi\mathcal E'(H)>0.\] By Lemma 59 and Equation (146), this says that the right derivative of \(16\pi(\mathcal E(s)-\sqrt{a(s)/(16\pi)})\) is strictly negative. The expression is zero at \(s=0\), whereas the supporting-energy inequality and Proposition 51 make it nonnegative for small positive \(s\). This is a contradiction.

The image of the linear test map is therefore disjoint from the open cone of strictly positive continuous functions. The separating form of the Hahn–Banach theorem (Rudin 1991, Theorem 3.4(a), pp. 59–60) gives a nonzero continuous linear functional, nonnegative on nonnegative functions, which annihilates the image. The Riesz representation theorem (Rudin 1987, Theorem 2.14, pp. 40–41) represents it by nonnegative finite measures on the three compact pieces. Its mass on the objective piece \(\mathcal T\) cannot be zero: otherwise its value on the image of \(Q\), which is strictly positive on both remaining pieces including their end point, would be positive. Normalize this objective mass to one. On the finite active rays divide the corresponding measure by \(w\) and call the result \(\Lambda\). It is locally finite; the original finite measure makes its pairing with \(C'_H\) finite. Put the end-point mass, multiplied by eight, into \(z\). Rearranging the annihilation identity gives Equation (162) with the stated signs. ◻

Proof. Apply Lemma 14 on a relatively compact open neighborhood of \(\operatorname{supp}s\) in the open exterior. To match its geometric density convention, put \[\widehat\mu=8\pi\mu,\qquad \widehat J=8\pi J.\] Equation (124) gives \(2\widehat\mu=R_g+(\mathop{\mathrm{tr}}_gK)^2-|K|_g^2\) and \(\widehat J=\mathop{\mathrm{div}}_g(K-(\mathop{\mathrm{tr}}_gK)g)\), and the dominant energy condition gives \(\widehat\mu\ge|\widehat J|_g\). The data are smooth on this neighborhood, and the future-normal convention (130) is the one used in that lemma.

The ray constraints agree exactly: \[2\bigl(\widehat\mu+\widehat J(v)\bigr) =16\pi\bigl(\mu+J(v)\bigr)=C(x,v).\] This identity holds for the varied data as well, and both derivatives use \(v'_0=-\tfrac12h^\sharp v\). Thus the active sets and the derivatives \(\mathfrak C'(h,p;v)\) and \(C'_H(x,v)\) coincide. Taking the parameter \(\delta=8\pi\epsilon\) in Lemma 14 gives the claimed variations and uniform convergence. Their compact support in the open exterior makes them admissible in the variation space and makes every boundary and ADM variation vanish. This density conversion does not change the ADM energy, momentum, or mass. ◻

Proof. Use the variations of Lemma 61 in Equation (162). Their ADM and boundary variations vanish. Their supports lie in a fixed compact set, on which \(\Lambda\) is finite, so the uniform constraint-derivative limit gives \[ \int_{\mathcal T}\int_T s\mathop{\mathrm{tr}}_TK\,\mathrm dA_g\,\mathrm d\varpi(T)=0 \quad\hbox{for every }s\in C_c^\infty(\operatorname{Int}M_{\mathrm{ext}}). \tag{169}\] Here the metric component is \(2sK\), and the area coefficient \(c=1/(2m)\) is positive.

We apply Proposition 15 to this identity. Lemma 59 and its accompanying common-plane argument supply the required measurable compact family, smooth minimal free parts, and continuous common tangent plane. Its contact graphs are \(C^{1,1}\), hence \(C^{1,\alpha}\cap W^{2,2}\) locally. Each minimizing filled set has no bounded complementary pocket: filling such a pocket would remove positive perimeter. Its frontier therefore has a connected end side and is a full enclosing cut once smooth. Every frontier counts all its components, including coincidence with \(S\), and has area \(A=|S|_g\).

For Theorem 53, the finite marginal boundary and Definition 52 match case (b) of the proposition exactly. For Theorem 54, \(S\) is connected and its wholly-interior comparison condition matches case (a). Thus \(\varpi\) is concentrated on \(S\) in either case. The comparison classes remain distinct; the connected case uses no partial-coincidence outermostness assumption. ◻

Proof. For a compact variation \(p=\dot K\) with \(h=0\), direct differentiation and integration of the momentum divergence give the coefficient \[2\bigl[u(\tau g-K)-\mathop{\mathrm{sym}}\nabla X+(\mathop{\mathrm{div}}X)g\bigr]\] against \(p\). It is zero distributionally. Taking its trace gives \(\mathop{\mathrm{div}}X=-u\tau\), and substituting back proves Equation (148).

Next use the compact simultaneous conformal variation \((h,p)=(4\psi g,2\psi K)\). Its background scaling term is \(-64\pi\psi(u\mu+J(X))\), which vanishes by Equation (167). The remaining pairing is \[-8\langle u,\Delta\psi\rangle +8\langle X,K(\nabla\psi,\cdot)\rangle=0.\] This is Equation (149) in distributions. Taking a divergence of the symmetric-gradient equation and using \(\mathop{\mathrm{div}}X=-u\tau\) also gives \[ \begin{split} \Delta X_j+\mathop{\mathrm{Ric}}_{jk}X^k &=(\tau\delta_j{}^i-2K_j{}^i)\nabla_i u +u(\nabla_j\tau-2\nabla^iK_{ij}). \end{split} \tag{170}\] Together with the lapse equation this is a second-order elliptic system with scalar Laplacian principal part and smooth lower-order couplings. Locally the measure moments belong to some negative Sobolev space. The local Laplace estimate (Melrose n.d., chap. 4, Section 3, Theorem 3.1, p. 127) applied to this system improves their Sobolev order by one, since the right sides have at most one derivative of the unknowns. Iterating with nested compact cutoffs puts them in every local Sobolev space. Sobolev embedding then gives smoothness. The distributional inequalities and complementarity now hold pointwise in the interior.

We give the full metric calculation, including the dependence of the momentum test vectors on the metric. Put \(P=K-\tau g\). For compactly supported variations the constraint pairing is the first variation of \[ I(g,K)=\int \left[u(R+\tau^2-|K|^2)+2X^i\nabla^jP_{ij}\right]\,\mathrm dV_g, \tag{171}\] together with the frame correction \[ -8\pi\int h(X,J^\sharp)\,\mathrm dV_g. \tag{172}\] The integration may be localized to a fixed region containing the variation. Changing its volume form instead of retaining the original fixed measure adds one half of its background integrand times \(\mathop{\mathrm{tr}}h\). That integrand is \(16\pi(u\mu+J(X))=0\), so this replacement does not change the pairing. Equation (172) follows directly from Equation (156) and has the displayed negative sign.

Integrating the shift term once by parts makes it \(-\int P^{ij}(\mathcal L_Xg)_{ij}\,\mathrm dV_g\) up to a boundary term unaffected by these variations. In taking a pure metric variation, \(u\), contravariant \(X\), and covariant \(K\) are fixed. The elementary variation formulas used are \[\begin{align*} \dot R&=-\mathop{\mathrm{Ric}}^{ij}h_{ij} +\nabla^i\nabla^jh_{ij}-\Delta(\mathop{\mathrm{tr}}h),\\ \dot\tau&=-K^{ij}h_{ij},\qquad \bigl(|K|^2\bigr)^{\boldsymbol\cdot} =-2(K^2)^{ij}h_{ij},\qquad (\,\mathrm dV)^{\boldsymbol\cdot}=\tfrac12\mathop{\mathrm{tr}}h\,\mathrm dV,\\ (P^{ij})^{\boldsymbol\cdot} &=-h^i{}_aK^{aj}-h^j{}_aK^{ia} +(K^{ab}h_{ab})g^{ij}+\tau h^{ij},\\ (\mathcal L_Xg)^{\boldsymbol\cdot}&=\mathcal L_Xh. \end{align*}\] For example, integrating the last term by parts and combining it with the variation of \(P^{ij}\) gives the following coefficient from the entire shift integral: \[(\mathcal L_XK)_{ij}-(\mathop{\mathrm{div}}X)K_{ij} -\bigl[X(\tau)+K^{ab}\nabla_aX_b\bigr]g_{ij}.\] The scalar-curvature terms contribute \(\mathop{\mathrm{Hess}}u-(\Delta u)g-u\mathop{\mathrm{Ric}}\). Consequently the vanishing metric coefficient is the explicit equation \[ \begin{split} 0={}&\nabla_i\nabla_j u-(\Delta u)g_{ij}-u\mathop{\mathrm{Ric}}_{ij} +2u(K^2-\tau K)_{ij}\\ &+\frac u2(R+\tau^2-|K|^2)g_{ij} +(\mathcal L_XK)_{ij}-(\mathop{\mathrm{div}}X)K_{ij}\\ &-\bigl[X(\tau)+K^{ab}\nabla_aX_b\bigr]g_{ij} -8\pi X_{(i}J_{j)}. \end{split} \tag{173}\] This calculation can equally be read distributionally, since each coefficient is linear in the moments and their derivatives.

For completeness, its simplification uses no vacuum assumption. Equation (148) gives \[\mathop{\mathrm{div}}X=-u\tau,\quad K^{ab}\nabla_aX_b=-u|K|^2,\quad \mathop{\mathrm{tr}}(\mathcal L_XK)=X(\tau)-2u|K|^2.\] Taking the trace of Equation (173) and using \(16\pi\mu=R+\tau^2-|K|^2\) and \(J(X)=-u\mu\) yields \[ \Delta u+X(\tau)=\frac u2(R+\tau^2+|K|^2). \tag{174}\] Substitution of this identity into Equation (173) gives exactly Equation (150).

These equations have the asserted finite-type property. The right side of Equation (150) is linear in \(u,X,\nabla X\), with smooth background coefficients, since \[(\mathcal L_XK)_{ij} =X^k\nabla_kK_{ij} +K_{kj}\nabla_iX^k+K_{ik}\nabla_jX^k.\] The second derivatives of \(X\) are also fixed by first jets. To make this explicit, put \(B_{ij}=\nabla_{(i}X_{j)}=-uK_{ij}\). Commuting covariant derivatives gives \[ \begin{split} \nabla_i\nabla_jX_k ={}&\nabla_i B_{jk}+\nabla_j B_{ik}-\nabla_k B_{ij}\\ &+\tfrac12\left( [\nabla_i,\nabla_j]X_k -[\nabla_i,\nabla_k]X_j -[\nabla_j,\nabla_k]X_i\right). \end{split} \tag{175}\] The commutators are curvature times \(X\), and differentiating \(B=-uK\) uses only \(u,\nabla u\) and the background first derivative of \(K\). Thus every derivative of \((u,X,\nabla u,\nabla X)\) is a homogeneous linear expression in that same first jet. This proves the final assertion. ◻

Proof. In a smooth normal collar \((s,y)\) of \(S\), all background coefficients in the first-jet system of Lemma 63 are smooth up to \(s=0\). Along a normal line the system is an ordinary linear differential equation for the full first jet, with bounded coefficients. Solve it down to \(s=0\) from a fixed interior collar surface. Smooth dependence on the initial data and on \(y\) supplies a smooth extension, which agrees with the original solution by uniqueness along each normal line. In particular all its one-sided derivatives have the values supplied by the system. The same system also proves that an interior first jet which vanishes at one point vanishes everywhere on the connected interior, by continuation along piecewise smooth paths.

We next establish the asymptotic form without having prescribed growth to the measures. Let \(Y\) denote all coordinate components of \((u,X)\). The original \(O_2(r^{-q})\), \(O_1(r^{-1-q})\) bounds, the constraints, and Equations (150) and (175) give \[ |\partial^2Y| \le C r^{-1-q'}|\partial Y| +C r^{-2-q'}|Y|, \qquad \tfrac12<q'<\min\{q,1\}. \tag{176}\] Only the original derivative bounds occur here: the curvature and \(\partial K,J\) are \(O(r^{-2-q'})\), and the coefficients of first jets are \(O(r^{-1-q'})\).

Fix \(q'\) with \(1/2<q'<\min\{q,1\}\), and choose \(q'<\widehat q<\min\{q,1\}\). Equation (176) also holds with \(\widehat q\) in place of \(q'\), and \(g-\delta=O_2(r^{-\widehat q})\). The pair \(Y=(u,X)\) is smooth and satisfies \(u\geq|X|_g\). Lemma 32, with input exponent \(\widehat q\) and output exponent \(q'\), therefore gives one constant \(Y_\infty\) such that \[ Y=Y_\infty+O(r^{-q'}),\qquad \partial Y=O(r^{-1-q'}),\qquad \partial^2Y=O(r^{-2-q'}). \tag{177}\]

At this point the original measure moments could still have a component supported on \(S\). Subtract the integration by parts of the smooth interior fields from the compact identity (162). Test with \(p=0\) and a metric variation whose value and full first jet vanish on \(S\). The area and expansion variations and all the smooth integrated boundary terms then vanish. The remaining scalar-curvature variation on \(S\) contains \[-\partial_\nu^2(\mathop{\mathrm{tr}}_S h).\] This is an arbitrary smooth function on \(S\): in collar coordinates one may take the tangential trace to be a prescribed multiple of \(s^2\) times a cutoff. All momentum and frame terms involve only the value or first jet of \(h\) and hence vanish on \(S\). It follows that the scalar moment has no boundary measure. Domination \(|X|\le u\) as measures eliminates a vector boundary measure also.

We can therefore integrate the smooth adjoint by parts all the way to \(S\) and to a large coordinate sphere. Use a prototype from Equation (158), with its cutoff chosen to vanish near \(S\). Its area and expansion variations vanish, and \(a(H)=0\). Its constraint pairing is integrable by Equation (160) and Equation (177). Interior terms vanish by the adjoint equations. If the constants in the latter equation are \((u_\infty,X_\infty)\), its outer boundary term is \[ \begin{split} \int_{S_R}\bigl[&u_\infty(\partial_jh_{ij}-\partial_i h_{jj}) +2X_\infty^j(p_{ij}-(\mathop{\mathrm{tr}}_\delta p)\delta_{ij})\bigr] n^i\,\mathrm dA_\delta+o(1). \end{split} \tag{178}\] For clarity, every omitted term vanishes under the stated decay: the derivatives of \(u,X\) are \(O(r^{-1-q'})\) and multiply \(h=O(r^{-1})\); their nonconstant values multiply \(\partial h,p=O(r^{-2})\); and the metric-variation momentum terms contain \(Kh=O(r^{-2-q'})\). Multiplication by sphere area still leaves \(O(r^{-q'})\). Thus the limit of Equation (178) is \(16\pi(u_\infty E'+X_\infty^iP_i')\).

The scalar prototype has \(E'=1/2\) and \(P'=0\); for the momentum prototype in direction \(p\), \(E'=0\) and \(P'=2p/3\). The latter follows by integrating \(B(p)_{ij}n^j=r^{-2}(p_i+(p\cdot n)n_i)\). Equation (162) for these four independent fluxes consequently gives \[u_\infty=b^0,\qquad X_\infty^i=b^i.\] This proves Equation (147) and in particular shows that the adjoint is nontrivial.

If \(u\) vanished at an interior point, smooth nonnegativity would give \(\nabla u=0\) there. The inequality \(|X|\le u\) gives \(X=0\); since \(u\) has zero differential, it also forces \(\nabla X=0\) at that point. The whole first jet would be zero. Its continuation property would make \((u,X)\) identically zero, contradicting \(u_\infty=b^0>0\). Thus \(u>0\) in the interior. ◻

Proof. The hypotheses of Proposition 34 were verified above. It gives the connected positive set, vanishing twist, static equations, lapse range, boundary attachments, and completeness with the entire boundary included. Its null-set and horizon cutoff arguments apply to all the finitely many components; they impose no common choice of boundary lapse alternative. In particular, the proposition does not discard any possible complete end approaching an interior zero of \(N\). Closedness of \(A\) supplies local static time coordinates, but no global primitive is asserted at this stage.

Take two copies of the attached base and identify corresponding points of every \(S_i\) by its collar reflection. The result is connected because the base is connected and its boundary is nonempty. Giving the second copy the opposite orientation makes the double orientable. The reflected second jets agree, so the scalar and harmonic equations hold continuously across each join.

For completeness, fold the double onto the attached base. The fold is distance nonincreasing, so a Cauchy sequence projects to a Cauchy sequence in that complete base. A complete locally compact length space is proper; hence the projected sequence is contained in a compact base set. Its inverse image in the double is the quotient of two compact copies and is compact. The original sequence has a convergent subsequence and therefore converges. This proves completeness without a classification or simple-connectivity assumption. ◻

Proof. The three-dimensional conformal formulas applied to Equation (191) give \[ \mathop{\mathrm{Ric}}_\gamma=2\,\mathrm dU\otimes\,\mathrm dU,\qquad \Delta_\gamma U=0. \tag{197}\] Initially \(\gamma-\delta,U=O_2(r^{-q_0})\). We give the coordinate construction with this derivative count.

Move to a sufficiently distant end and extend \(\gamma\) to a metric on \(\R^3\), equal to \(\delta\) inside a large ball, using a fixed-ratio cutoff annulus. In the resulting coordinates write \[\Delta_\gamma=(\delta^{ab}+a^{ab})\partial_{ab}+c^a\partial_a .\] The coefficients satisfy \(a=O_2(r^{-q_0})\), \(c=O_1(r^{-1-q_0})\), and their scaled Hölder norms through the orders needed here are small after increasing the cutoff radius. The bounded second derivatives in the hypothesis give scaled Lipschitz bounds on first derivatives, so they suffice for any fixed Hölder exponent less than one at this stage.

Put \(s_0=1-q_0\in(0,1/2)\). For a source with weighted scaled \(C^{0,\alpha}\) size \(O(r^{s_0-2})\), the Euclidean inverse normalized to vanish at zero is \[ Pf(x)=-\frac1{4\pi}\int_{\R^3} \left(\frac1{|x-y|}-\frac1{|y|}\right)f(y)\,\mathrm dy. \tag{198}\] It maps that weighted norm boundedly to scaled \(C^{2,\alpha}\) size \(O(r^{s_0})\). To verify the zeroth-order bound, split at \(|y|=2|x|\). On the far part, the kernel difference is bounded by \(C|x||y|^{-2}\), and its integral is \(O(|x|^{s_0})\) because \(s_0<1\). On the near part the separate kernels have the same bound because \(s_0>0\). Interior Poisson estimates on rescaled annuli give the derivative and Hölder bounds. The ordinary local estimates also apply on the fixed inner ball.

The equations \[v^i=P\bigl(-c^i-a^{ab}\partial_{ab}v^i-c^a\partial_av^i\bigr)\] are contractions in that weighted space. Indeed the operator norm of the last two terms is bounded by a constant times the small scaled norms of \(a\) and \(rc\). They produce harmonic coordinates \(x^i+v^i\) with \(v=O_{C^{2,\alpha}}(r^{1-q_0})\) and \(Dv=O(r^{-q_0})\). These functions are coordinates sufficiently far out.

There is a derivative issue in changing metric coordinates which must be addressed before using a classical Ricci equation. The original assumptions bound \(a,c\), respectively, in scaled \(W^{2,\infty}\) and \(W^{1,\infty}\). Differentiate the equation for \(v\) once and apply local \(W^{2,p}\) estimates to \(Dv\) on rescaled annuli. All differentiated coefficients and forcing are controlled by these bounds and the preceding \(C^{2,\alpha}\) estimate. For every finite \(p\) this gives \[v=O_{W^{3,p}}(r^{1-q_0}).\] Consequently the transformed \(\gamma-\delta,U\) have scaled \(W^{2,p}\) size \(O(r^{-q_0})\). Take \(p>3\); Morrey embedding gives scaled \(C^{1,\alpha}\) control for some \(\alpha>0\). In the harmonic chart Equation (197) has the elliptic form \[-\tfrac12\gamma^{ab}\partial_{ab}\gamma_{ij} +Q_{ij}(\gamma^{-1},D\gamma)=2U_iU_j,\qquad \gamma^{ab}\partial_{ab}U=0,\] where \(Q_{ij}\) is quadratic in \(D\gamma\). Schauder estimates first give scaled \(C^{2,\alpha}\) control; differentiating this elliptic system then gives \(\gamma-\delta,U=O_k(r^{-q_0})\) for each finite \(k\). No higher original-coordinate derivative bound has been assumed.

We use two elementary exterior Poisson estimates. If \(w=o(1)\), \(\Delta_\delta w=f\), and \(f=O_k(r^{-2-t})\), \(1<t<2\), then \[ w=\frac ar+O_k(r^{-t}). \tag{199}\] To prove this, cut \(w\) off on a fixed inner annulus and extend it by zero; this changes \(f\) only on a compact set. The extended solution equals the Newton potential of its Laplacian, since their difference is an entire decaying harmonic function. The coefficient is \(a=-(4\pi)^{-1}\int f\). For \(|y|<r/2\), subtracting \(1/r\) from the kernel costs \[Cr^{-2}\int_{|y|<r/2}|y||f(y)|\,\mathrm dy=O(r^{-t}).\] The remaining region, including the integrable kernel singularity, has the same bound. Rescaled interior estimates prove the derivative version. If instead \(f=O_k(r^{-4-\epsilon})\), \(0<\epsilon<1\), its first moments converge and the same reasoning gives \[ w=\frac ar+\frac{d\cdot x}{r^3}+O_k(r^{-2-\epsilon}). \tag{200}\] Here subtract \(1/r+(x\cdot y)/r^3\) in the near region. The remainder costs \(Cr^{-3}\int_{|y|<r/2}|y|^2|f(y)|\,\mathrm dy =O(r^{-2-\epsilon})\); the far region has that order as well.

Set \(\epsilon=2q_0-1\in(0,1)\). The harmonic-coordinate equations and initial symbol bounds yield \[\Delta_\delta(\gamma_{ij}-\delta_{ij}), \ \Delta_\delta U=O_k(r^{-2-2q_0}).\] Equation (199) gives \[\gamma_{ij}=\delta_{ij}+\frac{A_{ij}}r +O_k(r^{-1-\epsilon}),\qquad U=\frac ar+O_k(r^{-1-\epsilon}).\] The harmonic-coordinate condition is \(\partial_j(\sqrt{\det\gamma}\,\gamma^{ij})=0\). Expanding it at the displayed order gives \[\left(A_{ij}-\tfrac12(\mathop{\mathrm{tr}}A)\delta_{ij}\right) \frac{x^j}{r^3}=O(r^{-2-\epsilon}).\] Taking a radial limit shows \(A=\tfrac12(\mathop{\mathrm{tr}}A)I\). Its trace in dimension three is zero, hence \(A=0\). Since \(D^2U=O_k(r^{-3})\), the scalar equation now improves to \[\Delta_\delta U =(\delta^{ab}-\gamma^{ab})\partial_{ab}U =O_k(r^{-4-\epsilon}).\] Equation (200) proves Equation (196), initially with a real \(M\).

Its positivity uses the lapse range from Lemma 66. The function \(-U>0\) is \(\gamma\)-harmonic. For \(0<\eta<1\) and a fixed positive \(C\), the function \[w=r^{-1}+Cr^{-1-\eta}\] is positive and strictly \(\gamma\)-subharmonic sufficiently far out: \(\Delta_\gamma r^{-1}=O(r^{-4-\epsilon})\), while the leading Laplacian of \(r^{-1-\eta}\) is \(\eta(1+\eta)r^{-3-\eta}\). Choose \(c>0\) so small that \(-U\geq cw\) on a large inner sphere. The difference tends to zero and is superharmonic, so the exterior minimum principle gives \(-U\geq cw\) throughout that end. Taking the radial limit proves \(M\geq c>0\). ◻

Proof. An orientable three-manifold is spin. Choose a spin structure and use a smooth background metric to identify its spinor bundle with that for \(\mathfrak g\). The positive square-root change of orthonormal frames has the stated metric regularity. The spin connection consequently has continuous locally bounded coefficients, with weak first derivatives in \(L^p_{\mathrm{loc}}\). Write \(\nabla^S\) for that connection, \(D\) for the Dirac operator, and \(c\) for Clifford multiplication.

The weak Lichnerowicz formula has the following integrated form for every compactly supported \(H^1\) spinor \(\phi\): \[ \|D\phi\|_2^2=\|\nabla^S\phi\|_2^2. \tag{203}\] Here and below norms and integrals are for \(\mathfrak g\). To check the formula at the indicated regularity, first use a smooth compactly supported spinor. Approximate the metric on its support in \(C^1\) and \(W^{2,p}\) by smooth metrics, using the same background-bundle identification. The smooth Lichnerowicz formula passes to the limit: connection coefficients converge uniformly and scalar curvatures converge in \(L^p\). The limiting scalar term is zero by the assumed distributional scalar flatness and the \(L^p\) curvature expression. Density then proves Equation (203) for compact \(H^1\) spinors. Thus no integration surface is retained around the compactified point.

We next establish the coercivity needed to complete compact spinors. For a compactly supported scalar function \(f\) on the AE end, one-dimensional integration by parts along coordinate rays gives \[ \int_{r_0}^\infty f^2\,\mathrm dr \leq4\int_{r_0}^\infty r^2|\partial_rf|^2\,\mathrm dr. \tag{204}\] The inner boundary term has the favorable sign. Apply this to \(f=|\phi|\) and integrate over angles. Metric comparability and \(|\,\mathrm d|\phi||\leq|\nabla^S\phi|\) give \[ \int_{\mathrm{AE}}r^{-2}|\phi|^2\,\mathrm dV_{\mathfrak g} \leq C\|\nabla^S\phi\|_2^2. \tag{205}\] For any fixed compact \(K\subset W\), connect it to a fixed end annulus by a relatively compact connected domain. Poincaré’s inequality with the norm on that annulus retained gives \[\||\phi|\|_{L^2(K)} \leq C_K\left( \|\,\mathrm d|\phi|\|_{L^2(\text{domain})} +\|\phi\|_{L^2(\text{annulus})}\right).\] For example, this version follows from the ordinary Poincaré inequality after subtracting the mean; the observed annulus controls that mean because it has positive measure. Equation (205) controls its last term. We have proved \[ \|\phi\|_{L^2(K)}\leq C_K\|\nabla^S\phi\|_2. \tag{206}\] Connectedness is the only global input in this propagation.

Let \(\mathcal H\) be the completion of compactly supported smooth spinors in the norm \(\|\nabla^S\phi\|_2\). Equation (206) embeds \(\mathcal H\) in \(L^2_{\mathrm{loc}}\), and the locally bounded connection then embeds it in \(H^1_{\mathrm{loc}}\). Both \(\nabla^S:\mathcal H\to L^2\) and \(D:\mathcal H\to L^2\) are well-defined, and Equation (203) makes the latter an isometry. Its range is therefore closed. Equation (205) also passes to this completion. The local constants may depend on the chosen compact set; no uniform estimate over the remaining ends is required.

Fix the ordinary orthonormalized coordinate frame on the distinguished end and a smooth cutoff \(\theta\), equal to one near infinity and zero before a slightly smaller end neighborhood. For each constant spinor \(z\in\mathbb C^2\), set \(\psi_0=\theta z\). Because the connection there is \(O(r^{-2-\epsilon})\), \(\nabla^S\psi_0,D\psi_0\in L^2\). Orthogonally project \(D\psi_0\) onto the closed subspace \(D\mathcal H\) and choose \(\eta\in\mathcal H\) such that \[D\eta=-\operatorname{proj}_{D\mathcal H}(D\psi_0).\] Set \(\psi=\psi_0+\eta\), \(w=D\psi\). Then \(w\in L^2\) and \[\int\langle w,D\phi\rangle\,\mathrm dV_{\mathfrak g}=0 \quad\text{for every compact smooth }\phi.\] Thus \(Dw=0\) weakly. Local elliptic regularity for a first-order elliptic operator with locally Lipschitz principal coefficients and bounded lower-order coefficients gives \(w\in H^1_{\mathrm{loc}}\). This regularity can also be seen by freezing the principal coefficients on small coordinate balls: the constant-coefficient Dirac estimate controls one derivative in \(L^2\), the oscillation is absorbed, and commutators of mollification are bounded by the local Lipschitz norm. Apply the estimate to regularized \(w\) and pass to a weak limit. The metric in the lemma has more than this coefficient regularity.

Completeness supplies compactly supported Lipschitz distance cutoffs \(\chi_R\), equal to one on the radius-\(R\) ball and with \(|\,\mathrm d\chi_R|\leq C/R\). Indeed a complete locally compact Riemannian length space is proper, and a piecewise linear cutoff of its distance has these properties. Equation (203) applies to \(\chi_Rw\), and the weak equation gives \[\|\nabla^S(\chi_Rw)\|_2^2 =\|D(\chi_Rw)\|_2^2 =\|c(\,\mathrm d\chi_R)w\|_2^2 \leq \frac C{R^2}\|w\|_2^2.\] On every fixed compact set the cutoff eventually equals one. It follows that \(\nabla^S w=0\). A parallel spinor has constant length on the connected manifold; the distinguished end has infinite volume, so \(w\in L^2\) forces \(w=0\). No condition at another end entered this argument.

It remains to show that \(\psi\), rather than just \(D\psi\), is parallel. Define the continuous sesquilinear form \[\mathcal B(a,b)= \int\left(\langle\nabla^Sa,\nabla^Sb\rangle -\langle Da,Db\rangle\right)\,\mathrm dV_{\mathfrak g}\] on fields for which the indicated derivatives are in \(L^2\). For compact \(\phi\), the polarized Lichnerowicz identity, with one slot compactly supported, gives \(\mathcal B(\psi_0,\phi)=0\). Approximate \(\eta\) in \(\mathcal H\) by compact spinors. These limits use the global derivative norms; no boundary flux of \(\eta\) at another end is invoked. Continuity and Equation (203) imply \[\mathcal B(\psi_0,\eta)=0,\qquad \mathcal B(\eta,\eta)=0,\qquad \mathcal B(\psi,\psi)=\mathcal B(\psi_0,\psi_0).\] The last expression is zero. To see this directly, integrate the Lichnerowicz identity for \(\psi_0\) up to a large end sphere. There is no contribution at another end because \(\psi_0\) vanishes there. On that sphere \(\psi_0=O(1)\) and \(\nabla^S\psi_0,D\psi_0=O(r^{-2-\epsilon})\); the boundary integral is \(O(r^{-\epsilon})\) and tends to zero. This is also the zero ADM boundary term of the spin proof. Since \(D\psi=w=0\), we conclude that \(\|\nabla^S\psi\|_2^2=0\).

The parallel equation gives each resulting spinor a \(C^{1,\beta}_{\mathrm{loc}}\) representative for some \(\beta>0\). Indeed, \(H^1\subset L^6\) and the locally bounded connection first give \(W^{1,6}\), hence local Hölder continuity; the locally Hölder connection coefficients then make its first derivatives Hölder. Write \(\psi_z\) for this parallel spinor. The fixed cutoff, orthogonal projection, and inverse of \(D:\mathcal H\to D\mathcal H\) make \(z\mapsto\psi_z\) complex linear. For any point \(p\in W\), consider \[E_p:\mathbb C^2\longrightarrow\mathbb S_p, \qquad E_p(z)=\psi_z(p),\] where \(\mathbb S_p\) is the complex spinor fiber. If \(E_p(z)=0\), parallelism and connectedness give \(\psi_z=0\) everywhere. Then \(\eta=-\theta z\), which for \(z\ne0\) has infinite \(r^{-2}\)-weighted \(L^2\) norm on the distinguished end, contrary to Equation (205). Thus \(E_p\) is injective for every \(p\), and it is an isomorphism because the complex spinor rank is two. The images of a basis of \(\mathbb C^2\) form a global parallel spinor frame.

Its Hermitian Gram matrix is constant and positive definite; a constant change of basis makes the frame orthonormal. The weak curvature equations annihilate this frame, so the spin curvature vanishes. Faithfulness of the spin representation on \(\mathfrak{spin}(3)\) gives zero Riemann curvature. Moreover, Clifford multiplication identifies the real tangent bundle with the traceless skew-Hermitian endomorphisms of the spinor bundle. Three fixed orthonormal matrices in this parallel spinor frame, with matrix inner product \(-\tfrac12\operatorname{tr}(AB)\), therefore give a global parallel orthonormal tangent frame. The dual coframe is closed by torsion-freeness and integrates locally to coordinates in which the metric is exactly Euclidean. This also removes any apparent metric singularity at the compactified point.

Finally, completeness makes the universal flat cover Euclidean \(\R^3\). The global parallel frame makes its deck transformations translations. A nontrivial discrete translation group has rank at least one, and a quotient by it has volume growth at most quadratic. The distinguished AE end gives a cubic lower bound: radial paths in that end put a Euclidean annulus of radius comparable to \(R\) inside a metric ball of radius \(CR\). The two volume bounds are incompatible. The deck group is therefore trivial, proving the lemma. ◻

Proof. The complete base of Lemma 66 reaches every component of \(S\). Take its complete two-copy double, use \(h_+\) on the plus copy and \(h_-\) on the minus copy, and identify corresponding boundary points. The signed reflected lapse expresses the joined metric as \[q=\left(\frac{1+\lambda_{\mathrm{signed}}}{2}\right)^4h_{\mathrm d}.\] It is smooth at each positive-lapse join and \(C^2\) at each zero-lapse join. The matched second jets make its scalar curvature zero across every join, as well as on the two open sides. Compactify the unique AF end of the minus copy by Lemma 68.

We verify completeness of this conformal manifold without placing any restriction on the number or regularity of the possible internal-null ends. On the plus copy the factor \((1+\lambda)/2\) is at least \(1/2\). Fix a sufficiently large original AF tail. On its complement the continuous function \(\sqrt N\) is defined on a compact subset of the original closed exterior, equals zero on \(S\) and the internal zero set, and is strictly less than one elsewhere by Lemma 66. Its maximum on this compact set is therefore some \(a_0<1\). Thus on the entire corresponding minus remainder \((1-\lambda)/2\geq(1-a_0)/2>0\) uniformly. On the double away from the minus AF tail the conformal metric is consequently bounded below by a fixed positive multiple of the complete doubled base metric. All joins are already attached. A finite-length escape can therefore only run down the minus AF tail, and its proved nondegenerate one-point compactification supplies its limit. More explicitly, truncate that tail at a large sphere. The other side of the sphere is complete with compact boundary by the uniform lower bound, and the compactified tail is a compact metric region with that same boundary; their gluing is complete.

The conformal manifold is connected, orientable, and without boundary. It has local \(C^{1,\alpha}\cap W^{2,p}\) regularity for some \(\alpha>0\) and \(p>3\): the finitely many joins are at least \(C^2\), and the sole compactification point has the regularity established in Lemma 68. Its scalar curvature is zero distributionally, including at that point, and its distinguished plus end has the fast decay required by Lemma 69. That lemma makes this entire manifold Euclidean space.

Choose a sufficiently large sphere in the distinguished plus end. Under the Euclidean isometry it is a compact embedded sphere. Its end tail is the unbounded component of the complement: paths from large end radii to that sphere have lengths tending to infinity, and the tail has precisely that sphere as frontier. The other component is bounded and has compact closure. A sequence in \(\mathcal E\) approaching an interior point with \(N=0\) lies outside the chosen end tail and escapes every compact set of the complete conformal manifold. Indeed it cannot converge at an ordinary plus point by continuity of \(N\), cannot converge on any joined component because every original boundary collar has no interior zero, and cannot converge at a point in the open minus copy or its compactified end. Such a sequence is impossible in the compact Euclidean complement. Thus \(\mathcal E\) has no interior boundary in \(\Omega\).

It is a nonempty open and closed subset of connected \(\Omega\), and hence equals \(\Omega\). Vacuum now follows from Proposition 58, since \(N>0\) everywhere in \(\Omega\). ◻

Proof. The completed conformal double in the proof of Lemma 70 is Euclidean. In particular its plus metric \(q=h_+\) is flat up to every original boundary component. Put \(\phi=(1+\lambda)/2\). On each \(S_i\), Lemma 66 gives \[\phi=\tfrac12,\qquad \partial_{\nu_h}\log\phi=\kappa,\qquad \mathrm{II}_h=0,\qquad h|_{TS_i}=g|_{TS_i}.\] The conformal second-fundamental-form formula, with normal pointing into the plus side, is therefore \[\mathrm{II}_q =\phi^2\bigl(\mathrm{II}_h+ 2\partial_{\nu_h}\log\phi\,h|_{TS_i}\bigr) =8\kappa\,q|_{TS_i}, \qquad q|_{TS_i}=\tfrac1{16}g|_{TS_i}.\] Let \(F_i\) be the Euclidean position vector on \(S_i\) and \(\nu_i\) its unit normal into the plus side. Its shape operator is \(8\kappa I\), so the tangential differential of \(F_i-a\nu_i\) is zero for \(a=(8\kappa)^{-1}\). Connectedness makes \(F_i-a\nu_i\) a constant point \(c_i\). Thus the image lies on the sphere of radius \(a\) centered at \(c_i\), with \(\nu_i\) its outward radial normal. The image is open in that sphere by local invertibility and closed by compactness, hence is the whole sphere. The Euclidean embedding therefore identifies \(S_i\) diffeomorphically with it. Since \(q|_{TS_i}=g|_{TS_i}/16\), its area satisfies \[ |S_i|_g=16|S_i|_q=64\pi a^2 =\frac{\pi}{\kappa^2}=16\pi m^2=|S|_g. \tag{207}\] Summing over the finite family yields \(|S|_g=\sum_{i=1}^{\ell}|S_i|_g=\ell|S|_g\). The boundary is nonempty and \(|S|_g>0\), hence \(\ell=1\). In particular no connectedness or spherical topology assumption was needed before this step. ◻

Proof. Lemma 71 identifies the boundary in the Euclidean conformal double as one round sphere of radius \(a=(8\kappa)^{-1}\). The plus side is its outside: its normal is the outward radial normal, its entire boundary is this sphere, and it contains the distinguished unbounded end. Center Euclidean coordinates at this sphere and write \(\rho\) for their radius.

In the resulting Euclidean coordinates, write \(\psi=2/(1+\lambda)\), so \(h=\psi^4\delta\). The scalar-flat conformal equation gives \(\Delta_\delta\psi=0\) outside the sphere, with \(\psi=2\) on it and \(\psi\to1\) at infinity. The function \(1+a/\rho\) has the same data. Their difference vanishes by the exterior maximum principle. This proves Equation (208). Its Schwarzschild mass is \(2a=(4\kappa)^{-1}=m\), since \(\kappa=1/(4m)\). In particular \(a=m/2\).

The regular base collar has the same underlying boundary topology as the original collar, even when its coordinate is \(\lambda=\sqrt N\) instead of the original defining function \(N\). Together with the unchanged interior this identifies its completion homeomorphically with \(M_{\mathrm{ext}}\). An interior base isometry extends uniquely to the metric completions. On the regular boundary structures it is smooth: its boundary restriction preserves induced lengths, hence is a smooth boundary isometry, and normal geodesic coordinates extend it smoothly to a collar. Thus the completion has the claimed global topology, and \(\Omega\cong(a,\infty)\times S^2\) is simply connected. ◻

Proof of Proposition 65. Lemma 71 proves \(\ell=1\) and the spherical topology of \(S\). Lemma 70 proves \(N>0\) on \(\Omega\), and Lemma 72 gives the global static base with mass \(m\). For the reconstruction write \(M=m\), so \(\kappa=1/(4M)\). Simple connectivity and Lemma 66 give a global smooth function \(H\) with \[ \,\mathrm dH=-A. \tag{209}\] In static Schwarzschild coordinates use time \(T=t+H(x)\). Equations (189) and (209) give the exact identity \[ -N\,\mathrm dT^2+h =-u^2\,\mathrm dt^2+g_{ij}(\,\mathrm dx^i+X^i\,\mathrm dt) (\,\mathrm dx^j+X^j\,\mathrm dt). \tag{210}\] The graph \(T=H(x)\), corresponding to \(t=0\), therefore induces the original metric \(g\). The stationary future unit normal is \(n=(\partial_t-X)/u\). With the second-fundamental-form convention of the theorem, \[ K_{\mathrm{graph}} =\frac1{2u}\bigl(\partial_tg-\mathcal L_Xg\bigr) =-\frac1{2u}\mathcal L_Xg =K. \tag{211}\] This calculation recovers the original \(K\), including a non-time-symmetric slice.

We next establish smoothness in the original boundary structure. If \(u|_S=0\), Equation (195) gives \[A=\frac{Y^\flat}{a^2-s^2|Y|_g^2}.\] This is smooth in the original collar. The primitive \(H\) therefore extends smoothly and finitely to \(S\). For instance, fix its values on one interior collar section and integrate its smooth normal derivative to the boundary. Tangential derivatives of this extension agree with those prescribed by \(\,\mathrm dH=-A\), by continuity from the interior.

If \(u|_S>0\), the numerator \(X^\flat-(2\kappa)^{-1}\,\mathrm dN\) vanishes as a one-form on \(S\), by Equation (193). Smooth division by the defining function \(N\) gives \[ A=\frac1{2\kappa}\,\mathrm d\log N+\beta,\qquad H=-\frac1{2\kappa}\log N+B, \tag{212}\] where \(\beta\) and \(B\) are smooth in the original collar. The smoothness of \(B\) follows by applying the same normal-integration argument to its differential \(-\beta\). In this case \(H\to+\infty\) at the boundary, so static time itself is not a regular attachment coordinate.

Let \(r\) be the Schwarzschild area radius. On the whole exterior \[N=1-\frac{2M}{r},\qquad r=\frac{2M}{1-N}.\] Normalize the tortoise coordinate and Kruskal coordinates by \[ \begin{aligned} r_*&=r+2M\log\left(\frac r{2M}-1\right),\\ \mathsf U&=-e^{-\kappa(T-r_*)},& \mathsf V&= e^{\kappa(T+r_*)}. \end{aligned} \tag{213}\] On the right exterior \(\mathsf U<0,\mathsf V>0\), and the Schwarzschild metric extends as \[\mathbf g_M =-\frac{32M^3}{r}e^{-r/(2M)} \,\mathrm d\mathsf U\,\,\mathrm d\mathsf V+r^2\,\mathrm d\omega^2.\] An exact useful form of Equation (213) is \[ \mathsf U=-Q(N)\sqrt N\,e^{-\kappa T},\qquad \mathsf V= Q(N)\sqrt N\,e^{\kappa T},\qquad Q(N)=\frac{\exp\!\bigl(1/[2(1-N)]\bigr)}{\sqrt{1-N}}. \tag{214}\] The function \(Q\) is smooth and positive near zero, with \(Q(0)=\sqrt e\).

If \(u|_S=0\), both \(H\) and \(\sqrt N=s\sqrt{a^2-s^2|Y|^2}\) are smooth. Substitution into Equation (214) gives \(\mathsf U=\mathsf V=0\) on \(S\) and \[\partial_s\mathsf U|_S =-\sqrt e\,\kappa e^{-\kappa H|_S}<0,\qquad \partial_s\mathsf V|_S =\sqrt e\,\kappa e^{\kappa H|_S}>0.\] This is a smooth attachment at the bifurcation sphere. If \(u|_S>0\), Equation (212) instead gives \[ \mathsf U=-Q(N)N e^{-\kappa B},\qquad \mathsf V=Q(N)e^{\kappa B}. \tag{215}\] These are smooth in the original collar; \(\mathsf U\) vanishes simply and \(\mathsf V|_S=\sqrt e\,e^{\kappa B|_S}>0\). Thus the attachment is on the future horizon, with each generator met once after the angular identification.

The angular regularity requires a further check when \(u|_S>0\), because the regular base coordinate was \(\lambda\), not the original \(N\). On the smooth reflected collar in Equation (194), normal projection onto its fixed boundary is invariant under \(\lambda\mapsto-\lambda\). This follows from uniqueness of the normal geodesics and reflection invariance of the metric. Schwarzschild angular coordinates are constant along these normal geodesics, and their boundary values are the smooth round-sphere identification from Lemma 72. They are therefore smooth even functions of \(\lambda\). A smooth even function, with smooth tangential parameters, is a smooth function of \(N=\lambda^2\) for \(N\geq0\): all odd Taylor coefficients vanish, and repeated application of \((2\lambda)^{-1}\partial_\lambda\) to the Taylor remainder gives continuous derivatives of every order at zero. This proves the required angular smoothness in the original collar. In the zero boundary-lapse case the original collar is already a smooth base collar, so its completed base isometry supplies the angular extension directly.

We have constructed a smooth map \(\iota:M_{\mathrm{ext}}\to\) Schwarzschild. Equation (210) extends to \(S\) by continuity, hence \(\iota^*\mathbf g_M=g\) there as well. The positive definiteness of \(g\) proves that \(\,\mathrm d\iota\) is injective at every boundary point. Interior injectivity follows from the global base isometry; boundary injectivity follows from the sphere identification. The boundary image, at \(r=2M\), is disjoint from the open image, where \(r>2M\). The map is proper: a compact subset of Schwarzschild has bounded area radius, and its inverse image is closed in the compact base region corresponding to \([2M,R]\times S^2\). Thus the injective immersion is an embedding.

The future normal extends smoothly even where \(u=0\). To avoid taking a limit of the expression involving \(u^{-1}\), choose the smooth future timelike Kruskal vector \(Z=\partial_{\mathsf U}+\partial_{\mathsf V}\) near the boundary. Let \(Z^\top\) be its tangential projection along the embedded spacelike hypersurface and set \[\widehat n= \frac{Z-Z^\top} {\sqrt{-\mathbf g_M(Z-Z^\top,Z-Z^\top)}}.\] The denominator is positive, and this is a smooth future unit normal. On the open exterior it agrees with the normal used in Equation (211). Its second fundamental form is smooth and equals \(K\) in the interior, so equality extends to \(S\).

There is also a smooth spacelike extension of the image through its boundary. In the positive boundary-lapse case, use \((\mathsf U,\omega)\) as hypersurface coordinates; its normal derivative is nonzero by Equation (215). The remaining null coordinate is a smooth graph function and can be extended across \(\mathsf U=0\). In the zero boundary-lapse case use \((\chi,\omega)\), where \(\chi=(\mathsf V-\mathsf U)/2\); the displayed normal derivatives give \(\partial_s\chi>0\). Again extend the remaining graph function across zero. Smooth one-sided graph functions extend across a smooth boundary, and compactness of \(S\), together with openness of the spacelike condition, preserves spacelikeness on a sufficiently short collar. This extension makes no assertion about the original data behind \(S\).

Finally, the exact inverse-metric identity gives \[ |\,\mathrm dH|_h^2 =\left|\frac{X^\flat}{N}\right|_C^2 =\frac{|X|_g^2}{u^2N} \longrightarrow\frac{|b|^2}{(b^0)^2}<1. \tag{216}\] Choose a constant \(a_0<1\) larger than the limiting slope norm. Along radial rays in the Schwarzschild base, uniformly in angle, \[|H(r,\omega)| \leq C+a_0\int_R^r \frac{\,\mathrm d\rho}{\sqrt{1-2M/\rho}} =a_0r+O(\log r).\] Since \(r_*=r+O(\log r)\), it follows that \[H-r_*\longrightarrow-\infty,\qquad H+r_*\longrightarrow+\infty.\] This is approach to the chosen Schwarzschild spatial infinity. It allows a nonzero asymptotic tilt and does not impose \(P=0\) or \(K=0\). All conclusions of Proposition 65 follow. ◻

Proof. Every singular two-cycle with coefficients in \(\mathbb Z/2\) has compact support and lies in the interior of a compact smooth three-dimensional submanifold \(C\subset W\) with boundary. The closed orientable double of \(C\) is parallelizable by (Benedetti and Lisca 2018, Theorem 1.1). Naturality therefore shows that \(w_2(TW)\) evaluates trivially on this cycle. Duality between cohomology and homology over the field \(\mathbb Z/2\) gives \(w_2(TW)=0\). Together with orientability this is the spin-structure criterion; see (Lawson and Michelsohn 1989, II, Sections 1–2). This argument applies directly to the possibly noncompact \(W\).

Fix one such spin structure. Positive square roots of the changes of metric identify the oriented orthonormal-frame bundles and their spinor bundles. We choose the lifts continuously in \(t\); on compact sets these identifications and the metric coefficients converge in \(C^2\) as \(t\uparrow1\). On the distinguished end the orthonormalized coordinate frame has a spin lift because that end is simply connected. We choose these lifts consistently throughout the family.

We use the Clifford convention \(\mathfrak c_t(Y)\mathfrak c_t(Z)+\mathfrak c_t(Z)\mathfrak c_t(Y) =-2q_t(Y,Z)\); Clifford multiplication by a real tangent vector is skew-adjoint and has norm \(|Y|_{q_t}\). The local spin-curvature formula and the curvature symmetries yield \[D_t^2=(\nabla^S_t)^*\nabla^S_t+\tfrac14R_{q_t};\] the standard computation and formal adjoint identities are given in (Lawson and Michelsohn 1989, II, Sections 4–5 and 8). Integrating this identity for a compact test spinor proves Equation (224) for smooth scalar-flat metrics. For the present \(C^2\) metric, smooth it in \(C^2\) on a neighborhood of the test support and use the positive-square-root bundle identifications. The connections and Dirac coefficients converge, and the scalar curvatures converge uniformly to zero. Passing to the limit proves the identity for smooth compact test spinors. Density in local \(H^1\) then proves the stated form. The joined spheres are interior surfaces of the \(C^2\) metric on \(W\), so this calculation has no boundary at a joined sphere. ◻

Proof. For a compactly supported absolutely continuous function on a ray, integration by parts gives \[\int_{r_0}^{\infty}f^2\,dr =-r_0 f(r_0)^2-2\int_{r_0}^{\infty}rff'\,dr \le2\left(\int f^2\,dr\right)^{1/2} \left(\int r^2|f'|^2\,dr\right)^{1/2}.\] Consequently \(\int f^2\le4\int r^2|f'|^2\); the inner boundary term has the favorable sign without a prescribed inner trace. Apply this to \(f=|\zeta|\), integrate over directions, and use uniform metric comparability from Equation (222) and the Kato inequality \(|d|\zeta||_{q_t}\le|\nabla^S_t\zeta|_{q_t}\). This proves Equation (225), first for smooth spinors and then by \(H^1\) approximation.

Choose a fixed end annulus \(\mathcal A\) and a relatively compact connected smooth domain \(G\subset W\) containing both \(K\) and \(\mathcal A\). The scalar Poincaré inequality with an anchored mean gives \[\|f\|_{L^2(G)}\le C_G \bigl(\|df\|_{L^2(G)}+\|f\|_{L^2(\mathcal A)}\bigr).\] For completeness, subtract the mean of \(f\) on \(G\) and apply the usual Poincaré inequality; its restriction to \(\mathcal A\) bounds that mean by the two terms on the right. Take \(f=|\zeta|\). Equation (225) bounds its annular term, and Kato bounds its derivative. All norms on this fixed domain are uniformly equivalent as \(t\uparrow1\), since \(q_t\to q_*\) positively in \(C^2\) there. This proves Equation (226). In a fixed local trivialization \(\partial\zeta\) differs from \(\nabla^S_t\zeta\) by a uniformly bounded connection matrix times \(\zeta\), which gives the last assertion. ◻

Proof. Project \(-D_t\psi_{0,t}\) orthogonally onto the closed image in Equation (227). The unique preimage is \(\zeta_t\), and the projection inequality gives the norm bound. The residual \(w=D_t(\psi_{0,t}+\zeta_t)\in L^2\) is orthogonal to \(D_t\mathcal H_t\), so formal symmetry against compact tests gives \(D_tw=0\) distributionally.

We give the regularity step needed to test this equation. In a coordinate spin trivialization, \(D_t=A^j(x)\partial_j+B(x)\), where the principal coefficients are locally Lipschitz and \(B\) is locally bounded. The Clifford relations make the principal symbol elliptic. Freezing \(A^j\) at a point and applying the Fourier transform gives, for a smooth compact test vector in a sufficiently small chart, \[\|v\|_{H^1}\le C\bigl(\|D_tv\|_{L^2}+\|v\|_{L^2}\bigr).\] Indeed the constant symbol bounds \(|\xi||\widehat v(\xi)|\); the small oscillation of \(A^j\) on the chart is absorbed into the left side, and the bounded zeroth-order coefficient contributes the \(L^2\) term. This estimate applies to a localized weak solution by mollification. To see the necessary uniform bound, the commutator of \(A^j\partial_j\) with a mollifier has kernels involving \((A^j(x)-A^j(y))\partial_j\rho_\varepsilon(x-y)\) and \((\partial_jA^j)(y)\rho_\varepsilon(x-y)\). Their \(L^1\) norms are uniformly bounded by the Lipschitz norm of \(A^j\). The commutator with \(B\) is bounded on \(L^2\) as well. For a compact localization \(v\) of \(w\), both \(v\) and \(D_tv\) are in \(L^2\), so the mollified fields are uniformly \(H^1\) on smaller charts and their weak limit is \(v\). Hence \(w\in H^1_{\mathrm{loc}}\). This is also the local regularity result in (Bartnik and Chruściel 2005, Theorem 3.7); the present \(C^2\) metric satisfies its coefficient hypotheses.

For the fixed complete metric \(q_t\), the length-space form of Hopf–Rinow makes closed metric balls compact (Burago et al. 2001, sec. 2.5). Distance from a fixed point therefore supplies compactly supported Lipschitz cutoffs \(\chi_R\) which equal one on the ball of radius \(R\), vanish outside the ball of radius \(2R\), and satisfy \(|d\chi_R|_{q_t}\le C/R\) almost everywhere. The product \(\chi_Rw\) is compactly supported \(H^1\). Since \(D_tw=0\), Equation (224) gives \[\|\nabla^S_t(\chi_Rw)\|_2^2 =\|\mathfrak c_t(d\chi_R)w\|_2^2 \le CR^{-2}\|w\|_2^2.\] Letting \(R\to\infty\) shows on each compact set that \(\nabla^S_tw=0\). Its length is constant on the connected \(W\), by Kato or the metric compatibility of the connection. The distinguished end has infinite \(q_t\) volume. Since \(w\in L^2\), this constant is zero, and \(w=0\) as required. ◻

Proof. On the fields with finite derivative and Dirac norms used here, define the continuous bilinear form \[\mathcal B_t(\xi,\eta) =\langle\nabla^S_t\xi,\nabla^S_t\eta\rangle_{L^2} -\langle D_t\xi,D_t\eta\rangle_{L^2}.\] The polarized compact identity gives \(\mathcal B_t(\xi,\eta)=0\) whenever one entry is compactly supported and both are locally \(H^1\): multiply the other entry by a compact cutoff which is one near that support and use Equation (224). Approximate \(\zeta_t\) by compact tests in \(\mathcal H_t\). Equation (228) permits passage to the limit in the mixed terms, giving \[ \mathcal B_t(\zeta_t,\zeta_t)=0, \quad \mathcal B_t(\psi_{0,t},\zeta_t)=0, \quad \mathcal B_t(\psi_t,\psi_t) =\mathcal B_t(\psi_{0,t},\psi_{0,t}). \tag{231}\] This use of compact approximation concerns the correction \(\zeta_t\).

To estimate the last expression, take a scalar cutoff \(\beta_R\) which is one up to radius \(R\) in the distinguished end and zero beyond \(2R\), with \(|d\beta_R|_{q_t}\le C/R\) there; extend it as one off the end. The spinor \(\beta_R\psi_{0,t}\) is compactly supported. The product rules for \(\nabla^S_t\) and \(D_t\) give \[\begin{align*} 0={}&\mathcal B_t(\psi_{0,t},\beta_R\psi_{0,t})\\ ={}&\int_W\beta_R (|\nabla^S_t\psi_{0,t}|^2-|D_t\psi_{0,t}|^2)\,dV_{q_t} +\mathcal E_{t,R}, \end{align*}\] where \[\begin{align*} |\mathcal E_{t,R}| &\le\frac C R\int_{R<r<2R}|\psi_{0,t}| (|\nabla^S_t\psi_{0,t}|+|D_t\psi_{0,t}|)\,dV_{q_t} \\ &\le C\bigl((1-t)+R^{1-2q_0}\bigr). \tag{232}\end{align*}\] Here \(|\psi_{0,t}|=1\) on this annulus for large \(R\), its volume is \(O(R^3)\) uniformly in \(t\), and Equation (228) supplies the two derivative rates. The integral on the preceding line converges to \(\mathcal B_t(\psi_{0,t},\psi_{0,t})\) by integrability of the two squared derivatives. Since \(2q_0>1\), letting \(R\to\infty\) in Equation (232) yields \(|\mathcal B_t(\psi_{0,t},\psi_{0,t})|\le C(1-t)\). Finally \(D_t\psi_t=0\) and Equation (231) prove Equation (230). All explicit flux estimates involve the reference field, whose support lies in the chosen end neighborhood; the correction was handled by its compact approximation. ◻

Proof. Use the fixed spin structure and bundle identifications with the positive \(C^2\) metric \(q_*\) on compact subsets of \(W\). Equations (229) and (226) give uniform local \(H^1\) bounds for \(\zeta_t\). The reference spinors \(\psi_{0,t}\) converge locally with their coefficients, so \(\psi_t\) also has uniform local \(H^1\) bounds. A diagonal subsequence as \(t\uparrow1\) converges weakly in \(H^1\) on every compact subdomain to a spinor \(\psi_*\); write \(\zeta_*:=\psi_*-\psi_{0,*}\). The connection coefficients converge locally, and Equation (230) implies \[ \nabla^S_*\psi_*=0 \quad\hbox{in distributions on }W. \tag{233}\]

It remains to prove that this limit does not vanish. Equations (225) and (229) give the uniform estimate \[\int_{r\ge r_0}r^{-2}|\zeta_t|^2\,dV_{q_t}\le C.\] On every bounded end annulus, local weak lower semicontinuity and the converging metric and bundle coefficients pass this bound to \(\zeta_*\). Exhausting the end by such annuli gives \[ \int_{r\ge r_0}r^{-2}|\zeta_*|^2\,dV_{q_*}\le C. \tag{234}\] If \(\psi_*\) vanished identically on \(\mathcal P\), then \(\zeta_*=-\psi_{0,*}\) on this end. The reference spinor has unit length for \(r\ge2r_0\), and \(q_*\) is uniformly comparable to the Euclidean metric there, so the integral in Equation (234) would be bounded below by a positive constant times \(\int_{2r_0}^{\infty}dr=\infty\). Thus \(\psi_*\) is nonzero on the plus interior.

The metric is smooth on \(\mathcal P\). The parallel equation there first improves the local Sobolev regularity of \(\psi_*\) by differentiating in smooth local trivializations, and iteration gives a smooth parallel spinor. Metric compatibility shows that its length is constant; connectedness of \(\mathcal P\) makes that constant positive. Its spin curvature therefore annihilates it. The Clifford contraction of the spin-curvature formula, using the first Bianchi identity, is \[\sum_j\mathfrak c_*(e_j)R^S_*(e_j,Y)\psi_* =\tfrac12\mathfrak c_* \bigl(\mathop{\mathrm{Ric}}_{q_*}(Y,\cdot)^{\sharp}\bigr)\psi_*=0.\] Changing both curvature conventions changes the displayed sign but leaves the conclusion unchanged. Clifford multiplication by a nonzero vector is invertible, since its square is minus its squared length. Hence \(\mathop{\mathrm{Ric}}_{q_*}=0\) on \(\mathcal P\). In three dimensions the Riemann tensor is determined algebraically by the Ricci tensor, so \(q_*\) is flat there. Its \(C^2\) extension across the joined spheres has continuous curvature, which proves the last assertion. ◻

Proof. Apply Proposition 8 with \(n=4\) and obstacle \(B\). Its \(C^{1,1}\) conclusion applies to the whole frontier in this dimension. Its outward collar isotopy supplies the smooth-cut approximation, preserving the connected exterior. If \(H_B<0\), the collar flux inequality (17) removes all contact uniformly. The final part of the same proposition proves intrinsic completeness and outer area-minimization of the detached closed exterior. ◻

Proof. Smoothly extend the path through the fixed boundary in one collar. The common mean-convex tail gives uniform confinement directly by the clipping inequality (15): the divergence of the outward unit radial normal is positive for every metric in the path beyond the same radius. This is the common-sphere alternative in Proposition 9; the scaled first-derivative bounds used in its separate density-estimate alternative are unnecessary here. On the resulting fixed compact region, the smooth path has uniform local bounds. The proposition’s localized polar-measure continuity and uniform Grassmann-bundle Taylor expansion give all the assertions, retaining any mass on \(B\). ◻

Proof. All free frontiers are smooth in dimension four. Proposition 10 gives local agreement at a meeting point and continuity as both the point and the minimizing frontier vary. Its proof uses the small-excess theorem on the whole support, with a smooth-ambient mean-curvature bound after rescaling. This supplies the pointwise plane control beyond the integral convergence in Proposition 84. ◻

Proof. Put \(\varphi=\log f\). The scalar-curvature and tensor transformations in dimension four are \[R_{f^2g}=f^{-2}(R_g-6\Delta_g\varphi-6|d\varphi|_g^2), \quad \tau_f=f^{-1}\tau, \quad |K_f|_{f^2g}^2=f^{-2}|K|_g^2.\] Since \(\Delta\varphi+|d\varphi|^2=f^{-1}\Delta f\), they give \(f^2\mu_f=\mu-3f^{-1}\Delta_gf\). For the momentum, set \(\pi=K-\tau g\), so \(\pi_f=f\pi\). The connection difference is \[\widehat\Gamma^a_{ij}-\Gamma^a_{ij} =\delta_i^a\varphi_j+\delta_j^a\varphi_i-g_{ij}\varphi^a.\] Contracting the covariant derivative of \(f\pi\) yields \[fJ_{f,i}=J_i+3\pi_{ij}\varphi^j-(\mathop{\mathrm{tr}}_g\pi)\varphi_i =J_i+3K_{ij}\varphi^j,\] because \(\mathop{\mathrm{tr}}_g\pi=-3\tau\). In particular every trace-gradient term cancels; no extra asymptotic condition on \(\tau\) is used. Evaluating on the identified vector \(f^{-1}v\) gives \[f^2J_f(f^{-1}v)=J(v)+3f^{-1}K(\nabla f,v).\] Adding twice the density and momentum identities proves (250). Finally \(\nu_f=f^{-1}\nu\), \(H_{f^2g}=f^{-1}(H_g+3\partial_\nu\log f)\), and \(\mathop{\mathrm{tr}}_{TS,f^2g}(fK)=f^{-1}\mathop{\mathrm{tr}}_{TS,g}K\). Their sum proves (251) with the same oriented normal. ◻

Proof. Choose a smooth majorant \(d_0\ge |K|_g\) which on the end is a sufficiently large radial multiple of \(r^{-1-q_0}\). It can be chosen with all symbol bounds there. We solve \[ -\Delta_g\phi =d_0\sqrt{|\,\mathrm d\phi|_g^2+r^{-6}}+w, \qquad \partial_\nu\phi=-1\text{ on }S, \qquad \phi\longrightarrow0\text{ at infinity}. \tag{254}\]

First truncate at a large coordinate sphere \(S_R\), imposing \(\phi=0\) on \(S_R\), and replace \(d_0\) in the equation by \(t d_0\), \(0\le t\le1\). The inner and outer boundary components are disjoint smooth hypersurfaces; there is no corner where the two boundary conditions meet. The \(t=0\) problem is the invertible mixed Laplace problem. Every linearization in \(\phi\) adds to \(-\Delta_g\) a smooth drift of norm at most \(d_0\), with homogeneous Neumann data at \(S\) and homogeneous Dirichlet data at \(S_R\). The maximum and boundary point principles give uniqueness, and the mixed elliptic Fredholm theory then gives invertibility. These are the usual local elliptic and boundary estimates on a fixed smooth truncated domain; see (Gilbarg and Trudinger 2001).

Solutions are nonnegative. Indeed a negative minimum cannot occur in the interior because the right side of the equation is positive. At \(S\), the outward normal of the truncated domain is \(-\nu\), so the prescribed outward derivative is \(+1\), also excluding such a minimum by the boundary point principle. The outer Dirichlet value is zero.

For completeness, the continuation bounds do not require a monotonicity assumption in the unknown. Fix \(p>4\). The mixed \(W^{2,p}\) estimate, the linear growth of the right side in \(\,\mathrm d\phi\), and first-derivative interpolation give, on this fixed truncation, \[\|\phi\|_{W^{2,p}}\le C\bigl(1+\|\phi\|_{L^\infty}\bigr),\] uniformly in \(t\). If the supremum were unbounded, division by that supremum and compactness in \(C^1\) would produce a nonnegative function \(z\) with supremum one satisfying \[-\Delta_g z=t_*d_0|\,\mathrm dz|_g, \qquad \partial_\nu z=0\text{ on }S, \qquad z=0\text{ on }S_R\] for some \(t_*\in[0,1]\). At points where \(\,\mathrm dz\ne0\) the right side is a bounded drift applied to \(\,\mathrm dz\), and it is zero otherwise. The strong maximum and boundary point principles for this bounded-drift equation contradict the positive maximum. The resulting \(W^{2,p}\) and Schauder bounds close the continuation argument and give a smooth solution at \(t=1\) on every sufficiently large truncation.

We next make the bounds independent of \(R\). For \(W=r^{-2}-r^{-2-\delta}\), the Euclidean identity \[-\Delta_\delta W=(2+\delta)\delta r^{-4-\delta}\] and the AF estimates imply, beyond a fixed radius \(R_1\), \[ -\Delta_g W-d_0|\,\mathrm dW|_g\ge c_1w>0. \tag{255}\] The metric errors and the drift term are smaller than \(w\) because \(\delta<q_0<q\). Also \(d_0r^{-3}=o(w)\). If \(M_R\) is the supremum of the truncated solution on the fixed core bounded by \(S_{R_1}\), a sufficiently large fixed multiple of \((1+M_R)W\) is a supersolution on \(R_1\le r\le R\). To see this, use \(\sqrt{a^2+b^2}\le a+b\) and (255); choose the multiple also to dominate \(M_R\) on \(S_{R_1}\). It dominates the zero outer value. Comparison is valid by writing the difference of the two gradient nonlinearities as a bounded drift. Therefore \[ 0\le\phi_R\le C(1+M_R)W\qquad (R_1\le r\le R). \tag{256}\]

If \(M_R\) diverged along a sequence, normalize by \(M_R\). Local mixed estimates near \(S\) and interior estimates elsewhere give a subsequential limit \(z\) on every fixed compact set. More precisely, (256) bounds \(\phi_R/M_R\) on a neighborhood of the fixed closed core. Local Neumann \(W^{2,p}\) estimates at \(S\), and interior \(W^{2,p}\) estimates away from it, give uniform bounds there for \(p>4\). Compactness in \(C^1\) on the closed core retains both \(\sup_{\mathrm{core}}z=1\) and \(\partial_\nu z=0\). The limit satisfies \(-\Delta_gz=d_0|\,\mathrm dz|_g\), homogeneous Neumann data on \(S\), and, by (256), \(z\to0\) at infinity. Its positive global maximum is attained in a compact set. The same strong maximum and boundary point principles again give a contradiction. Thus \(M_R\) is bounded, and exhaustion produces a smooth solution of (254) with \(\phi=O(r^{-2})\).

Here is the derivative control needed later, using only the stated derivatives of the original data. On a fixed annulus in the \(y\) variables put \(g_\rho(y)=(g_{ij}(\rho y))\), \(U_\rho(y)=\rho^2\phi(\rho y)\), and \(a_\rho(y)=\rho d_0(\rho y)\). The exact rescaled equation is \[ -\Delta_{g_\rho}U_\rho =a_\rho\sqrt{|\,\mathrm dU_\rho|_{g_\rho}^2+|y|^{-6}} +\rho^{-\delta}|y|^{-4-\delta}. \tag{257}\] The metric components have uniform \(C^{1,1}\) bounds from \(g-\delta=O_2\). Consequently both the principal coefficients of this Laplacian and its first-order coefficients have uniform \(C^{0,\alpha}\) bounds for every fixed \(\alpha<1\). No third derivative of \(g\) is needed. The chosen radial \(d_0\) gives \(a_\rho=O(\rho^{-q_0})\) with uniform symbol bounds, independently of derivatives of \(K\). First-derivative interpolation in the local \(W^{2,p}\) estimates gives a uniform \(W^{2,p}\) bound on a smaller annulus. Taking \(p>4\) controls \(\,\mathrm dU_\rho\) in \(C^{0,\alpha}\) for some \(\alpha>0\). The positive \(|y|^{-6}\) term is bounded away from zero there, so the entire right side of (257) is uniformly \(C^{0,\alpha}\). The usual Schauder estimate, including the first-order Laplacian coefficients with these \(C^{0,\alpha}\) bounds, now gives a uniform \(C^{2,\alpha}\) bound on a further smaller annulus. Scaling back proves \(\phi=O_2(r^{-2})\) without an additional metric or tensor falloff derivative. The equation now yields \[d_0\sqrt{|\,\mathrm d\phi|_g^2+r^{-6}}=O(r^{-4-q_0})=o(w),\] which proves both the differential inequality and the last limit in (252). The right side of (254) is integrable. The divergence theorem consequently gives a finite limiting physical sphere flux. Its difference from the Euclidean sphere flux tends to zero, since \(g-\delta=O(r^{-q})\) and \(\,\mathrm d\phi=O(r^{-3})\).

Finally, differentiating Lemma 86 at \(f=1+s\phi\) gives \[C'_Q=-2\phi C+6[-\Delta_g\phi+K(\nabla\phi,v)], \qquad \theta'_Q=-\phi\theta_++3\partial_\nu\phi.\] On \(\mathcal A\) the first term is zero, and on \(S\) we have \(\theta_+=0\). This proves (253). On the full end ball bundle, \(C=O(r^{-2-q})\), \(\phi C=O(r^{-4-q})=o(w)\), and \(K(\nabla\phi,v)=O(r^{-4-q})=o(w)\). Thus the normalized derivative tends uniformly to six. ◻

Proof. Trace reversal on symmetric tensors in dimension four is invertible. The right side of (258), written as \[D_{ij}=r^{-4}\bigl(e_i x_j+x_i e_j-(e\cdot x)\delta_{ij}\bigr),\] satisfies \(\partial_jD_{ij}=0\) and \(D_{ij}\widehat x^j=r^{-3}e_i\). The scalar prototype satisfies the Euclidean linearized scalar constraint because \(\Delta_\delta r^{-2}=0\) off the origin. With the ADM normalizations (239)–(240), the scalar prototype has \(E'=1\), \(P'=0\), and the momentum prototype for \(e\) has \(E'=0\), \(P'=e/3\). Background corrections to these fluxes vanish by the AF estimates, so the derivatives are independent.

On the end, \(h_0=O_2(r^{-2})\) and \(p_0=O_1(r^{-3})\). Their Euclidean linearized scalar and momentum constraints vanish. Each remaining linear constraint term has a factor from the decaying background: for example, \((g-\delta)\partial^2h_0\), \(\partial g\,\partial h_0\), \(\partial^2g\,h_0\), \(Kp_0\), or \((g-\delta)\partial p_0\) has order at most \(r^{-4-q}\). The variation of the vector identification contributes \(-J(h_0^\sharp v)\), of the same order. This proves (259). Lemma 87 then proves (260) and the uniqueness of \(a\): an element of \(\mathcal V_0\) cannot have the normalized end derivative of \(Q\).

Compact variations have zero charge derivatives. The prototypes have the fluxes just computed. For \(Q\), the momentum derivative is zero and \[E'_Q=-\frac1\omega\lim_{R\to\infty} \int_{|x|=R}\partial_r\phi\,\,\mathrm dA_\delta,\] which is finite by Lemma 87. Products with background errors have zero limiting flux. Linearity completes the proof. ◻

Proof. Positivity of the metric and uniform comparability hold for small \(s\), since \(h_0\) and \(\phi\) are bounded relative to \(g\). They also give completeness with the boundary included. The topology, smoothness, end count and orientability remain those of \(\Omega\).

We check DEC on the whole ball of test vectors. First use the intermediate metric and tensor \(\overline g_s=g+sh_0\), \(\overline K_s=K+sp_0\), identifying its vector ball with that of \(g\) by \(\overline I_s\). For sufficiently large \(r\), Taylor expansion and the Euclidean cancellations in Lemma 89 give, uniformly in \(|v|_g\le1\), \[\overline C_s(x,\overline I_sv) =C(x,v)+O(sr^{-4-q})+O(s^2r^{-6}).\] Here the remainder is uniform in \(r\), \(s\) in a fixed small interval, and the whole vector ball. To check all its components, the coordinate constraint expressions have the schematic forms \[\begin{align*} R_g&=g^{-1}\partial^2g+\operatorname{smooth}(g^{-1})(\partial g)^2,\\ J&=\operatorname{smooth}(g^{-1})\partial K +\operatorname{smooth}(g^{-1})(\partial g)K, \end{align*}\] and \(2\mu-R_g\) is quadratic in \(K\), with smooth coefficients in \(g^{-1}\). Each second \(s\) derivative on the intermediate path is \(O(r^{-6})\): the largest terms are \(h_0\partial^2h_0\), \((\partial h_0)^2\), \(p_0^2\), \(h_0\partial p_0\), and \((\partial h_0)p_0\). Terms containing an original \(K\), \(\partial K\), or background metric derivative decay faster. This calculation includes all occurrences of \(\tau=g^{ij}K_{ij}\), without a separate trace estimate. The vector identification also has the uniform matrix expansion \[\overline I_s=(\mathop{\mathrm{Id}}+s h_0^\sharp)^{-1/2} =\mathop{\mathrm{Id}}-\tfrac{s}{2}h_0^\sharp+O(s^2r^{-4});\] the eigenvalues lie in a fixed positive interval, so the matrix-function Taylor bound is uniform. Multiplication by the original or varied momentum covector keeps the same remainder order. In fact the preceding calculation gives the componentwise estimates \[ \begin{aligned} \overline\mu_s&=\mu+O(sr^{-4-q})+O(s^2r^{-6}),\\ \overline J_s&=J+O(sr^{-4-q})+O(s^2r^{-6}). \end{aligned} \tag{263}\]

Apply Lemma 86 to these intermediate data with \(f_s=e^{as\phi}\). After multiplication by the positive factor \(f_s^2\), the constraint for the final data is \[ C(x,v)+s\bigl(6aw+o(w)\bigr)+O(s^2r^{-6}), \tag{264}\] again uniformly for the full vector ball. Indeed, \(-\Delta_g\phi/w\to1\) and \(K(\nabla\phi,v)=o(w)\). Changing the metric, tensor, and vector identification in these conformal differential terms adds only the above decay orders, while the extra exponential term uses \(|\,\mathrm d\phi|^2=O(r^{-6})\). Since \(a>0\) and \(r^{-6}=O(w)\), one can first choose a fixed large radius so the coefficient of \(s\) in (264) is positive, and then choose \(s\) small so its remainder is absorbed. DEC follows on this fixed far end, using \(C\ge0\).

The remaining base region and its closed unit ball bundle are compact. The derivative of the constraint is strictly positive on its original zero set. By continuity it is positive on a neighborhood of that zero set; on the complement the original constraint has a positive minimum. A uniform Taylor expansion therefore gives \(C_s\ge0\) everywhere in the compact region for small positive \(s\). The same compactness argument on \(S\), where \(\theta_+=0\), gives \(\theta_{+,s}<0\).

The new densities are integrable componentwise, not merely after contraction with active vectors. The exact conformal formulas give \[\begin{align*} f_sJ_s&=\overline J_s+ 3as\overline K_s(\nabla_{\overline g_s}\phi,\cdot),\\ f_s^2\mu_s&=\overline\mu_s-3as\Delta_{\overline g_s}\phi -3a^2s^2|\,\mathrm d\phi|_{\overline g_s}^2. \end{align*}\] The extra covector in the first line is \(O(sr^{-4-q})+O(s^2r^{-6})\). Also \(\Delta_{\overline g_s}\phi=\Delta_g\phi+O(sr^{-6})\), with \(\Delta_g\phi=O(w)\). Together with (263), these identities express the new densities as bounded smooth scaling factors times the original integrable densities, plus integrable errors of orders \(r^{-4-\delta}\), \(r^{-4-q}\), and \(r^{-6}\). Uniform metric comparability controls both norms and volume elements. The differentiated falloff has exponent \(\min(q,2)>1\), uniformly for \(s\) in a small fixed interval, so large coordinate spheres remain mean convex uniformly.

The charge limits can also be checked before differentiation. Write \(\pi=K-\tau g\). Since conformal trace reversal gives \(\pi_s=f_s(\overline K_s-\overline\tau_s\overline g_s)\), expansion on the end yields \[\pi_s=\pi+s\bigl(p_0-(\mathop{\mathrm{tr}}_\delta p_0)\delta\bigr) +O(sr^{-3-q})+O(s^2r^{-5}).\] The terms with the original trace are of size \(h_0K=O(r^{-3-q})\); multiplication by \(f_s-1=O(sr^{-2})\) adds only the displayed error orders. All errors have zero limiting sphere flux. Therefore \(P_s=P+sP'_{H_0}\) in the prescribed coordinates, without an extra decay assumption on \(\tau\). Similarly, \[g_s-g-s(h_0+2a\phi g)=O_1(s^2r^{-4}),\] whose energy flux vanishes. The finite flux of \(\phi\) and the scalar prototype thus give \(E_s=E+sE'_H\). In particular both charge functions exist and are differentiable with the derivatives computed in Lemma 89. The path and its first two parameter derivatives are uniformly bounded relative to \(g\), as required by Proposition 84. This verifies all claims and all hypotheses of the numerical theorem. ◻

Proof. Compactify \(\mathcal A\) by one end point. If \(\mathcal A\) is already compact, adjoin an isolated point; if it is empty, use just that point. The ball fibers are compact and the base has one end, so this is a compact Hausdorff space, denoted \(\overline{\mathcal A}\). By Lemma 89, \(C'_H/w\) extends continuously to it with value \(6a\) at the added point. On the disjoint compact union \[\mathcal K=\overline{\mathcal A}\sqcup S\sqcup\mathcal T\] consider the linear map to \(C(\mathcal K)\) whose three components are \[ H\longmapsto \left(\frac{C'_H}{w},\ -\theta'_H,\ c a'_T(h)-6\omega\mathcal E'(H)\right). \tag{266}\]

Its image does not meet the open cone of functions strictly positive everywhere. Otherwise its value at the added point gives \(a>0\), and Lemma 90 supplies actual feasible data (262). Let \(a(s)\) be their enclosing infimum. The area derivative formula in Proposition 84 gives \[a'(0+)=\min_{T\in\mathcal T}a'_T(h).\] Strict positivity of the third component in (266), and compactness of \(\mathcal T\), imply \[\left.\frac{\,\mathrm d}{\,\mathrm ds}\right|_{0+} 6\omega\bigl(\mathcal E(g_s,K_s)-\mathfrak m(a(s))\bigr) =6\omega\mathcal E'(H)-c\min_{T\in\mathcal T}a'_T(h)<0.\] The expression starts at zero and is nonnegative by (248), a contradiction.

Hahn–Banach separation of a linear subspace from this open convex cone gives a nonzero positive continuous functional on \(C(\mathcal K)\) annihilating the image. The subspace need not be closed: its closure is still disjoint from the open cone. The Riesz representation theorem identifies the functional with a nonzero finite positive measure on \(\mathcal K\).

Its mass on \(\mathcal T\) is positive. If that mass were zero, testing \(Q\) would give a strictly positive value on every remaining component: \(C'_Q/w\ge6\) on \(\mathcal A\), the added-point value is six, and \(-\theta'_Q=3\) on \(S\). This would contradict annihilation. Normalize the mass on \(\mathcal T\) to one and call the resulting measure there \(\varpi\). Call its restriction to \(S\) \(\eta\). On \(\mathcal A\), divide the remaining measure by the positive function \(w\) to obtain \(\Lambda\); this is locally finite and has finite weighted mass. If the normalized added-point mass is \(\lambda_\infty\), put \(z=6\lambda_\infty\). Rearranging annihilation of (266) is exactly (265). ◻

Proof. Choose a relatively compact open neighborhood of \(\operatorname{supp}\ell\) in \(\operatorname{int}\Omega\) and apply Lemma 14 there in spatial dimension four, with \(s=\ell\) and \(\delta=1/j\). The present densities already satisfy \(2\mu=R_g+(\mathop{\mathrm{tr}}_gK)^2-|K|_g^2\) and \(J=\operatorname{div}_g(K-(\mathop{\mathrm{tr}}_gK)g)\), with \(\mu\ge|J|_g\). The smoothness and future-normal convention (237) are precisely those of the lemma. Equation (249) defines the same constraint \(C=2(\mu+J(v))\) and the same isometric ray transport, so its derivative is the derivative \(\mathfrak C'\) in that lemma.

The resulting pairs have metric component \(2\ell K\) and fixed compact support in the open exterior. They therefore belong to \(\mathcal V_0\) by Definition 88, and the lemma gives the asserted uniform convergence, including active rays at which \(\mu=0\). Their charge, boundary, and end-coefficient variations vanish because the support stays strictly inside the open exterior. ◻

Proof. Apply Equation (265) to the variations in Lemma 92. Their charge, boundary, and end-coefficient terms vanish, and \(\Lambda\) is finite over their fixed compact support. Since the metric component is \(2\ell K\) and \(c>0\), uniform convergence of the active constraint derivatives gives \[ \int_{\mathcal T}\int_T\ell\,\mathop{\mathrm{tr}}_T K\,\,\mathrm dA_g\,\,\mathrm d\varpi(T)=0 \qquad\text{for every }\ell\in C^\infty_c(\operatorname{int}\Omega). \tag{269}\]

Propositions 83 and 85 give exactly the geometry needed for Proposition 15: the full frontiers have area \(A\), lie in one compact set, have smooth minimal free parts and \(C^{1,1}\) contact graphs, and carry a continuous common tangent plane away from \(S\). The surface-area kernel is measurable by the tangent-plane measure topology. Their end sides are connected, and each entire smooth frontier is an admissible full cut; no component or contact area is discarded.

Definition 81 supplies connectedness and marginality of \(S\), outer area-minimality, and the wholly-interior outermostness condition in case (a) of Proposition 15. Hence almost every frontier equals \(S\). Its area-derivative conclusion gives Equation (267); substituting this into Equation (265) gives Equation (268). ◻

Proof. For a variation supported in the open exterior, every term of (280) except the constraint pairing vanishes. For a pure tensor variation \(p=\dot K\), that pairing is \[2\bigl\langle u,\tau\mathop{\mathrm{tr}}_g p-K:p\bigr\rangle +2\bigl\langle X,\nabla^j(p_{ij}-(\mathop{\mathrm{tr}}_g p)g_{ij})\bigr\rangle.\] Its distributional adjoint equation is \[ u(\tau g-K)-\nabla_{(i}X_{j)}+(\operatorname{div}X)g=0. \tag{281}\] Taking the trace in four dimensions gives \(3u\tau+3\operatorname{div}X=0\). Substitution proves (271).

For the conformal variation \((h,p)=(2\psi g,\psi K)\), the constraint transformation already established in Section 16 gives, on active rays, \[C'_H(x,v)=6\{-\Delta\psi+K(\nabla\psi,v)\}.\] Its adjoint is (272). This calculation is valid for measure moments, so both equations hold in distributions before any regularity assertion.

Diverge (271) and substitute \(\operatorname{div}X=-u\tau\). Commuting covariant derivatives gives a Laplace equation for \(X\) whose right side consists of smooth coefficients times \(u,X,\nabla u\). In (272), the right side consists of smooth coefficients times \(X,\nabla X\). Thus the coupled system has diagonal Laplace principal part on the scalar and vector bundles; all couplings have order at most one. Distributional interior elliptic regularity, iterated with the smooth coefficients, makes both densities smooth. The measure inequalities and complementarity now give the asserted pointwise statements.

It remains to verify the full metric equation, including its matter term. We may now work with smooth \(u,X\). For a pure metric variation \(h\), keep covariant \(K\), scalar \(u\), and contravariant \(X\) fixed. The constraint pairing is the variation of \[ \int\bigl[u(R+\tau^2-|K|^2) +2X^i\nabla^j(K_{ij}-\tau g_{ij})\bigr]\,\mathrm dV_g \tag{282}\] minus \(\int h(X,J^\sharp)\,\mathrm dV_g\). The latter is precisely the contribution of \(\dot v=-\tfrac12h^\sharp v\). In using the varying volume form in (282), no term has been added: its original integrand is \(2u\mu+2J(X)=0\) pointwise.

For completeness, put \(\pi^{ij}=K^{ij}-\tau g^{ij}\). Integrating the shift term once by parts gives \(-\int\pi^{ij}(\mathcal L_Xg)_{ij}\,\mathrm dV_g\), with unchanged boundary terms outside the variation support. At fixed covariant \(K\), \[\delta\pi^{ij} =-h^i{}_aK^{aj}-h^j{}_aK^{ia} +(K:h)g^{ij}+\tau h^{ij}.\] Variation of the last integral, followed by integration of the term \(-\pi^{ij}(\mathcal L_Xh)_{ij}\), has coefficient \[ \mathcal L_XK-(\operatorname{div}X)K -\{X(\tau)+K^{ab}\nabla_aX_b\}g. \tag{283}\] One can see the cancellations directly: if \(B=\mathcal L_Xg\), the variation of \(\pi\) contributes \(2\mathop{\mathrm{Sym}}(KB)-(\mathop{\mathrm{tr}}B)K-\tau B\); integration of \(\mathcal L_Xh\) contributes \((\mathcal L_X\pi)^{ij}+(\operatorname{div}X)\pi^{ij}\) with indices lowered; and the volume variation contributes \(-\tfrac12(\pi:B)g\). Together they are (283).

The scalar and quadratic terms in (282) have coefficient \[\nabla^2u-(\Delta u)g-u\mathop{\mathrm{Ric}}_g +2u(K^2-\tau K)+u\mu g.\] Adding (283) and the test-vector correction therefore gives \[\begin{align*} 0={}&\nabla^2u-(\Delta u)g-u\mathop{\mathrm{Ric}}_g+2u(K^2-\tau K)+u\mu g \\ &+\mathcal L_XK-(\operatorname{div}X)K -\{X(\tau)+K^{ab}\nabla_aX_b\}g-X_{(i}J_{j)}. \tag{284}\end{align*}\] Equation (271) gives \(K^{ab}\nabla_aX_b=-u|K|^2\). Moreover, using \(\nabla^jK_{ij}=J_i+\nabla_i\tau\) in (272) gives \[ \Delta u+X(\tau) =u|K|^2-J(X)=u(|K|^2+\mu). \tag{285}\] Together with \(\operatorname{div}X=-u\tau\), these identities cancel the scalar multiples of \(g\) in (284) and give (273). ◻

Proof. Put \(B_{ij}=-uK_{ij}\). Differentiate \(\nabla_{(i}X_{j)}=B_{ij}\) in three arrangements, add two of the resulting equations, and subtract the third. Commuting derivatives gives \[\begin{align*} \nabla_i\nabla_jX_k &=\nabla_iB_{jk}+\nabla_jB_{ik}-\nabla_kB_{ij} +\mathcal R_{ijk}(X),\tag{286}\\ \mathcal R_{ijk}(X) &=-\tfrac12\bigl( ([\nabla_j,\nabla_i]X)_k +([\nabla_i,\nabla_k]X)_j +([\nabla_j,\nabla_k]X)_i\bigr). \end{align*}\] Every commutator is a curvature contraction with \(X\). Thus this equation uses only \(K\nabla u\), \(u\nabla K\), and curvature times \(X\). Expanding the Lie derivative in (273) gives \[\begin{align*} \nabla_i\nabla_ju ={}&u(\mathop{\mathrm{Ric}}_{g,ij}+\tau K_{ij}-2(K^2)_{ij})-X^a\nabla_aK_{ij} \\ &-K_{aj}\nabla_iX^a-K_{ia}\nabla_jX^a+X_{(i}J_{j)}. \tag{287}\end{align*}\] Consequently \[\mathcal J=(u,X,\nabla u,\nabla X)\] obeys a homogeneous linear first-order system, with smooth coefficients built from \(g,K\), their required derivatives, and \(J\). Along any smooth curve, this is a linear ODE for \(\mathcal J\). Uniqueness along curves proves the first-jet assertion.

In a compact collar of \(S\), start on an interior collar section and solve this ODE along the normal segments down to \(S\). Its coefficients are smooth up to the one-sided boundary. Smooth dependence on the initial point and on the normal parameter provides the smooth extension, agreeing with the original solution for positive collar parameter. The pointwise inequalities and complementarity extend by continuity.

The original moments could still have parts supported on \(S\); the interior regularity alone does not exclude them. Let \(s\) be geodesic collar distance, let \(f\in C^\infty(S)\) be arbitrary and extended constantly along the normal segments, and take a collar cutoff \(\chi\) equal to one near \(S\). Choose the compact pure metric variation \[ h=\frac{s^2\chi(s)f}{6}\bigl(g-\mathrm ds\otimes\mathrm ds\bigr), \qquad p=0. \tag{288}\] This is one of the allowed smooth compact variations, with strict-direction coefficient \(a=0\). Its value and first jet vanish on \(S\). The trace along the three-dimensional collar slices is \(s^2\chi f/2\), so its second normal derivative at \(S\) is \(f\). At \(S\), the scalar variation \[R'_g(h)=\nabla^i\nabla^jh_{ij}-\Delta(\mathop{\mathrm{tr}}_gh)-\mathop{\mathrm{Ric}}_g:h\] therefore equals \(-f\): the only nonzero second jets are normal-normal derivatives of tangential components, whereas \(h_{ss}=h_{sA}=0\). The variation of \(\tau^2-|K|^2\) uses only \(h\), that of \(J\) at fixed covariant \(K\) uses only \(h,\nabla h\), and the test-vector correction uses only \(h\). Consequently \(C'_H|_S=-f\), independently of the active ray \(v\).

The area derivative vanishes because \(h|_S=0\); the expansion derivative vanishes because \(h|_S=\nabla h|_S=p|_S=0\); and the charge derivative vanishes by compact support. Integrating the smooth interior densities by parts gives zero as well: the adjoint vanishes, and its scalar boundary terms involve only \(h,\nabla h\), while its momentum boundary terms involve only \(h\). This is ordinary integration up to \(S\) using the one-sided smooth fields, not distributional differentiation of a zero extension across \(S\). Thus (280) says that the scalar boundary measure annihilates every \(f\in C^\infty(S)\), and that measure is zero. The inequality \(|X|_g\le u\) as measures eliminates the vector boundary part. Moreover, since \(u\) is the pushforward of the positive measure \(\Lambda\), the entire measure \(\Lambda\) has zero mass over \(S\), not merely zero first moment there. ◻

Proof. Fix a sufficiently large coordinate sphere of radius \(R\) and put \[D(r)=\sup_{\omega\in S^3}|\partial Y(r\omega)|, \qquad \mathcal M(r)=\sup_{R\le t\le r}D(t).\] Along each ray, \(|Y(t\omega)|\le C+\int_R^tD(a)\,\mathrm da\le C+t\mathcal M(t)\), uniformly in \(\omega\). Integrating \(\partial_r(\partial Y)\) using (289), and then taking the supremum over directions and radii at most \(r\), gives \[\mathcal M(r)\le C+C\int_R^r t^{-1-q_0}\mathcal M(t)\,\mathrm dt.\] The kernel is integrable. Gronwall therefore bounds \(\mathcal M\) uniformly, so \(\partial Y=O(1)\) and \(Y=O(r)\). Equation (289) now gives \(\partial^2Y=O(r^{-1-q_0})\). Each coordinate derivative has a radial limit \(L(\omega)\), with error \(O(r^{-q_0})\). Any two points on the sphere of radius \(r\) can be joined by a spherical arc of length at most \(\pi r\); integrating the Hessian along this arc gives an \(O(r^{-q_0})\) difference between their gradients. Letting \(r\to\infty\) shows that \(L(\omega)\) is one constant matrix \(L\). Radial integration of \(\partial Y-L\) then gives \[\partial Y=L+O(r^{-q_0}),\qquad Y=Lx+O(1),\] where boundedness of the second error uses \(q_0>1\). Since \(u\ge0\) in every direction, the linear part of \(u\) vanishes. Uniform comparability of \(g\) with the Euclidean metric and \(|X|_g\le u\) then force the linear part of \(X\) to vanish as well. Thus \(Y=O(1)\) and \(\partial Y=O(r^{-q_0})\). Substitution in (289), using \(q_0>1\), improves the Hessian bound to \(O(r^{-2-q_0})\). Integrating each gradient component radially toward its zero limit gives \(\partial Y=O(r^{-1-q_0})\). A further radial integration gives \(Y=B(\omega)+O(r^{-q_0})\); comparison of values along the same spherical arcs now gives an \(O(r^{-q_0})\) difference, so \(B\) too is independent of direction. This proves the asserted constant limit and both differentiated estimates. ◻

Proof. Let \(Y\) denote all coordinate components of the pair \((u,X)\). The prolonged system and the original \(O_2\) metric and \(O_1\) tensor decay imply Equation (289). Indeed curvature, \(\nabla K\), and \(J\) have order \(-2-q_0\), while \(K\) and the Christoffel symbols have order \(-1-q_0\). Equations (286) and (287) use no higher background derivatives. Converting their covariant Hessians to coordinate Hessians adds terms with coefficients \(\Gamma\), \(\partial\Gamma\), and \(\Gamma^2\), controlled by the stated \(O_2\) metric decay.

Lemma 97 now gives constants with \[ (u,X)=(u_\infty,X_\infty)+O_2(r^{-q_0}). \tag{290}\]

Use the five end prototypes of Definition 88, whose charge derivatives are computed in Lemma 89, in (280). Their strict-direction coefficient is \(a=0\), so the term \(za\) is absent; their boundary and area terms vanish because the prototypes are supported away from \(S\). Integration by parts using the interior adjoint equations leaves only the outer boundary flux. By (290), its limit is \[ 6\omega\bigl(u_\infty E'+X_\infty^iP'_i\bigr). \tag{291}\] For clarity, the leading integrand is \[u_\infty(\partial_jh_{ij}-\partial_i h_{jj}) +2X_\infty^j\bigl(p_{ij}-(\mathop{\mathrm{tr}}_\delta p)\delta_{ij}\bigr).\] This is exactly (291) with the energy and momentum normalizations of Equations (239)–(240). Every other boundary term contains a decaying background or adjoint coefficient, or its derivative, paired with \(h=O_2(r^{-2})\) or \(p=O_1(r^{-3})\) at the corresponding order. Multiplication by the sphere area \(O(r^3)\) makes each such flux tend to zero. The volume pairing converges as well: the prototype constraint derivatives are \(O(r^{-4-q})\) and the moment densities are bounded.

The scalar prototype \(h=r^{-2}\delta\), \(p=0\), has \((E',P')=(1,0)\). For the tensor prototype indexed by a constant vector \(e\), its Euclidean trace reversal contracts with the sphere normal to \(r^{-3}e\), and thus has \((E',P')=(0,e/3)\). These independent charges identify \((u_\infty,X_\infty)=(b^0,b)\) from (280) and (291). This proves (274).

If \(u\) vanished at an interior point, its nonnegativity would give \(\mathrm du=0\) there. Taylor’s theorem and \(|X|\le u\) would give \(X=0\) and \(\nabla X=0\) at the same point. The whole first jet would vanish, contradicting Lemma 96 and \(u_\infty=b^0>0\). ◻

Proof. Take arbitrary compact pure tensor variations up to \(S\). The actual outward normal of the exterior domain at its inner boundary is \(-\nu\). Integration by parts in (280) therefore gives \[-2\int_S\bigl[p(X,\nu)-(\mathop{\mathrm{tr}}_gp)X_\nu\bigr]\,\mathrm dA_g -\int_S(\mathop{\mathrm{tr}}_Sp)\,\mathrm d\eta=0.\] Arbitrary mixed components \(p_{\nu A}\) imply \(X_{TS}=0\), and arbitrary tangential trace implies \(\mathrm d\eta=2X_\nu\,\mathrm dA_g\).

Now take \((h,p)=(2\psi g,\psi K)\) with arbitrary compact support up to \(S\). At a MOTS, its expansion derivative is \(3\partial_\nu\psi\), while its area derivative is \(3\int_S\psi\,\mathrm dA_g\). Integrating its constraint pairing with (272) gives \[\begin{align*} -3c\int_S\psi\,\mathrm dA_g ={}&6\int_S\left[u\partial_\nu\psi -\{\partial_\nu u+K(X,\nu)\}\psi\right]\,\mathrm dA_g -3\int_S\partial_\nu\psi\,\mathrm d\eta. \end{align*}\] The boundary value and normal derivative of \(\psi\) can be prescribed independently. Their coefficients give \(\mathrm d\eta=2u\,\mathrm dA_g\) and \(\partial_\nu u+K(X,\nu)=c/2\). Together with the tensor-variation conclusions these are exactly (275). ◻

Proof. Let \(L_S\) be the expansion variation at \(S\) under normal motion with speed \(s\nu\). Its principal part is \(-\Delta_S\). More explicitly, in a geodesic normal extension of \(\nu\), if \(\mathrm{II}\) is the second fundamental form of \(S\) in \((\Omega,g)\), \[L_Ss=-\Delta_Ss+2K(\nu,\nabla_Ss) +\left[-|\mathrm{II}|^2-\mathop{\mathrm{Ric}}_g(\nu,\nu) +\mathop{\mathrm{tr}}_{TS}(\nabla_\nu K)\right]s.\] Only the smoothness of the lower coefficients and the displayed principal part will be needed.

Extend \(s\nu\) to a smooth compactly supported vector field \(Y_1\) up to the boundary, and use \(H=(\mathcal L_{Y_1}g,\mathcal L_{Y_1}K)\) in (280). These are legitimate one-sided tensor variations; one may compute their jets using any smooth extension across \(S\). Their area and expansion derivatives are respectively \(\int_SHs\,\mathrm dA_g\) and \(L_Ss\).

We check the active-ray derivative with the specified vector identification. At a fixed interior point put \(\mathcal A(v)=\nabla_vY_1\), so \(h^\sharp=\mathcal A+\mathcal A^*\), where the adjoint uses \(g\). Let \(\phi_t\) be the local flow of \(Y_1\) and let \(v_t\) be the isometrically identified test vector for \(g_t=\phi_t^*g\). Naturality gives \[C_{\phi_t^*g,\phi_t^*K}(x,v_t) =C_{g,K}(\phi_t x,\mathrm d\phi_t(v_t)).\] At \(t=0\), the covariant velocity of the vector on the right, relative to parallel transport along \(\phi_t x\), is \[\mathcal A(v)+\dot v =\mathcal A(v)-\tfrac12(\mathcal A+\mathcal A^*)v =\tfrac12(\mathcal A-\mathcal A^*)v.\] It is perpendicular to \(v\). The base derivative of \(C\), with \(v\) parallel transported, vanishes at an interior active ray because it differentiates a nonnegative function at a two-sided interior minimum. The remaining vector derivative is \(2J(\tfrac12(\mathcal A-\mathcal A^*)v)\), also zero: at an active unit ray \(J=-\mu v^\flat\), and at an active ray with \(|v|<1\) we have \(\mu=J=0\). Therefore \(C'_H=0\) at every interior active ray.

The vector field \(Y_1\) need not preserve \(S\): its compact Lie derivatives are permitted tensor variations, independently of feasibility of a flow on the whole exterior. No normal derivative at a one-sided boundary minimum is being set equal to zero. Such boundary rays contribute no integral because \(\Lambda\) has zero mass over \(S\) by Lemma 96. Thus the entire constraint pairing vanishes. The multiplier identity and \(\mathrm d\eta=2u\,\mathrm dA_g\) now read \[-c\int_S Hs\,\mathrm dA_g=-2\int_SuL_Ss\,\mathrm dA_g,\] and hence give \[ 2L_S^*u=cH. \tag{292}\] Here the adjoint is with respect to \(\mathrm dA_g\) on the closed manifold \(S\).

Outer area minimization, tested against every nonnegative outward normal speed, gives \(H\ge0\) pointwise. Hence \(L_S^*u\ge0\) and \(u\ge0\) on \(S\). Increase the zeroth coefficient of \(L_S^*\) by a constant until that coefficient is nonnegative. The inequality remains valid because \(u\ge0\). The strong minimum principle for this operator with principal part \(-\Delta_S\) shows that a zero of \(u\) forces \(u\) to vanish on the connected surface \(S\). Otherwise it is everywhere positive. ◻

Proof. The interior positivity of \(u\) makes (276) a smooth Lorentzian metric, with future normal \(n=u^{-1}(\partial_t-X)\) after choosing the indicated time orientation. In the convention of the problem, the second fundamental form of a lapse-shift metric is \[\frac1{2u}\bigl(\partial_tg-\mathcal L_Xg\bigr).\] Its coefficients here are independent of \(t\), and (271) identifies this tensor with the original \(K\). The vector field \(\xi=\partial_t\) is Killing by construction.

We record the curvature computation to fix both the matter term and the sign convention. The Gauss and normal-variation identities for a general lapse-shift metric, with positive second fundamental form convention, are \[\begin{align*} \mathop{\mathrm{Ric}}_{\mathbf g,ij} &=\mathop{\mathrm{Ric}}_{g,ij}+\tau K_{ij}-2(K^2)_{ij} +u^{-1}\bigl(\partial_tK_{ij} -(\mathcal L_XK)_{ij}-(\nabla^2u)_{ij}\bigr),\\ \mathop{\mathrm{Ric}}_{\mathbf g}(n,n) &=-u^{-1}(\partial_t\tau-X\tau)-|K|^2+u^{-1}\Delta u. \end{align*}\] Tracing these equations uses \[g^{ij}\bigl(\partial_tK_{ij}-(\mathcal L_XK)_{ij}\bigr) =\partial_t\tau-X\tau+2u|K|^2.\] In the stationary case this yields \[\begin{align*} \mathop{\mathrm{Ric}}_{\mathbf g,ij} &=\mathop{\mathrm{Ric}}_{g,ij}+\tau K_{ij}-2(K^2)_{ij} -u^{-1}\bigl((\mathcal L_XK)_{ij}+(\nabla^2u)_{ij}\bigr), \tag{293}\\ R_{\mathbf g} &=R_g+\tau^2+|K|^2-2u^{-1}(\Delta u+X\tau). \tag{294}\end{align*}\] Equation (285) and complementarity show that \[R_{\mathbf g}=2\mu+\frac{2J(X)}u=0.\] Writing \(\mathbf G\) for the Einstein tensor, the constraints and (273) consequently give \[ \mathbf G(n,n)=\mu,\qquad \mathbf G(n,e_i)=J_i,\qquad u\mathbf G_{ij}=-X_{(i}J_{j)}. \tag{295}\] These identities can also be compared with the transversal Killing construction in (Beig and Chruściel 1997); its momentum symbol has the opposite sign to the one used here. The null-fluid structure of a modified constraint adjoint is discussed in (Huang and Lee 2024, sec. 6.1). We use the displayed direct calculation, in four spatial dimensions, rather than importing a stationarity or dimensional-transfer theorem.

If \(\mu=0\), the dominant energy condition gives \(J=0\), so all entries in (295) vanish. If \(\mu>0\), the inequalities \[0=u\mu+J(X)\ge u\mu-|J||X|\ge0\] force \(|J|=\mu\), \(|X|=u\), and \(J=-\mu X^\flat/u\). Since \(\mathbf g(\xi,n)=-u\) and \(\mathbf g(\xi,e_i)=X_i\), the tensor with components (295) is exactly \[ \mathbf G=\frac\mu{u^2}\,\xi^\flat\otimes\xi^\flat \quad\hbox{where }\mu>0. \tag{296}\] The Killing one-form in this equation is the spacetime one-form. It is null wherever the displayed tensor is nonzero. Thus \(\mathbf G(\xi,\cdot)=0\) everywhere. Since \(R_{\mathbf g}=0\), this is the first assertion of (277). If \(N>0\), then \(|X|<u\), so complementarity forces \(\mu=J=0\); all the Ricci components vanish. This proves the second assertion. An internal null region has not been excluded or assumed vacuum.

It remains to examine the original boundary and the end. Equation (274) gives \(N\to(b^0)^2-|b|^2=1\). At \(S\), Equation (275) gives \(X=u\nu\), so \(N=0\) and all tangential derivatives of \(N\) vanish. If \(u>0\) on \(S\), Equation (271) gives \(\langle\nabla_\nu X,\nu\rangle=-uK(\nu,\nu)\). Hence \[\partial_\nu N =2u\bigl(\partial_\nu u+uK(\nu,\nu)\bigr)=2u\kappa,\] which is (278). Compactness of \(S\) supplies a full collar on which \(N>0\).

In the other case, \(u|_S=0\) and \(X|_S=0\). Tangential derivatives of \(X\) vanish. The normal-normal and normal-tangential components of (271) then give \(\nabla_\nu X=0\) on \(S\), so every first derivative of \(X\) vanishes there. Smooth one-sided division by normal distance gives \(u=sd\) and \(X=s^2Y_2\). The boundary condition gives \(d|_S=\partial_\nu u|_S=\kappa>0\). This is (279), and once again compactness gives a full collar with \(N>0\). Together with positivity far out on the unique end, these collars place all interior zeros of \(N\) in an interior compact set. ◻

Proof of Theorem 94. Combine Lemmas 95, 96, 98, 99, 100, and 101. ◻

Proof. We give the coordinate and expansion arguments, since the initial falloff alone would not justify the compactification below.

Extend \(\gamma\) from a sufficiently distant region to a smooth metric on \(\R^4\), using a cutoff interpolation with \(\delta\). Write its Laplacian as \[\Delta_\gamma =\Delta_\delta+D^{ij}\partial_i\partial_j+D^i\partial_i.\] By making the interpolation sufficiently far out, the scaled \(C^{0,\alpha}\) norms of \(D^{ij}\) and \((1+r)D^i\) are as small as desired, for any fixed \(\alpha\in(0,1)\). Here a scaled norm means the ordinary norm after rescaling each annulus to fixed size. The initial differentiated falloff gives the needed Hölder estimates: second derivatives bound the scaled Hölder seminorm of first derivatives. Moreover \(D^i=O(r^{-1-q_0})\) in these scaled norms.

Put \(a=1-q_0\in(-1,0)\). On all of \(\R^4\), with weight \(1+r\), the Euclidean Newton operator maps scaled \(C^{0,\alpha}\) functions of order \(a-2\) to scaled \(C^{2,\alpha}\) functions of order \(a\). For completeness, the zeroth-order estimate follows from the kernel \(C|x-y|^{-2}\) by splitting the integral into \(|y|<r/2\), a comparable annulus, and \(|y|>2r\); convergence uses \(-2<a<0\). Interior elliptic estimates after rescaling give the two derivative and Hölder bounds. The small coefficient norms therefore make the map obtained from \[\Delta_\delta v^i =-D^i-D^{ab}\partial_a\partial_b v^i-D^a\partial_a v^i\] a contraction in that weighted space. Its solution satisfies \(v=O(r^{1-q_0})\) with the stated scaled \(C^{2,\alpha}\) bounds, and \(x^i+v^i\) are \(\gamma\)-harmonic coordinates on the far end. They are indeed coordinates there: their first derivative tends to the identity, and a cutoff of the correction can be chosen to have globally small first derivative. Local elliptic regularity makes the coordinates smooth wherever the original metric is smooth.

In the harmonic chart, \(\gamma-\delta\) and \(U\) initially have order \(-q_0\) in scaled \(C^{1,\alpha}\): one derivative of the transformed metric uses two derivatives of the coordinate correction. We do not assume a weighted second-derivative estimate at this stage. The chart and the fields are nevertheless smooth locally, so Equations (305) take the coupled elliptic form \[ -\frac12\gamma^{ab}\partial_a\partial_b\gamma_{ij} +Q_{ij}(\gamma,\partial\gamma) =\frac32 U_iU_j, \qquad \gamma^{ab}\partial_a\partial_bU=0, \tag{308}\] where \(Q\) is quadratic in first derivatives, with smooth coefficients depending on \(\gamma\). On an annulus rescaled to unit size, the initial \(C^{1,\alpha}\) norms of \(\gamma-\delta\) and \(U\) are \(O(R^{-q_0})\). Both quadratic right sides have \(C^{0,\alpha}\) norm \(O(R^{-2q_0})\), and the principal coefficients are uniformly elliptic with bounded \(C^{1,\alpha}\) norm. Interior Schauder estimates on a smaller annulus therefore restore \(C^{2,\alpha}\) bounds of order \(R^{-q_0}\) for both unknowns. Differentiating this coupled system now inductively gives \(\gamma-\delta,U=O_k(r^{-q_0})\) for every fixed \(k\). Thus no higher derivative falloff is being assumed in the original coordinates. It follows in particular that \[ \Delta_\delta(\gamma-\delta),\ \Delta_\delta U =O_k(r^{-2-2q_0}). \tag{309}\]

We use the following elementary multipole fact in dimension four. If a decaying smooth function \(w\) on an end satisfies \[\Delta_\delta w=O_k(r^{-4-j-\eta}), \qquad j\in\{0,1,2,\ldots\},\quad 0<\eta<1,\] then \[ w=\sum_{d=0}^j\frac{P_d(x)}{r^{2+2d}} +O_k(r^{-2-j-\eta}), \tag{310}\] with \(P_d\) homogeneous harmonic of degree \(d\). To prove this, cut \(w\) off to the end and represent the result by the whole-space Newton potential of its Laplacian. The difference is an entire harmonic function tending to zero, hence vanishes. The source has convergent moments through degree \(j\). Taylor expansion of the kernel in the region \(|y|<r/2\) has remainder bounded by \(C r^{-3-j}|y|^{j+1}\); integration, and estimation of the complementary region after subtraction of the same moments, gives the error in (310). The local singularity of the Newton kernel is integrable. Rescaled elliptic estimates supply the differentiated error bounds. This also proves the statement simultaneously for each fixed finite number of derivatives.

Choose \(\epsilon\in(0,1)\) with \(4+\epsilon<2+2q_0\). Applying (310) first with \(j=0\) gives monopole expansions for \(\gamma-\delta\) and \(U\). Write the metric monopole as \(D_{ij}/r^2\). The harmonic-coordinate condition is \[\partial_j\bigl(\sqrt{\det\gamma}\,\gamma^{ji}\bigr)=0.\] Its leading homogeneous term implies \[\bigl(D-\tfrac12(\mathop{\mathrm{tr}}D)\mathop{\mathrm{Id}}\bigr)x=0 \quad\hbox{for every }x.\] Taking the trace in dimension four gives \(D=0\). This proves (306). Since \(U=O_k(r^{-2})\), its equation now improves to \(\Delta_\delta U=O_k(r^{-6-\epsilon})\). The case \(j=2\) of (310) gives (307), with the sign of its constant temporarily undetermined. In particular, the strict exponent in this source estimate excludes a logarithmic term at quadrupole order. The first use of (310) likewise accounts for the entire order \(r^{-2}\) tensor term, rather than only its spherical average. The metric equation in (308) has Euclidean source \(O_k(r^{-6})\) at this stage. The case \(j=1\) of (310), with any smaller positive exponent if necessary, gives \(\gamma-\delta=O_k(r^{-3})\). No expansion of the metric beyond this order is asserted or needed.

Finally \(-U\) is a positive \(\gamma\)-harmonic function. On a sufficiently distant region the function \(r^{-2}+r^{-2-\epsilon}\) is positive and subharmonic for \(\gamma\); its favorable Euclidean Laplacian dominates the metric error. A small positive multiple lies below \(-U\) on a fixed large sphere, and both tend to zero at infinity. The maximum principle therefore keeps it below \(-U\) throughout the region. Taking the limit of \(r^2(-U)\) proves \(M_1>0\). ◻