Proof. Let \(C_t\) be the compact region inside \(S_t\), including the filling. On a sufficiently distant end the smooth unit field \(Z=\nabla r/|\nabla r|_g\) satisfies \[ \operatorname{div}_g Z=H_{S_r}=\frac2r+O(r^{-1-q})>0. \tag{20}\] Increase \(R_0\) so this holds for \(r>R_0\) and \(F\subset\operatorname{int}C_{R_0}\).

For any bounded finite-perimeter set \(M\), the slicing and Gauss–Green formulas give, for almost every \(t>R_0\), \[\begin{align*} \int_{M\cap\{r>t\}}\operatorname{div}_g Z\,dV_g &=\int_{\partial^*M\cap\{r>t\}}g(Z,\nu_M)\,dA_g-b(t), \tag{21}\\ P_g(M\cap C_t)&=P_g(M;\{r<t\})+b(t), \tag{22}\end{align*}\] where \(b(t)=\int_{S_t}\mathbf1_M\,dA_g\) uses the common almost-everywhere trace. In Equation 21, cut off \(Z\) beyond the bounded set \(M\) if necessary. Since \(|Z|_g=1\), \[ P_g(M\cap C_t)\leq P_g(M)- \int_{M\cap\{r>t\}}\operatorname{div}_g Z\,dV_g. \tag{23}\] Clipping strictly decreases perimeter if the discarded volume is positive.

Fix \(R_1>R_0\) and clip each member of a minimizing sequence at a good radius in \((R_0,R_1)\). The resulting sets lie in \(C_{R_1}\), contain \(F\), and have bounded perimeter and volume. Compactness in the space \(BV\) of functions of bounded variation gives an \(L^1\) convergent subsequence, and lower semicontinuity gives a minimizer. The obstacle condition is closed under \(L^1\) convergence. These compactness and lower-semicontinuity results apply in finitely many coordinate charts on a larger compact neighborhood (Ambrosio et al. 2000, Theorem 3.23 and Proposition 3.6). Equation 23 equates this confined infimum with the unrestricted bounded-set infimum. Applying it to a minimizer at good radii decreasing to \(R_0\) proves that the minimizer has zero volume outside \(C_{R_0}\). Its perimeter support therefore lies strictly inside \(C_{R_1}\).

Uniform end bounds make the error in Equation 20 uniform, so the same construction works for the stated family. Only boundedness and the closedness in \(L^1\) of the containment condition were used for the obstacle; a large compact truncation is always a finite-perimeter competitor. This proves the last assertion. ◻

Proof. Let \(M\) minimize and let \(M'\) be a bounded finite-perimeter local competitor, without an obstacle restriction. The set \(M'\cup F\) is admissible. Perimeter submodularity gives \[ P_g(M)-P_g(M')\leq P_g(F)-P_g(M'\cap F). \tag{25}\] Choose a smooth compactly supported vector field \(V\) with \(|V|_g\leq1\) which agrees with the outward unit normal of \(F\) on \(\partial F\). The signed-distance collar and a cutoff provide such a field. Gauss–Green gives \[P_g(F)=\int_F\operatorname{div}_gV\,dV_g,\qquad \int_{M'\cap F}\operatorname{div}_gV\,dV_g\leq P_g(M'\cap F).\] Consequently, for \(\Lambda=\|\operatorname{div}_gV\|_\infty\), \[ P_g(M)-P_g(M')\leq \Lambda\operatorname{Vol}_g(F\setminus M') \leq\Lambda\operatorname{Vol}_g(M\mathbin\triangle M'). \tag{26}\] The unchanged perimeters outside the variation neighborhood cancel, so this is also a local inequality.

The exact obstacle regularity result we use is Huisken–Ilmanen’s Regularity Theorem 1.3(iii): in a smooth Riemannian domain of ambient dimension less than eight, a pure perimeter minimizer respecting a \(C^2\) obstacle has \(C^{1,1}\) boundary, smooth off contact (Huisken and Ilmanen 2001). Here \(M\) minimizes against every compactly supported competitor containing the open obstacle \(\operatorname{int}F\); the metric and obstacle are smooth, the dimension is three, and the bulk term is zero. Lemma 9 keeps its frontier away from the auxiliary truncation boundary. The regular representative is the closure of its measure-theoretic interior, whose boundary equals the perimeter support. Away from \(F\), both signs of every compact normal variation are allowed, so this smooth boundary has mean curvature zero. The stated Sobolev regularity follows from \(C^{1,1}\).

A compact embedded frontier has finitely many components: it has a finite cover by connected graph neighborhoods. If its complement had a bounded open component, that component’s boundary would be a nonempty union of frontier components. Adding it to \(M\) removes their positive area without adding a new frontier and contradicts minimality. There is only one unbounded complementary component, because \(\widehat\Omega\) has one end and \(M\) is bounded.

Given an admissible smooth exterior \(D\) with full boundary \(\Gamma\), the bounded set \(M_D=\widehat\Omega\setminus\operatorname{int}D\) contains \(F\) and has boundary exactly \(\Gamma\). Therefore \(P_g(M_D)=\operatorname{Area}_g(\Gamma)\), including every coincident portion on \(S\). In particular \(D=\Omega\) gives \(M_D=F\). This proves \(p_g(F)\leq a_g(S)\).

Conversely, flow a minimizing set \(M\) for small time \(s>0\) by a compactly supported smooth field pointing strictly outward on \(S\). The image \(M_s\) contains \(F\) in its interior. Indeed it contains the flowed obstacle, whose boundary in the signed-distance collar has moved strictly outside \(S\). The flow’s surface Jacobians converge uniformly to one, so \(P_g(M_s)\to P_g(M)\).

A compact \(C^1\) embedded boundary has a \(C^1\) defining function with nonzero differential near its boundary. Approximate this function in \(C^1\) by smooth functions; the implicit function theorem yields smooth embedded boundaries converging in \(C^1\), with the same enclosed side. Apply this to \(\partial M_s\). Its positive distance from \(F\) preserves enclosure under sufficiently close approximation, and surface Jacobians give area convergence. Fill bounded complementary components of each approximation. This deletes components and does not increase area. The closure of the remaining exterior is connected, contains the whole distant end, and has smooth compact two-sided intrinsic boundary in \(\operatorname{int}\Omega\). It is an admissible full cut. Letting the approximation error and then \(s\) tend to zero proves \(a_g(S)\leq p_g(F)\). ◻

Proof. At contact, write frontier and obstacle as graphs \(u\geq\psi\), with \(u\in C^{1,1}\) and \(\psi\) smooth. On the contact set \(Du=D\psi\) and \(D^2u=D^2\psi\) almost everywhere. The second statement follows from the fact that the weak derivative of a Sobolev function vanishes almost everywhere on a level set, applied to the Lipschitz functions \(D_i(u-\psi)\). Off contact \(u\) solves the minimal graph equation; at almost every contact point its second derivatives equal those of the minimal graph \(\psi\). Thus it solves the same equation weakly throughout the patch. The bounded gradient makes this a uniformly elliptic equation, so interior elliptic regularity and bootstrapping give smoothness. The strong maximum principle for the difference of the two smooth minimal graph equations gives local coincidence at each contact point. Contact is also closed, so connectedness extends the coincidence over the obstacle component. ◻

Proof. Two-dimensional area elements satisfy \(dA_{g_j}\geq c_jdA_g\), so \(a_{g_j}(S)\geq c_ja_g(S)\). For the upper bound, test with any fixed smooth cut of \(g\)-area less than \(a_g(S)+\varepsilon\), use compact convergence on that cut, and then let \(\varepsilon\downarrow0\). ◻

Proof. Confinement supplies one compact set \(C\) containing every minimizer. Uniform metric comparison on \(C\) gives \(\varepsilon_j\downarrow0\) such that for every set \(M'\) whose perimeter is supported there, \[ (1-\varepsilon_j)P_g(M')\leq P_{g_j}(M') \leq(1+\varepsilon_j)P_g(M'). \tag{29}\] Taking infima, using confinement for both metrics, proves \(p_{g_j}(F)\to p_g(F)\), and hence \(P_g(M_j)\to p_g(F)\). Compactness in \(BV\) and lower semicontinuity give a subsequential \(L^1\) limit \(M\) containing \(F\) and supported in \(C\), with \(P_g(M)\leq p_g(F)\). Therefore it minimizes, proving Equation 27.

The vector-measure continuity needed here can also be proved directly. Use the fixed metric \(g\), and write \[\lambda_j=\nu_j\,dA_g|_{\partial M_j},\qquad \lambda=\nu\,dA_g|_{\partial M},\] where \(\nu_j,\nu\) are outward unit normals. Gauss–Green and \(L^1\) convergence give weak convergence of these vector measures. Their total variation masses converge by Equation 27. Lower semicontinuity with nonnegative continuous weights and convergence of the total masses also imply \(|\lambda_j|\rightharpoonup|\lambda|\).

Approximate the measurable unit normal \(\nu\) in \(L^2(|\lambda|)\) by a continuous vector field \(V\) on \(C\) with \(|V|_g\leq1\). One obtains such approximations in finitely many bundle charts, joins them by a partition of unity, and projects onto the unit ball. They make \(\int(1-g(V,\nu))\,d|\lambda|\) arbitrarily small. For each such \(V\), \[\begin{align*} \int|\nu_j-V|_g^2\,d|\lambda_j| &\leq2\left(|\lambda_j|(C)-\int g(V,d\lambda_j)\right),\\ \limsup_j\int|\nu_j-V|_g^2\,d|\lambda_j| &\leq2\int(1-g(V,\nu))\,d|\lambda|. \end{align*}\] A continuous function of a unit normal extends continuously to the closed unit-ball bundle and is uniformly continuous on that compact bundle. The displayed estimate, Cauchy–Schwarz, and weak convergence of \(|\lambda_j|\) give convergence of its integrals, first at a fixed approximation \(V\), then as the approximation error tends to zero. Apply this to the even function \(\Psi(x,\nu^{\perp_g})\) to obtain Equation 28. This is the needed case of Reshetnyak’s Continuity Theorem (Ambrosio et al. 2000, Theorem 2.39).

The constant metric sequence gives compactness of \(\mathcal M_g(F)\). For a continuous symmetric tensor \(h\), the function \(\Psi(x,T)=\tfrac12\operatorname{tr}_T h\) is continuous, giving the last assertion. ◻

Proof. The determinant formula for a two-plane area gives, uniformly over planes in the compact truncation, \[dA_{g_s}=\left(1+\frac s2\operatorname{tr}_T h+o(|s|)\right)dA_g.\] For every finite-perimeter competitor \(M'\) in that truncation, \[ P_{g_s}(M')=P_g(M')+sL_h(M')+o(|s|)P_g(M'), \tag{31}\] with the same remainder modulus. By Proposition 13, the minimum of \(L_h\) exists.

Testing \(a(s)\) with any \(M\in\mathcal M_g(F)\) gives \[\limsup_{s\downarrow0}\frac{a(s)-a(0)}s\leq L_h(M).\] In the other direction, take a minimizer \(M_s\) for \(g_s\). Its \(g\)-perimeter is bounded and at least \(a(0)\), so Equation 31 gives \[\frac{a(s)-a(0)}s\geq L_h(M_s)+o(1)\qquad(s>0).\] Along a sequence realizing the lower limit, strict compactness gives a subsequence converging to a member of \(\mathcal M_g(F)\); continuity of \(L_h\) yields the opposite inequality. No differentiable choice of minimizer is needed. Since \(F\subset M\), the frontier does not enter \(\operatorname{int}F\), which proves the final assertion. ◻

Proof. If \(M_1,M_2\) minimize, their union and intersection contain \(F\). Submodularity gives \[2p_g(F)\leq P_g(M_1\cup M_2)+P_g(M_1\cap M_2) \leq P_g(M_1)+P_g(M_2)=2p_g(F).\] Thus both also minimize. If their frontier planes were distinct at an intersection, the tangent half-spaces would meet transversely. The boundary of their union or intersection would have a corner: its tangent cone would contain two distinct half-planes. This contradicts its \(C^1\) regularity from Proposition 10.

Here is the required closedness argument. On the common compact truncation, Equation 26 has a fixed constant. It yields uniform density bounds at every frontier point, for sufficiently small coordinate balls centered there: \[ \begin{gathered} c r^3\leq\operatorname{Vol}_g(M\cap B_r),\qquad c r^3\leq\operatorname{Vol}_g(B_r\setminus M),\\ c r^2\leq P_g(M;B_r)\leq C r^2. \end{gathered} \tag{33}\] For clarity, compare with \(M\setminus B_r\) and \(M\cup B_r\). For either phase volume \(v(r)\), the isoperimetric inequality for its intersection with the ball and these comparisons give \[v(r)^{2/3}\leq C\bigl(2v'(r)+\Lambda v(r)\bigr) \quad\text{for almost every }r.\] The coarea formula identifies \(v'\) with the slice area up to fixed coordinate constants. Since \(v(r)\leq Cr^3\), the last term is absorbed for small \(r\). Integrating \((v^{1/3})'\geq c\) gives \(v(r)\geq cr^3\); both phases are nonempty at every scale at a perimeter-support point. Relative isoperimetry gives the perimeter lower bound; the same competitors and the coordinate-sphere area bound give the upper bound. Smooth compact geometry makes these constants uniform.

Suppose now \(M_j\to M\) in \(\mathcal M_g(F)\) and \(x_j\in\partial M_j\) tend to \(x\). The two volume bounds pass to every fixed small ball about \(x\) under \(L^1\) convergence; hence \(x\in\partial M\). Huisken–Ilmanen’s Regularity Theorem 1.3(ii) gives uniform local \(C^{1,\alpha}\) estimates for a fixed \(0<\alpha\leq1/2\) (Huisken and Ilmanen 2001). Its dependencies are the distance to the ambient domain boundary, obstacle \(C^{1,\alpha}\) bounds, metric \(C^1\) bounds and positive definiteness, and the bulk bound, here zero. All are fixed in this application. Arzelà–Ascoli in local graph charts gives subsequential \(C^{1,\beta}\) convergence, \(0<\beta<\alpha\). The density bounds and the \(L^1\) limit identify the limiting sheet with \(\partial M\). Thus \[T_{x_j}\partial M_j\longrightarrow T_x\partial M.\] The incidence set \(\{(M,x,T_x\partial M):M\in\mathcal M_g(F),\,x\in\partial M\}\) is therefore compact. Its spatial projection \(\mathcal Z\) is compact, and the previously proved uniqueness of the plane at each limit point makes \(T\) continuous.

By Proposition 13, the surface measures depend weakly continuously on \(M\), so their average defines the finite Borel measure \(\eta\). Each surface uses the same plane \(T(x)\) at its points. Integrating this identity proves Equation 32. ◻

Proof. The scalar-curvature formula in dimension three is \(R_{u^4g}=u^{-5}(-8\Delta_g u+R_gu)\). Both \(\tau^2\) and \(|K|^2\) acquire the factor \(u^{-4}\). This proves the first identity. Write \(\pi=K-\tau g\) and \(f=\log u\). The transformed trace reversal is \(\pi'=u^2\pi\). In the divergence of this tensor use \[(\nabla'_A-\nabla_A)B =2\{df(A)B+df(B)A-g(A,B)\nabla f\}.\] Contracting the two covariant tensor slots gives \(u^2\operatorname{div}_{g'}(u^2\pi) =\operatorname{div}_g\pi+4K(\nabla f,\cdot)\); the terms containing \(\tau\,df\) cancel. This proves the momentum identity. Since \(dV_{g'}=u^6dV_g\), fixed flux forms give \(X_i'=u^{-6}X_i\). In an orthonormal frame their components therefore scale by \(u^{-4}\), whereas \(u^{-2}Y\) has the same orthonormal components as \(Y\). The electromagnetic identity and Equation 36 follow. Its last term is nonnegative for \(u\geq1\), and \(K(\nabla u,Y)\geq-|K||du|\) proves Equation 37. Finally \(H'=u^{-2}(H+4\partial_\nu\log u)\) and \(\operatorname{tr}'_S K'=u^{-2}\operatorname{tr}_S K\). ◻

Proof. Indices in the following calculation are raised with \(\eta\). Define \[ \begin{split} \mathcal F_{lab}={}&\partial_a h_{lb}-\partial_l h_{ab} +\eta_{lb}(\partial^d h_{ad}-\partial_a\operatorname{tr}_\eta h)\\ &-\eta_{ab}(\partial^d h_{ld}-\partial_l\operatorname{tr}_\eta h). \end{split} \tag{40}\] It is skew in \(l,a\). Expanding its divergence and collecting the two Hessians, wave operator, and trace terms of the linearized Ricci tensor gives \[ \partial^l\mathcal F_{lab}=2\operatorname{Ein}^{\mathrm{lin}}_{ab}(h) =O(r^{-2-2q_0})+O(r^{-4}). \tag{41}\] The last estimate follows because the difference between exact and linearized Einstein curvature is bounded by \(C(|h||\partial^2h|+|\partial h|^2)\).

In coordinates adapted to a background plane \(x^0=0\), direct substitution in Equation 40 gives \[ \begin{split} \mathcal F_{i00}&=\partial_jh_{ij}-\partial_ih_{jj},\\ \mathcal F_{i0k}&=2(K^{\mathrm{lin}}_{ik} -\delta_{ik}\operatorname{tr}K^{\mathrm{lin}}) +\partial_kh_{0i}-\delta_{ik}\partial_jh_{0j},\\ K^{\mathrm{lin}}_{ik} &=\tfrac12(\partial_0h_{ik}-\partial_ih_{0k} -\partial_kh_{0i}). \end{split} \tag{42}\] Here \(j\) is spatial. The last expression for \(K^{\mathrm{lin}}\) follows by expanding the future unit normal and its connection; in Gaussian normal coordinates it reduces to \(K^{\mathrm{lin}}=\tfrac12\partial_0h\). For fixed \(k\), the extra vector in the middle line has identically zero spatial divergence. Its flux through any exterior coordinate sphere is zero: extend \(h_{0i}\) smoothly across the ball and integrate that identity. The value of the extension cannot affect the flux. The nonlinear errors in \(K\) and in its trace reversal are \(O(r^{-1-2q_0})\) and hence have integrated flux \(O(R^{1-2q_0})\). Consequently the plane fluxes \(\int\mathcal F_{i0b}n^i\,dA/(16\pi)\) equal \((E,P_i)\).

For a fixed translation vector \(a\), regard \(\tfrac12\mathcal F_{lab}a^b dx^l\wedge dx^a\) as a two-form and take its background Lorentzian Hodge dual. With orientation \(dx^0\wedge dx^1\wedge dx^2\wedge dx^3\), its integral on an outward plane sphere is the flux just computed, with the common orientation chosen to give positive Schwarzschild energy. Equation 41 bounds its exterior derivative. Two large plane sections can be joined along a cylinder in \(r\asymp R\) with absolute three-dimensional coordinate volume \(O(R^3)\). Stokes’ Theorem bounds the difference of the integrals by \[C(R^{1-2q_0}+R^{-1})\longrightarrow0.\] Replacing the sections by spheres or uniformly comparable ellipsoids in their planes has the same estimate, using the intervening annular three-region. The limiting flux is thus independent of the plane for fixed \(a\). Tensor covariance under constant Lorentz changes proves Equation 39, including the plus sign of its spatial components. Substitution of \(b=(E/m,-P/m)\) gives \(m\).

For the last assertions, write the stationary metric in a rest coordinate system as \[G=\gamma_{ij}dy^i dy^j+2\beta_i dy^i dt-Wdt^2, \qquad W\longrightarrow1,\quad\beta=O_1(r^{-q_0}).\] The orbit metric is \(h_s=\gamma+W^{-1}\beta\otimes\beta\), so \(h_s-\gamma=O_1(r^{-2q_0})\). Its extra ADM flux is \(O(R^{1-2q_0})\) and vanishes. Finally let \(\gamma=\delta+k\) with \(k=O_2(r^{-q_0})\), and express its pullback under \(y=x+\zeta(x)\). Taylor expansion with one derivative gives \[\gamma_{\mathrm{new}} =\delta+k+2\operatorname{sym}D\zeta+\mathcal R, \qquad \mathcal R=O_1(r^{-2q_1}+r^{-q_0-q_1}).\] The linear ADM integrand of \(2\operatorname{sym}D\zeta\) is \(\Delta\zeta_i-\partial_i\operatorname{div}\zeta =\partial_j(\partial_j\zeta_i-\partial_i\zeta_j)\). Its flux is zero by a smooth interior extension and skew symmetry. The remainder flux is \(O(R^{1-2q_1}+R^{1-q_0-q_1})\to0\). This proves coordinate invariance in the precise class used here. A change of stationary time coordinate changes the threading one-form but leaves \(h_s\) itself unchanged. ◻

Proof. Choose smooth majorants and positive interpolations with the stated tails; the bound on \(K\) permits a constant radial majorant \(Cr^{-1-q}\) outside a compact set. When \(K\) is compactly supported, choose \(b\) with compact support instead. On \(\Omega_L=\{r\leq L\}\), including the compact core, impose the additional boundary condition \(\phi=0\) at \(r=L\). Continue from \(t=0\) to \(t=1\) in \[-\Delta_g\phi=t b\sqrt{|d\phi|_g^2+a^2}+w, \qquad \partial_\nu\phi=-1,\quad \phi|_{r=L}=0.\] The linearization has principal part \(-\Delta_g\) and drift \(-t b\nabla\phi/\sqrt{|d\phi|^2+a^2}\) of norm at most \(b\). The inner Neumann and outer Dirichlet boundaries are disjoint smooth components. The homogeneous mixed problem has zero kernel: a nonconstant positive maximum or negative minimum is excluded by the strong maximum principle and the boundary point lemma, and a constant is excluded by the Dirichlet condition. The mixed elliptic Fredholm index is zero, by continuation from the mixed Laplacian, so the linearization is invertible.

Here are bounds closing this continuation. On a fixed \(\Omega_L\), mixed \(W^{2,p}\) estimates, \(p>3\), and \(\sqrt{|d\phi|^2+a^2}\leq |d\phi|+a\) give \[\|\phi\|_{W^{2,p}} \leq C_L(1+\|d\phi\|_{L^p}+\|\phi\|_{L^p}) \leq \tfrac12\|\phi\|_{W^{2,p}} +C'_L(1+\|\phi\|_{L^p}).\] The second inequality is gradient interpolation, with its small parameter chosen after \(C_L\). If the supremum norms were unbounded, divide a sequence by its supremum norm. Compactness in \(C^1\) gives a function of supremum one solving \(-\Delta z=t b|dz|\), with zero mixed boundary data. This is a homogeneous equation with bounded drift, by taking the drift to be \(t b\nabla z/|dz|\) where \(dz\ne0\) and zero elsewhere. The same maximum principles give a contradiction. The resulting \(W^{2,p}\) bounds, followed by elliptic regularity, close the continuation. The solution is nonnegative: an interior negative minimum contradicts \(-\Delta\phi>0\), and a negative minimum at \(S\) contradicts the boundary point lemma, since the outward domain derivative is \(-\partial_\nu\phi=1\).

The bounds can be made independent of \(L\) on each compact subset. To see this without a bounded-domain constant depending on \(L\), put \(f(r)=r^{-1}(1-r^{-\delta_0})\) on a sufficiently distant tail. Direct differentiation gives \[ -\Delta_g f-|K|_g|df|_g =\delta_0(1+\delta_0)r^{-3-\delta_0} +O(r^{-3-q}). \tag{44}\] The same estimate holds with \(b\) in place of \(|K|\). Since \(ba=O(r^{-3-q})\) and \(\delta_0<q\), a sufficiently large multiple \(A f\) satisfies \(-\Delta_g(Af)\geq b\sqrt{|d(Af)|^2+a^2}+w\). Comparison on the exterior annulus, choosing its inner boundary value to dominate \(\phi\), gives \[ 0\leq\phi\leq C(1+\sup_{\Omega_{L_*}}\phi)r^{-1} \quad(r\geq L_*), \tag{45}\] with fixed \(L_*\). If the core supremum diverged along an exhaustion, normalize by it and apply the preceding local estimates, including the inner boundary estimates. A subsequence converges locally in \(C^1\) to a nonzero nonnegative solution of \(-\Delta z=b|dz|\), with \(\partial_\nu z=0\) and \(z=O(r^{-1})\). The tail estimate forces its positive maximum to occur in a compact set; the strong and boundary maximum principles again exclude it. Thus exhaustion gives a positive smooth solution of Equation 43.

Rescale each exterior annulus to unit size. The metric has uniformly bounded \(C^2\) coefficients, the scaled drift has norm \(O(r^{-q})\), and the already proved bound is \(\phi=O(r^{-1})\). Interior \(W^{2,p}\) estimates and then Schauder estimates give \(d\phi=O(r^{-2})\), \(d^2\phi=O(r^{-3})\). In the compact-\(b\) case the equation is the linear Poisson equation on a smooth symbol end; differentiating its rescaled form gives the estimates at every order. Its right side is integrable in either case. The divergence theorem therefore gives a finite limit of \(\int_{r=L}\partial_{\nu_L}\phi\,dA_g\). Its difference from the Euclidean gradient flux is \(O(L^{-q})\), so that limit also exists. For \(u=1+s\phi\), \(-\Delta u-|K||du|\geq s w\). Apply Equations 37 and 38. The scalar-curvature conclusion follows from its conformal formula. The energy increment is finite and tends to zero with \(s\), because its ADM flux contribution is \(-(s/2\pi)\lim\int\partial_{\nu_L}\phi\,dA_g\). ◻

Proof. Choose \(0<\delta_0<\min(q,1)\) and \(f\) as in Equation 44. Far enough out \(f'<0\) and \(-\Delta_g f-|K||df|\geq0\). Let \(\chi(r)\) be a smooth nondecreasing cutoff from zero to one supported in this valid range. Define \[\phi(r)=\int_r^\infty\chi(t)(-f'(t))\,dt\] there, and extend it constantly over the compact core. It is bounded, positive, smooth, and equals \(f\) far out. The radial differentiation formula shows \[-\Delta_g\phi-|K||d\phi| =\chi(-\Delta_g f-|K||df|)-\chi' f'|dr|_g^2\geq0.\] For each \(s\geq0\) set \(u_s=1+s\phi\) and use \((u_s^4g,u_s^2K)\) with the original forms. Charged DEC and all boundary signs persist. Completeness persists, since \(u_s\geq1\), and the full-cut infimum can only increase. The data still have decay exponent greater than \(1/2\). The new constraint terms have orders \(r^{-3-\delta_0}\) and \(r^{-3-q}\), so their absolute integrability follows from that of the old sources and Equation 35.

The asymptotic expansion \(u_s=1+s/r+o_2(r^{-1})\) gives, directly in the original chart, \[ E_s=E+2s,\qquad P_s=P,\qquad Q_{E,s}=Q_E, \qquad Q_{B,s}=Q_B. \tag{46}\] Indeed the leading new metric term is \(4s\delta_{ij}/r\) and its energy flux is \(32\pi s\); the factor \(1/(16\pi)\) yields \(2s\). The trace reversal is multiplied by \(u_s^2\), whose additional momentum flux is \(O(r^{-q})\) and tends to zero.

The assumed timelike numerical conclusion implies \(r_{A,s}\leq2\sqrt{(E+2s)^2-|P|^2}\). By Proposition 8, \(A_{\min,g}(S)>0\), while \(r_{A,s}\geq\sqrt{A_{\min,g}(S)/(4\pi)}\). If \(E\leq|P|\), take \(s\downarrow(|P|-E)/2\) from above. The right side tends to zero and contradicts this fixed positive lower bound. Thus \(E>|P|\). Applying the assumed timelike result to the original data finishes the proof. No common limiting boost is used. ◻

Proof. We first specify the ordinary divergence inverse used in the proof. For a mean-zero smooth compactly supported source in a ball, the support-preserving Bogovskiı̆ operator gives a smooth compactly supported vector field, with divergence equal to the source and a gain of one derivative in \(W^{\ell,p}\). The regularized integral operators and their support and Sobolev mapping properties are given in (Costabel and McIntosh 2010, Theorems 3.2 and 4.9, Corollary 4.11). To use balls on the annulus, cover the prescribed source support by finitely many balls whose closures lie inside the annulus and join them by a finite connected family of such balls. Split a source by a partition of unity. Along a spanning tree of overlapping balls transfer the integral of each leaf piece to its parent using a fixed unit-integral bump in the overlap. Removing leaves successively leaves each piece with integral zero, since the total integral is zero. Applying the ball operators and adding the results gives a divergence inverse with fixed compact support. The finite cutoffs and bumps preserve all the stated Sobolev estimates. Use successively larger fixed compact supports for successive applications below.

For a scalar \(f\) with zero affine moments solve \(\partial_iV_i=f\). Compact support and integration by parts imply \(\int V_i=-\int z_i f=0\). Solve \(\partial_j B_{ij}=V_i\) componentwise. Then the symmetric part \(P=(B+B^{\mathsf T})/2\) satisfies \(\partial_i\partial_jP_{ij}=f\). In dimension three the inverse of trace reversal is \[H=P-\tfrac12\operatorname{tr}P\,\delta, \qquad H-\operatorname{tr}H\,\delta=P.\] This gives the scalar solution and gains two derivatives.

For momentum, zero translation moments permit a componentwise solve \(\partial_jB_{ij}=V_i\). Put \(S_{ij}=(B_{ij}-B_{ji})/2\). The rotation moments give \[0=\int(z_iV_j-z_jV_i)=2\int S_{ij}.\] Solve \(\partial_kD_{ijk}=-S_{ij}\), choosing \(D\) skew in \(i,j\), and define \[C_{ijk}=D_{ijk}-D_{ikj}-D_{jki}.\] Directly using that skew symmetry gives \(C_{ijk}=-C_{ikj}\) and \((C_{ijk}-C_{jik})/2=D_{ijk}\). It follows that \(P_{ij}=B_{ij}+\partial_kC_{ijk}\) is symmetric and \(\partial_jP_{ij}=V_i\). Set \(k=P-\tfrac12\operatorname{tr}P\,\delta\). There is one net derivative of gain: the second divergence solve gains a derivative and the displayed correction differentiates it once. All supports remain compact in the prescribed larger subannulus. Integration against the adjoint kernels also proves the necessity of the stated moment conditions. ◻

Proof. Use \(z=x/R\) on \(A=\{1<|z|<4\}\) and regard the scaled fields as \(g(Rz)\) and \(RK(Rz)\). The constraints scale by \(R^2\). Write \(H=g(Rz)-g_*(Rz)\) and \(k=R(K(Rz)-K_*(Rz))\). These have \(C^2\times C^1\) norm \(O(R^{-\beta})\). If \(\lambda\) equals one near the inner boundary, zero near the outer boundary, and takes values in \([0,1]\), blend the metric and second form using \(\lambda\). The linearized constraint error beyond the blend of the two linear sources is \[f_R=[\mathcal L,\lambda]H,\qquad V_R=[\mathcal D,\lambda]k.\] If \(P(H)=H-\operatorname{tr}H\,\delta\), the scalar commutator is \[ f_R=(\partial_i\partial_j\lambda)P(H)_{ij} +2(\partial_i\lambda)\partial_jP(H)_{ij}, \qquad (V_R)_i=(\partial_j\lambda)P(k)_{ij}. \tag{49}\] Thus \(\|f_R\|_{W^{1,p}}+\|V_R\|_{W^{1,p}}=O(R^{-\beta})\) for every fixed finite \(p\). Only the second derivatives of the original metric and first derivatives of its second form are used.

We next estimate all adjoint-kernel moments. Put \(h=g-g_*\), \(\kappa=K-K_*\) in the physical \(x\) coordinates, and \(s=\mathcal L_xh\), \(V=\mathcal D_x\kappa\). They belong to \(L^1\). In fact the scalar linear source for either datum equals its \(2\mu\) plus an error bounded by \[C\bigl(|g-\delta||\partial^2g|+|\partial g|^2+|K|^2\bigr),\] and the momentum linear source equals \(J\) plus an error bounded by \[C\bigl(|g-\delta||\partial K|+|\partial g||K|\bigr).\] These errors are \(O(r^{-2-2q})\) for the original data and \(O(r^{-4})\) for the reference. They are integrable because \(q>1/2\). For any integrable source \(F\) on the end, \[ \int_{r\leq4R}r|F|\,dx=o(R). \tag{50}\] Indeed splitting at a fixed \(M\) and dividing by \(R\) bounds the outer contribution by \(4\int_{r>M}|F|\), while the inner contribution tends to zero; then let \(M\to\infty\).

For an affine scalar \(p\) define its linear constraint boundary flux \[B_p(t;h)=\int_{r=t} \{p\,\partial_jP(h)_{ij}-(\partial_jp)P(h)_{ij}\} n^i\,dA.\] For a Euclidean Killing field \(Z\), define \(B_Z(t;\kappa)=\int_{r=t}P(\kappa)_{ij}Z_i n^j\,dA\). Integration by parts gives, for either type of test, the difference of the two boundary fluxes as the integral of the test times the linear source; the adjoint-kernel term is zero. For \(p=1\) and constant \(Z\), the boundary flux differences tend to zero as \(t\to\infty\), since the two ADM vectors agree. For momentum the difference between Euclidean and geometric trace reversal has integrated error \(O(t^{1-2q})+O(t^{-1})\to0\). For homogeneous linear \(p\) or rotational \(Z\), integration from a fixed sphere and Equation 50 instead give \[ B_p(t;h)=o(t),\qquad B_Z(t;\kappa)=o(t). \tag{51}\] No limit for these first-moment fluxes is required.

For clarity, the scaling factors in the commutator moments are as follows. A constant test gives \[\int_A f_R\,dz =-R^{-1}\left\{B_1(R;h) +\int_{R<r<4R}\lambda(x/R)s(x)\,dx\right\}=o(R^{-1}).\] A homogeneous linear test \(p(z)\) gives \[\int_Ap(z)f_R(z)\,dz =-R^{-2}\left\{B_{p(x)}(R;h) +\int_{R<r<4R}\lambda(x/R)p(x)s(x)\,dx\right\} =o(R^{-1}).\] For momentum the identical calculation uses \(R\kappa(Rz)\); translation tests have factor \(R^{-1}\), rotation tests factor \(R^{-2}\). Equations 50 and 51 therefore give \[ \int_Ap f_R=o(R^{-1}),\qquad \int_A Z\cdot V_R=o(R^{-1}) \tag{52}\] for every fixed affine \(p\) and every fixed Killing field \(Z\).

Choose a fixed nonnegative bump \(\chi\) supported in the annulus and positive on a ball. The Gram matrix of the affine basis with weight \(\chi\) is positive definite. Subtract the linear combination of the smooth functions \(\chi p\) having the same affine moments as \(f_R\). Likewise use the Gram matrix of the Killing-field basis and the vector bumps \(\chi Z\) for \(V_R\). Equation 52 says that the coefficients of all these bumps are \(o(R^{-1})\). Their norms of every fixed order therefore have this size. The remaining sources have zero moments and \(W^{1,p}\) norm \(O(R^{-\beta})\). Apply Lemma 20 with the negative signs to cancel them. The resulting compact corrections have norms \[\|H_R^{\mathrm c}\|_{W^{3,p}} +\|k_R^{\mathrm c}\|_{W^{2,p}}=O(R^{-\beta}).\] Taking \(p>3\) controls their \(C^2\times C^1\) norms and ensures positivity of the corrected metric for large \(R\).

The nonlinear constraint map about \((\delta,0)\) has quadratic remainder bounded in \(C^0\) by \(C R^{-2\beta}\) on these scaled data. Its sources consequently differ from the convex blend of the original and vacuum sources by \(o(R^{-1})+O(R^{-2\beta})\) in \(C^0\). The original scaled momentum is \(O(R^{-\beta})\); changing its norm from the original metric to the corrected metric costs only \(O(R^{-2\beta})\). Convexity of the cone \(\{(\mu,J):\mu\geq|J|\}\) therefore proves a scaled DEC deficit of size \(o(R^{-1})+O(R^{-2\beta})=o(R^{-1})\). Undoing the scaling multiplies sources by \(R^{-2}\) and gives Equation 48. All corrections are supported away from the two annular boundaries, so the asserted exact matching and smoothness follow. ◻

Proof. The reference plane. Since \(m>0\), use the exterior Schwarzschild metric \[ G_*=-\left(\frac{1-m/(2r_y)}{1+m/(2r_y)}\right)^2dT^2 +(1+m/(2r_y))^4|dy|^2 \tag{54}\] at sufficiently large radius. Set \(v=P/E\), so \(|v|<1\), and \(A=(I-vv^{\mathsf T})^{-1/2}\). Parametrize the plane \(T=v\cdot y\) by \(y=Ax\). Its induced background Minkowski metric is exactly \(|dx|^2\). The unit normal and an adapted spatial frame are related to the horizontal ones by the Lorentz boost with velocity \(v\). By Lemma 17, the plane data have \[(E_*,P_*)=(\gamma m,\gamma m v)=(E,P), \qquad\gamma=(1-|v|^2)^{-1/2}.\] They are vacuum, with \(g_*-\delta=O_\ell(r_x^{-1})\) and \(K_*=O_\ell(r_x^{-2})\) at every order. Identify their \(x\) coordinates with the original end coordinates and apply Lemma 21.

Bending the reference end. Beyond \(r_x=4R\) replace the plane by the graph \[ T(y)=(v\cdot y)\psi\left(\frac{\log(r_y/R_b)}{L}\right), \tag{55}\] where \(\psi\) is smooth, is one on \((-\infty,0]\), is zero on \([1,\infty)\), and has values in \([0,1]\). Choose \(L\) sufficiently large that \(|v|(1+\|\psi'\|_\infty/L)<1\); then choose \(R_b\) to be a sufficiently large fixed multiple of \(R\). The bend starts in the exact reference portion. Its background gradient has norm at most the preceding strict bound, so it is uniformly spacelike in \(G_*\) once \(R\) is large. It becomes a horizontal Schwarzschild slice for \(r_y\geq R_b e^L\). Denote the resulting data before conformal repair by \((g_R,K_R)\).

For fixed \(L\), the graph has \(|dT|\leq c<1\), \(|d^2T|\leq C/r_y\), and derivatives of the background Schwarzschild coefficients are \(O(r_y^{-2})\). The graph formulas thus give uniform comparability of \(g_R\) with \(\delta_y\), \(|\partial g_R|\leq C/r_y\), and \(|K_R|\leq C/r_y\) throughout the bend. The annular corrections have stronger corresponding bounds. It follows that, throughout these large-radius ranges, \[ c\leq |dr_y|_{g_R}\leq C, \qquad |\Delta_{g_R}r_y|\leq C/r_y, \qquad |K_R|_{g_R}\leq C/r_y. \tag{56}\] Here and below constants may depend on \(L\) and \(v\), but not on \(R\).

Retain the original forms on the same manifold during the replacement and bend. Since \(y=Ax\) is fixed and all these metrics are uniformly comparable, their norms are \(O(r_y^{-2})\) and their electromagnetic ball term is \(O(r_y^{-4})\). The bend is vacuum for the total constraints. Thus Lemma 21, together with \(I_{g_R}(Y)=O(r_y^{-4})\), implies the following estimate for the charged deficit. With \(\xi=r_y/R\) there are fixed \(0<a<\xi_0<b\) such that the data are original for \(\xi\leq\xi_0\) and horizontal Schwarzschild for \(\xi\geq b\), and \[ \left(-\inf_{|Y|_{g_R}\leq1}D_{g_R}(Y)\right)_+ \leq \begin{cases} 0,&\xi\leq\xi_0,\\ \eta_RR^{-3},&\xi_0\leq\xi\leq b,\\ C r_y^{-4},&\xi\geq b, \end{cases} \qquad \eta_R\to0. \tag{57}\] The constant radial formulas used next are extended across the core.

Repair of the charged deficit. Choose a smooth \(\omega\geq0\) vanishing near \(a\) and at least one on \([\xi_0,b+1]\). On \([a,b+1]\) solve \[z_R'=C_0z_R+\omega,\qquad z_R(a)=0,\] and put \(z_R=0\) below \(a\). The constant \(C_0\) will be chosen large using Equation 56. On the horizontal tail let \[A_R(t)=(t+m/(2R))^2z_R(t).\] Starting at \(b\), smoothly continue its derivative to \(t^{-2}\) over \([b,b+1]\), using a convex interpolation of the positive derivative already prescribed and \(t^{-2}\). Thus \[ A_R'(t)\geq c t^{-2}\quad(t\geq b),\qquad A_R'(t)=t^{-2}\quad\hbox{eventually}. \tag{58}\] The quantities \(A_R(\infty)\) and \(F_R(\xi)=\int_\xi^\infty z_R(t)\,dt\) are uniformly bounded for fixed \(L\). Define \[u_R=1+\frac{e_R}{R}F_R(r_y/R),\qquad e_R>0.\] It is smooth, at least one, and constant on the compact region \(r_y\leq aR\). Direct differentiation, with \(F_R'=-z_R\), gives \[ \begin{split} -\Delta_{g_R}u_R-|K_R||du_R| =\frac{e_R}{R^3}\bigl(&z_R'|dr_y|_{g_R}^2 +Rz_R\Delta_{g_R}r_y\\ &-R|K_R|z_R|dr_y|_{g_R}\bigr). \end{split} \tag{59}\] On \([a,b]\) this is at least \(e_RR^{-3}(c z_R'-C z_R)\). Choose \(C_0\) sufficiently large to make it nonnegative everywhere there and at least \(c'e_RR^{-3}\) on \([\xi_0,b]\). On the horizontal tail \(K_R=0\). Writing \(w_*=1+m/(2r_y)\), the radial conformal Laplacian formula is \(\Delta_{w_*^4\delta}u=w_*^{-6}r_y^{-2} \partial_{r_y}(r_y^2w_*^2\partial_{r_y}u)\). It yields the exact identity \[ -\Delta_{g_R}u_R =e_RR^{-3}(1+m/(2R\xi))^{-6}\xi^{-2}A_R'(\xi) \geq c''e_RR^{-3}\xi^{-4}. \tag{60}\] Choose \(e_R\to0\) with \(e_R/(\eta_R+R^{-1})\to\infty\). For fixed \(L\), Equations 57, 59, and 60, followed by Equation 37, show that \((g_0,K_0)=(u_R^4g_R,u_R^2K_R)\) satisfies charged DEC. All boundary signs persist, since \(u_R\) is constant near \(S\).

By Equation 58, \[u_R=1+\frac{e_RA_R(\infty)}{r_y}+O_\ell(r_y^{-2}) \quad\hbox{for every }\ell\] for each fixed \(R,L\). Its ADM energy and momentum are therefore \[ E_0=m+2e_RA_R(\infty),\qquad P_0=0. \tag{61}\] The second form is compactly supported and the scalar curvature has \(O_\ell(r_y^{-4})\) decay on the tail. Both assertions follow from the exact horizontal form and Equation 60.

Strictification and the limiting comparisons. Fix this repaired exterior and apply the compact-\(K\) part of Lemma 18, with a fixed \(0<\delta_0<1\). A further conformal change by \(1+s\phi\), \(s>0\), gives the strict inequalities and every stated decay and integrability property. Its energy tends to \(E_0\) as \(s\downarrow0\), its momentum remains zero, and it can be made arbitrarily small in any prescribed finite collection of compact smooth norms.

It remains to check the area comparison, including cuts that escape into the bend. On the Schwarzschild graph, the spatial Schwarzschild metric dominates \(|dy|^2\) and its lapse squared is at most one. Moreover Equation 55 gives \[dT=\psi(v\cdot dy) +(v\cdot y)\psi'\frac{dr_y}{Lr_y}.\] Since \(0\leq\psi\leq1\), the tensor square of this expression is bounded above by \((v\cdot dy)^2+C L^{-1}|dy|^2\). Consequently, on the plane and bend, \[ g_R\geq|dy|^2-dT^2 \geq |dy|^2-(v\cdot dy)^2-C L^{-1}|dy|^2 \geq(1-C' L^{-1})|dx|^2. \tag{62}\] On the replacement annulus \(g_R=\delta_x+O(R^{-\beta})\), while the original metric has \(g=\delta_x+O(R^{-q})\) throughout the distant end. Inside the replacement radius the metrics agree. Both conformal changes increase the metric. Therefore the final metric has a global lower bound \(g_{R,L,s}\geq(1-C'/L-o_R(1))g\).

Choose \(L_j\to\infty\) first. For each \(L_j\) choose \(R_j\) so large that all the preceding constructions apply, their error bounds are at most \(1/j\), and the energy increment in Equation 61 is at most \(1/j\). The choices are possible because for fixed \(L\) the repair constants and \(A_R(\infty)\) are bounded independently of \(R\). Then choose \(s_j>0\) small enough that the strictification energy increment and the errors in the first \(j\) compact smooth seminorms are at most \(1/j\). The replacement and bend move to infinity; \(u_R\to1\) uniformly and is constant on every fixed compact set eventually. Thus \(g_j,K_j\) converge locally smoothly to \(g,K\). The field convergence follows from the unchanged forms and the smooth convergence of the volume forms. This proves the energy, momentum, and local convergence assertions. The global lower metric bound proves completeness and gives, for every full cut \(\Gamma\), \[\operatorname{Area}_{g_j}(\Gamma) \geq(1-\epsilon_j)\operatorname{Area}_g(\Gamma).\] Taking infima yields the required lower limit for the areas. For the upper limit choose, for any \(\eta>0\), a fixed smooth full cut \(\Gamma_\eta\) with \(\operatorname{Area}_g(\Gamma_\eta)\leq A_{\min,g}(S)+\eta\). It is compact, so local convergence gives \[\limsup_j A_{\min,g_j}(S) \leq\lim_j\operatorname{Area}_{g_j}(\Gamma_\eta) \leq A_{\min,g}(S)+\eta.\] Let \(\eta\downarrow0\). This proves Equation 53 for precisely the original full cut definition. If one uses the perimeter formulation, Proposition 10 identifies the same infimum, including the coincident obstacle frontier. At every stage the charge fluxes were preserved as integrals of the two original closed forms. ◻

Proof. Every connected component of the boundary is an orientable closed surface, hence is diffeomorphic to the boundary of a handlebody of its genus. Attach a separate such handlebody on each component. The collar theorem gives signed collar coordinates, increasing toward the original exterior. Extend the metric and tensor smoothly a short distance inward. The strict expansion inequality persists on a shorter collar by continuity.

On the next fixed collar, interpolate the metric to a product metric. The metrics in this interpolation are positive definite and form a fixed compact family; their cross-sectional mean curvatures are therefore bounded. Before making this interpolation, subtract \(\sigma_i Lg\) from \(K\), with \(L\ge0\) increasing smoothly from zero. This changes \(\sigma_i\operatorname{tr}_{\rm tan}K\) by \(-2L\). Start the change within the region where the expansion is already strict; after \(L\) has become large, it dominates the bounded mean curvature and the bounded tensor terms in the interpolation. One may then interpolate the remaining tensor to the pure trace \(-\sigma_iL_0g\), still retaining strictness. Finish with an exactly product segment. This gives the asserted fixed transition.

Choose a smooth metric on each handlebody extending its product boundary collar, and put \(K=-\sigma_iL_0g\) on that cap. Insert between the transition and cap as many copies of the fixed product segment as are needed. A length bounded by a fixed multiple of \(1+N\) is available, with the multiple enlarged when the height barriers are chosen. There are finitely many fixed transition and cap pieces. Product translation identifies all local neck charts, so all asserted local bounds and the local injectivity radius are uniform. Only the cylinder volume grows with its length. The original end is unaltered, proving completeness and the one-end assertion. Extending \(r\) as a positive constant along the inserted pieces proves the weight assertion. This construction concerns the metric and tensor only; all subsequent uses of the flux forms take place on the original exterior, where they remain closed with their prescribed two fluxes. ◻

Proof. First obtain the scalar identity before conformal multiplication. On \(\Omega_N\times\mathbb R\) consider the stationary Lorentzian metric \[\mathbf G=-l^2(dz-df)^2+\bar g, \qquad \bar g=g+l^2df^2.\] Written in the coordinates \((x,z)\), it has spatial metric \(g\), lapse \(U=l\sqrt D\), and shift \(\gamma=l^2\nabla f=Uw\). Its future unit normal to the constant-\(z\) slices is \(\mathbf n=U^{-1}(\partial_z-\gamma)=U^{-1}\partial_z-w\). With the convention for \(K\) in Definition 1, stationarity gives \[ b=-U^{-1}\operatorname{sym}\nabla\gamma =-H^f-p(dt\otimes w+w\otimes dt),\qquad \operatorname{div}\gamma=-U H_b, \quad H_b=\operatorname{tr}b. \tag{74}\] Here \(\operatorname{sym}\) means the average of a two-tensor and its transpose. The contracted Gauss equation and normal variation of the second fundamental form give, with this sign convention, \[\mathbf R=R_g+H_b^2-|b|^2-2\mathbf{Ric}(\mathbf n,\mathbf n), \qquad \mathbf{Ric}(\mathbf n,\mathbf n) =-\mathbf n(H_b)-|b|^2+U^{-1}\Delta_g U.\] Since \(H_b\) is independent of \(z\), \(\mathbf n(H_b)=-w(H_b)\). Substituting and using \(\operatorname{div}(Uw)=-UH_b\) yields \[ \mathbf R =R_g+\mathsf B(b,b) -2U^{-1}\operatorname{div}(\nabla U+H_bUw). \tag{75}\] In the coordinates \(z-f(x)\) the same metric is the warped static metric \(-l^2d(z-f)^2+\bar g\). Its scalar curvature is \(\mathbf R=R_{\bar g}-2l^{-1}\Delta_{\bar g}l\). The inverse and determinant of \(\bar g\) are \(A_d\) and \(D\det g\), respectively. Hence \[ l^{-1}\Delta_{\bar g}l =U^{-1}\operatorname{div}\bigl((U/l)A_d\nabla l\bigr), \qquad \nabla U-(U/l)A_d\nabla l=-Ub(w,\cdot). \tag{76}\] For the second identity, divide by \(U\) and use \(d\log U=p(1+v)dt+H^f(w,\cdot)\) and \(d\log l=pdt\); the resulting difference is \(H^f(w,\cdot)+p\{vdt+w(t)w\}=-b(w,\cdot)\). Equations 75 and 76 now give \[ \tfrac12R_{\bar g} =\tfrac12R_g+\tfrac12\mathsf B(b,b) +U^{-1}\operatorname{div}(U\mathsf P_b w). \tag{77}\]

For clarity, the insertion of the original constraint tensor is an exact cancellation. Namely, \[U^{-1}\operatorname{div}(U\mathsf P_Kw) =J(w)+\mathsf P_K:U^{-1}\nabla(Uw) =J(w)-\mathsf B(K,b),\] where symmetry of \(\mathsf P_K\) permits use of Equation 74. Moreover, \(\mu=R_g/2-\mathsf B(K,K)/2\) and \(\mathsf Q=K-b\). Expanding \(\mathsf B(\mathsf Q,\mathsf Q)\) in Equation 77 therefore proves \[ \tfrac12R_{\bar g} =\mu+J(w)+\tfrac12\mathsf B(\mathsf Q,\mathsf Q) -U^{-1}\operatorname{div}(U\mathsf P_{\mathsf Q}w). \tag{78}\]

In dimension three the conformal scalar law is \[\tfrac12e^{2t}R_{\widehat g} =\tfrac12R_{\bar g}-2\Delta_{\bar g}t-|dt|_{A_d}^2.\] The same determinant calculation as above gives \(\Delta_{\bar g}t =U^{-1}\operatorname{div}(UA_d\nabla t)-p|dt|_{A_d}^2\). Set \(W=\mathsf P_{\mathsf Q}w+2A_d\nabla t\). Since \(u=e^tU\), \[-U^{-1}\operatorname{div}(UW) =-u^{-1}\operatorname{div}(uW)+W(t).\] It follows that \[ \begin{aligned} \tfrac12e^{2t}R_{\widehat g} ={}&\mu+J(w)+\tfrac12\mathsf B(\mathsf Q,\mathsf Q) +\mathsf P_{\mathsf Q}(w,\nabla t)\\ &+2s_p|dt|_{A_d}^2-u^{-1}\operatorname{div}(uW). \end{aligned} \tag{79}\]

Differentiating Equation 66 gives the useful exact one-form identity \[ 2\,dZ=(2+pv)dt+H^f(w,\cdot). \tag{80}\] In the block notation, \(\operatorname{tr}T=F-\chi j\), and \(\mathsf P_{\mathsf Q}w\) has transverse component \(a_wM+pv y\) and axial component \(-a_w(F-\chi j)\). Thus both sides of the first identity below have transverse component \(u(qy+a_wM)\) and axial component \(u(2dx+\chi a_wj-a_wF)\): \[ uW=V-uFw,\qquad u^{-1}\operatorname{div}(uw) =F+(1-\chi)j-\tau+2s_pa_wx. \tag{81}\] For the second identity, use \(u^{-1}\operatorname{div}(uw)=-H_b+w(t) =\operatorname{tr}H^f+(2p+1)w(t)\). Replacing the flux in Equation 79 and differentiating \(uFw\) gives Equation 72 with Equation 71.

To check the full expansion, put \(z_T=\operatorname{tr}T=F-\chi j\). The transverse, mixed, and axial blocks of \(\mathsf Q\) are \(T\), \(M+pa_wy\), and \(j+2pa_wx\). Therefore \[\tfrac12\mathsf B(\mathsf Q,\mathsf Q) =\tfrac12|T^0|^2-\tfrac14z_T^2 +|M+pa_wy|^2-z_Tj-2pa_wxz_T,\] and \[\mathsf P_{\mathsf Q}(w,\nabla t) =a_w\langle M,y\rangle+pv|y|^2-a_wxz_T.\] Combining the terms in \(M,y\) completes the square \(|M+s_pa_wy|^2-v|y|^2/4\). The remaining linear terms in \(x\) equal \(2s_pa_wx(F-z_T)=2s_p\chi ja_wx\); the remaining terms in \(j,F\) equal \(-z_Tj+F(F+(1-\chi)j)=\chi j^2-\chi Fj+F^2\). These are exactly all the terms in Equation 73. ◻

Proof. Completing first the \(x\) square and then the \(F\) square in Equation 73 gives the exact expression \[ \begin{aligned} \mathcal T={}&\tfrac12|T^0|^2+|M+s_pa_wy|^2 +(2s_p-v/4)|y|^2\\ &+2s_pd\left(x+\frac{a_wj}{q}\right)^2 +\tfrac34\left(F-\frac{\chi j}{3}\right)^2 +\frac{d(8+v)}{3q^2}j^2. \end{aligned} \tag{84}\] Indeed, the residual coefficient of \(j^2\) is \[\chi-\frac13\chi^2-s_p\frac{\chi^2v}{2d} =\frac{\chi}{q}\left(\frac43+\frac v6\right) =\frac{d(8+v)}{3q^2}.\] Every displayed square has a positive coefficient, and \(2s_p-v/4\ge3s_p/2\ge3/4\). The last square implies \(dj^2\le3(N+2)^2\mathcal T/8\) and \(\chi^2j^2\le3\mathcal T/2\). The \(F\) square then gives \(F^2\le3\mathcal T\); hence \(|T|^2=|T^0|^2+(F-\chi j)^2/2\le13\mathcal T/2\). The transverse square gives \(|y|^2\le4\mathcal T/3\). Undoing the \(x\) shift gives \[dx^2\le2d\left(x+\frac{a_wj}{q}\right)^2 +\frac{2dv}{q^2}j^2 \le\frac{8}{3}\mathcal T.\] Likewise, \[|M|^2\le2|M+s_pa_wy|^2+2s_p^2v|y|^2 \le\left(2+\frac43(N+\tfrac12)\right)\mathcal T.\] Adding these bounds proves Equation 82. In particular the coercivity loses no exponential factor from \(l\) or \(d\).

When \(dt=0\), Equation 73 reduces to \[\mathcal T=\tfrac12|T^0|^2+|M|^2 +\tfrac34F^2-\tfrac12\chi Fj +(\chi-\tfrac14\chi^2)j^2.\] The matrix of the last two variables \((F,\sqrt\chi j)\) is \[\begin{pmatrix}3/4&-\sqrt\chi/4\\ -\sqrt\chi/4&1-\chi/4\end{pmatrix}.\] Its eigenvalues are \(1\) and \((3-\chi)/4\in[1/2,3/4]\). This proves Equation 83. ◻

Proof. Use the graph of \(f\) in the Riemannian warped product \(g+l^2dz^2\). At \(dt=0\) its second fundamental form, up to its irrelevant overall normal sign, is \(H^f\); its induced metric is \(\bar g\) and its inverse is \(A_d\). At that point \(l^{-1}\nabla^2l=p\nabla^2t\). The horizontal Ricci tensor of the ambient metric is \(\operatorname{Ric}_g-p\nabla^2t\), its unit vertical Ricci component is \(-p\Delta_gt\), and its mixed Ricci components vanish. Its scalar curvature is \(R_g-2p\Delta_gt\). The graph unit normal has horizontal part \(-w\) and vertical part of length \(\sqrt d\). The contracted graph Gauss equation therefore gives \[\tfrac12R_{\bar g} =\tfrac12R_g-v\operatorname{Ric}_g(e,e)+\mathcal S(H^f) -vp\operatorname{tr}_{e^\perp}\nabla^2t.\] At a stationary point the conformal law contributes exactly \(-2\operatorname{tr}_{A_d}\nabla^2t\). This proves Equation 86, including all the Hessian terms.

To prove the inequality, first use \(C=H^f+K\). Its transverse trace is \(F-\chi j\), so direct contraction gives \[\mathcal S(C)=\tfrac14(F-\chi j)^2-\tfrac12|T^0|^2 +dj(F-\chi j)-d|M|^2.\] Subtracting this from the stationary expression for \(\mathcal T\) gives the exact formula \[ \begin{aligned} \mathcal T-\mathcal S(C) ={}&|T^0|^2+(1+d)|M|^2+\tfrac12F^2-dFj\\ &+b_*j^2,\qquad b_*=\chi(1+d)-\tfrac12\chi^2. \end{aligned} \tag{88}\] Since \(\chi\le d\), we have \(b_*\ge\chi(1+d/2)\ge\chi\) and \[\frac{d^2}{2b_*} \le\frac{dq}{2(2+d)} \le\frac13+\frac{pv}{6}.\] Young’s inequality, in the form \(d|Fj|\le b_*j^2/2+d^2F^2/(2b_*)\), thus proves \[ \mathcal T-\mathcal S(C) \ge\tfrac16\mathcal E_0-\frac p6vF^2. \tag{89}\]

For the remaining polarization, let \(X=A_d^{1/2}CA_d^{1/2}\) and \(Y=A_d^{1/2}KA_d^{1/2}\), regarding these as symmetric matrices in a \(g\)-orthonormal frame. Then \[|X|^2=|T^0|^2+\tfrac12(F-\chi j)^2+2d|M|^2+d^2j^2 \le\frac{N+4}{2}\mathcal E_0, \qquad |Y|\le|K|.\] Here \(d^2/\chi=dq/2\le(N+2)/2\) controls the only axial loss. The trace-reversal map on symmetric three-by-three matrices has operator norm two, whence \[|(\operatorname{tr}X)(\operatorname{tr}Y)-\langle X,Y\rangle| \le\sqrt{2(N+4)}\sqrt{\mathcal E_0}|K|.\] Also \(\mathcal S(K)\le|K|^2\). As \(\mathcal S(C-K)=\mathcal S(C) -\{(\operatorname{tr}X)(\operatorname{tr}Y)-\langle X,Y\rangle\} +\mathcal S(K)\), Young’s inequality with coefficient \(1/12\) of \(\mathcal E_0\) and Equation 89 imply \[\mathcal T-\mathcal S(H^f) \ge\tfrac1{12}\mathcal E_0-\tfrac p6vF^2-(6N+25)|K|^2.\] Use \(\mathcal E_0\ge\mathcal T\) and \(0\le p\le N\) to obtain Equation 87. ◻

Proof. At a positive interior maximum of \(f\), the left side of Equation 93 is nonpositive and the right side is strictly positive. A negative minimum is excluded in the same way. The zero boundary value therefore gives \(f=0\). The defining formulas now give \(t=Z\), \(\mathcal T=2s_p|dZ|^2\), \(u=e^Zl(Z)\), and \(V=2e^Zl(Z)\nabla Z\). Differentiating this flux proves Equation 95. For \(b=0\), define \(\Psi(z)=\int_0^z e^sl(s)\,ds\). The equation is \(\Delta_g\Psi(Z)=0\) with zero boundary value, so \(\Psi(Z)=0\) by the maximum principle. Since \(\Psi'>0\), \(Z=0\). This identifies the starting solution for continuation without assuming existence at any other parameter value. ◻

Proof. At an interior maximum of \(h\), its gradient vanishes, and hence \(D=1\), \(A_\chi=I\). The trace equation there reads \[h=\operatorname{tr}_gK_a+\frac l\eta\Delta_g h \le\operatorname{tr}_gK_a.\] At an interior minimum the reverse inequality holds. The boundary height is zero and \(|\operatorname{tr}_gK_a|\le C\) uniformly on the filled manifolds. This proves the first assertion. It also proves \(h=0\) when \(K_a=0\).

Choose \(r_0\) outside the support of \(K\) and large enough for the following radial calculation. At a contact point with \(H_R=C_0(r^{-b_1}-R^{-b_1})\), the slope axis of \(h\) is the radial axis. The transverse eigenvalues of \(A_\chi\) are one and its axial eigenvalue is \(0<\chi\le1\). The asymptotic bounds for the prepared metric imply, uniformly for this entire range of \(\chi\), \[ \operatorname{tr}_{A_\chi}\nabla^2 H_R =C_0 b_1 r^{-b_1-2} \bigl(-2+(1+b_1)\chi+o(1)\bigr)<0. \tag{100}\] Indeed, in the Euclidean metric the radial and transverse Hessian eigenvalues are respectively \(C_0b_1(1+b_1)r^{-b_1-2}\) and \(-C_0b_1r^{-b_1-2}\); the metric and connection errors are \(o(r^{-b_1-2})\). Since \(1-b_1>0\), one choice of \(r_0\) makes the inequality strict for every slope and every \(N\). Choose \(C_0\) so that \(H_R\ge C\) at \(r=r_0\) for every \(R\ge2r_0\). If \(h-H_R\) had a positive interior maximum, the matching gradients would determine the same \(t\), \(l\), \(D\), and \(A_\chi\) in evaluating the trace at that point, while \(\nabla^2h\le\nabla^2H_R\). The trace equation, \(K=0\), and (100) would give \(0<h\le(l/(\eta\sqrt D)) \operatorname{tr}_{A_\chi}\nabla^2H_R<0\). Applying the same argument to \(-h\) proves the end bound. The two barriers vanish at \(r=R\); their one-sided derivatives there, together with the zero tangential derivative of \(h\), imply (99).

For a fixed boundary slope the function \[t\longmapsto t+\tfrac14\log(1+l(t)^2|df|_g^2)\] has derivative \(1+pv/2\ge1\). At \(t=-\epsilon\) its value is at most \[-\epsilon+\tfrac14\log (1+C^2\ell^2\eta^{-2}R^{-2b_1-2}) =-\epsilon+\tfrac14\log (1+C^2\ell^{-1}R^{-2b_1-2}).\] This is negative for sufficiently large \(R\) at fixed \(N\); for example it suffices to require \(C^2\ell^{-1}R^{-2b_1-2}<e^{4\epsilon}-1\). Since its value at the actual boundary \(t\) is \(Z=0\), strict monotonicity gives \(t>-\epsilon\). ◻

Proof. The outer boundary is excluded by Lemma 28. Consider first \(b=1\) and an interior minimum with \(t=-\epsilon\). Write \(e\) for the slope axis, arbitrary at zero slope. The stationary-point calculation of Lemma 26 gives \[\begin{align*} \tfrac12e^{2t}R_{\widehat g} &=\tfrac12R_g-v\operatorname{Ric}_g(e,e) +\mathcal S(H^f) -vp\operatorname{tr}_{e^\perp}\nabla^2t -2\operatorname{tr}_{A_d}\nabla^2t \\ &\le\tfrac12R_g-v\operatorname{Ric}_g(e,e) +\mathcal S(H^f), \tag{101}\end{align*}\] where \(\mathcal S(B)=\tfrac12((\operatorname{tr}_{A_d}B)^2 -|B|^2_{A_d\otimes A_d})\). The signs in this inequality use \(p,v\ge0\), \(A_d>0\), and \(\nabla^2t\ge0\). The same lemma supplies a constant \(c_1>0\), independent of \(N\), and the algebraic estimate \[ \mathcal T-\mathcal S(H^f) \ge c_1\mathcal T-P_N(vF^2+|K_a|^2). \tag{102}\] Both formulas concern the actual tensor \(K_a\); no energy condition on the filling or on the scaled tensor is used.

By the scalar identity (72) and \(F=\eta f\), \(w(F)=G\), so comparison with (101) yields \[\begin{align*} u^{-1}\operatorname{div}V &\ge \mathcal T-\mathcal S(H^f)+G +(\mu_a-R_g/2)+J_a(w)-F\operatorname{tr}_gK_a +v\operatorname{Ric}_g(e,e)\\ &\ge c_1\mathcal T+G-B_N(\rho_0+v\rho_1) \end{align*}\] with \(B_N\le C(1+N)\), using the coefficient \(6N+25\) in Lemma 26. To justify the last bound throughout the noncompact range, observe that \(\mu_a-R_g/2=((\operatorname{tr}K_a)^2-|K_a|^2)/2\), and that this term, \(J_a\), \(K_a\), and \(F\operatorname{tr}K_a\) are supported in the enlarged core, where they are bounded and \(\rho_0\) is bounded below. On the end, \(|F|^2=|h|^2\le C\rho_1\) by Lemma 28, while \(|\operatorname{Ric}_g|\le C r^{-3}\le C\rho_1\). The same estimates on a fixed connecting annulus are absorbed into the constant. This proves the displayed bound with constants independent of \(R\) and \(a\).

At the assumed minimum \(m_0=1\). Equating the last lower bound with (97) gives \[ 0\ge(c_1-\delta_0)\mathcal T+(1-\delta_0)G +(C_N-B_N)(\rho_0+v\rho_1)-c_*\rho_0. \tag{103}\] Choose \(0<\delta_0<\frac12\min(c_1,1)\), and then choose the polynomial \(C_N\ge B_N+c_*+1\). The right side of (103) is positive because \(\rho_0>0\), a contradiction.

In the remaining homotopy segment \(a=0\), Lemma 28 gives \(f=0\). Consequently \(t=Z\), \(v=0\), \(A_\chi=I\), \(u=e^t l(t)\), and \(\mathcal T=(2p+1)|dt|^2\). At an interior minimum \(t=-\epsilon\), the equation is \[2u\Delta_g t=b u(\rho-C_N\rho_0).\] Its left side is nonnegative, whereas its right side is negative if \(b>0\). If \(b=0\), set \(\Phi_0(t)=\int_0^t e^s l(s)\,ds\). Then \(\Delta_g\Phi_0(t)=0\) with zero boundary data, hence \(\Phi_0(t)=0\) and \(t=0\), since \(\Phi_0\) is strictly increasing. This proves the assertion on the whole homotopy. ◻

Proof. Only the support of the penalty needs consideration for (105). There \(-\epsilon\le t<-\epsilon+1/N\le1\), and therefore, when \(Z\ge2\), \[G=\frac\eta l\left(e^{2(Z-t)}-e^{-2(Z-t)}\right) \ge c\eta e^{2Z}.\] The radius weights are bounded above, so the negative penalty has magnitude at most \(C C_N\). Thus one may take \[Z_0\ge2+\tfrac34N\epsilon+ \tfrac12\log(1+C C_N/\delta_0).\] Increasing this expression to a polynomial proves (105).

Choose a nondecreasing Lipschitz function \(q\) which is zero on \((-\infty,Z_0]\), one on \([Z_0+1,\infty)\), and has \(0\le q'\le2\). It vanishes on the outer boundary. Testing the divergence equation with \(q(Z)\) gives \[\begin{align*} \int u\Xi q(Z)\,dV_g &=-\int u q'(Z) \bigl(2|dZ|_A^2+A(K_a(w,\cdot),dZ)\bigr)\,dV_g \\ &\le C\int_{\{Z_0<Z<Z_0+1\}}u|K_a|^2\,dV_g. \tag{107}\end{align*}\] The last inequality is the scalar estimate \(-2x^2+yx\le y^2/8\), and \(A\le I\). The last integrand is supported in \(\mathcal C_N\), where \(u=l e^{2Z-t}\le e^{1+\epsilon+2(Z_0+1)}\) on the strip. Its integral is consequently at most \(\exp(P_N)\). All terms on the left are nonnegative by (105), and \(q=1\) above \(Z_0+1\), so \[ \int_{\{Z>Z_0+1\}}u(\delta_0\mathcal T+\rho+ \tfrac12\delta_0G)\,dV_g \le\exp(P_N). \tag{108}\] The same argument applies to any fixed enlargement of the core, since \(\rho\) has a positive lower bound there.

Here is a pointwise conversion of this estimate to the claimed moment. Since \(l\ge\ell\ge\eta\) and \(D\ge1\), \[\begin{align*} u(1+G) &=\sqrt D e^t\bigl[l+\eta(\sqrt D-D^{-1/2})\bigr]\\ &\ge\eta e^t D \ge\eta e^Z\sqrt D. \end{align*}\] The final inequality is \(\sqrt D=e^{2(Z-t)}\ge e^{Z-t}\). Thus (108), \(\rho\ge c\) on the core, and \(\eta^{-1}=e^{3N\epsilon/2}\) bound the integral in (106) above \(Z_0+1\). On the complementary set, \(\sqrt D=e^{2(Z-t)}\le e^{2(Z_0+1+\epsilon)}\) and \(e^Z\le e^{Z_0+1}\); multiplication by the core volume \(C(1+N)\) gives the same bound there. ◻

Proof. Use the graph of \(f\) in the product metric \(g+dz^2\) on a larger base patch times the entire real line. Put \[E_f=1+|df|_g^2,\qquad A_1=I-\frac{df\otimes df}{E_f}.\] Its measure is \(d\nu_1=\sqrt{E_f}\,dV_g\) and its inverse metric on base covectors is \(A_1\). Equations (104) give \[e^{-2}D\le E_f\le\ell^{-2}D.\] The axial eigenvalues of \(A_1,A_d,A\) are respectively \(E_f^{-1},d,\chi\), and \(2d/(2+N)\le\chi\le d\). Hence the measures and inverse metrics are pairwise comparable with factors bounded by \(C(2+N)\ell^{-2}\le\exp(P_N)\).

The scalar mean curvature of this graph in \(g+dz^2\) is \[H_1=E_f^{-1/2}\operatorname{tr}_{A_1}\nabla^2f =\frac{\sqrt D}{l\sqrt{E_f}} \operatorname{tr}_{A_1}H^f.\] Coercivity, Proposition 25, controls the transverse block of \(H^f+K_a\) and the axial block multiplied by \(\sqrt d\) by \(P_N\sqrt{\mathcal T}\). Since \(E_f^{-1}\le e^2d\le e^2\sqrt d\) and \(|K_a|\le C\), it follows that \[|\operatorname{tr}_{A_1}H^f| \le P_N(1+\sqrt{\mathcal T}),\qquad |H_1|\le\exp(P_N)(1+\sqrt{\mathcal T}).\] This estimate uses weighted axial contraction, rather than an unweighted bound for the full Hessian.

Each larger base patch can be smoothly isometrically embedded in a fixed Euclidean space. Only finitely many compact geometric models are needed: the fixed exterior and transition patches, the fixed caps, and translates of product-neck patches. One can obtain such embeddings by extending each model to a compact smooth Riemannian manifold and applying the smooth isometric embedding theorem (Nash 1956, Theorem 2, p. 59). The Euclidean dimensions and the second fundamental forms on the smaller patches therefore have fixed bounds. Taking the product of these embeddings with the identity of the real line gives an isometric embedding of the ambient product. The Euclidean mean curvature vector of the graph has norm at most \(|H_1|+C\), because the trace of the ambient second fundamental form over its three unit tangent vectors is bounded by a fixed constant.

The Euclidean submanifold Sobolev inequality of Michael and Simon (Michael and Simon 1973), in the form proved in (Simon 2018, chap. 4, Section 6, Theorem 6.7), applies to this smooth three-dimensional immersed graph and compactly supported test functions: if \(\zeta\ge0\), then \[\left(\int\zeta^{3/2}\,d\nu_1\right)^{2/3} \le C\int\bigl(|d\zeta|_{A_1}+|\mathbf H|\zeta\bigr)\,d\nu_1.\] There is no boundary term because the support is in the interior of the larger patch. Apply it to \(\zeta=|\varphi|^4\) and use Cauchy–Schwarz on \(4|\varphi|^3|d\varphi|\) and \(|\mathbf H||\varphi|^4\) to obtain \[\left(\int|\varphi|^6\,d\nu_1\right)^{1/3} \le C\int\bigl(|d\varphi|_{A_1}^2+ |\mathbf H|^2\varphi^2\bigr)\,d\nu_1.\] The zero function is harmless; otherwise this follows by dividing the intermediate inequality by \((\int|\varphi|^6\,d\nu_1)^{1/2}\) and squaring. The already established metric, measure, and mean curvature comparisons yield (109). ◻

Proof. Put \(B=e^t l\), \(\kappa=K_a(w,\cdot)\) and \(\gamma=d\log B=(1+p)dt\). Coercivity and \(A\le A_d\) show that \[ |\gamma|_A\le B_N\sqrt{\mathcal T},\qquad |\kappa|_A\le C, \tag{111}\] with \(B_N\) polynomial. The boundedness of the weights in the penalty gives, globally, \(\Xi\ge\delta_0\mathcal T-L_N\) for a polynomial \(L_N\). Test \(\operatorname{div}V=u\Xi\) with the nonnegative function \(\psi^2e^{2sZ}/B\). As \(u/B=\sqrt D\), exact differentiation gives \[\begin{align*} &\int\psi^2e^{2sZ}\Xi\,d\nu +4s\int\psi^2e^{2sZ}|dZ|_A^2\,d\nu\\ &=\int e^{2sZ}\bigl[ \psi^2 A(\gamma,2dZ+\kappa) -2\psi A(d\psi,2dZ+\kappa) -2s\psi^2A(\kappa,dZ)\bigr]\,d\nu. \tag{112}\end{align*}\] For clarity, set \(X=|dZ|_A\) and \(Y=|d\psi|_A\le|d\psi|_g\). Young’s inequality and (111) give the following pointwise bounds for the terms on the right: \[\begin{align*} 2B_N\psi^2\sqrt{\mathcal T}X &\le\tfrac14\delta_0\psi^2\mathcal T +C\delta_0^{-1}B_N^2\psi^2X^2,\\ C B_N\psi^2\sqrt{\mathcal T} &\le\tfrac14\delta_0\psi^2\mathcal T +C\delta_0^{-1}B_N^2\psi^2,\\ 4|\psi|YX&\le s\psi^2X^2+4s^{-1}Y^2,\\ 2C|\psi|Y&\le C(\psi^2+Y^2),\\ 2Cs\psi^2X&\le s\psi^2X^2+C s\psi^2. \end{align*}\] Choose the polynomial threshold \(s_0\ge1+C\delta_0^{-1}B_N^2\), with its constant large enough that the first \(X^2\) error is at most \(s\psi^2X^2\). Substituting these five estimates into (112) leaves at least \(\delta_0\mathcal T/2+sX^2\) on its left and at most \(P_N(1+s)(\psi^2+Y^2)\) on its right, all multiplied by \(e^{2sZ}\). Absorbing the fixed factor \(\delta_0/2\) proves (110). ◻

Proof. First take \(b=1\) and write \(Y=e^Z\). Choose nested core patches \(U_r\Subset U_R\) with \(1/2\le r<R\le1\), in uniform local coordinates, and cutoffs with \(|d\psi|_g\le C/(R-r)\). Combining Lemmas 31 and 32, applied to \(\varphi=\psi e^{sZ}\), gives \[ \|Y\|_{L^{6s}(U_r,d\nu)} \le\bigl[C S_NE_N(1+s)^2(1+(R-r)^{-2})\bigr]^{1/(2s)} \|Y\|_{L^{2s}(U_R,d\nu)}. \tag{114}\] Indeed, differentiation of \(\psi e^{sZ}\) contributes at most \(2e^{2sZ}|d\psi|_A^2+2s^2\psi^2e^{2sZ}|dZ|_A^2\); the latter integral is bounded by \(2s\) times (110). This accounts for the power \((1+s)^2\) in (114).

Set \(s_i=3^i s_0\), take successive spatial gaps proportional to \(b2^{-i-1}\), where \(0<b\le1\) is the total gap, and iterate (114). The product of constants has logarithm at most \[\sum_{i=0}^{\infty}\frac{ \log(C S_NE_N)+2\log(1+s_0)+C i+2\log(1/b)}{2s_0 3^i}.\] The identities \[\sum_{i\ge0}3^{-i}=\tfrac32, \qquad \sum_{i\ge0}i3^{-i}=\tfrac34\] therefore give \[ \log\sup_{U_r}Y \le\frac1{2s_0}\log\int_{U_R}Y^{2s_0}\,d\nu +\frac{D_N}{s_0} +\frac{C}{s_0}\log\frac1{R-r}, \tag{115}\] where \(D_N\le C(1+\log S_N+\log E_N+\log(1+s_0))\le P_N\). The limiting norm is the supremum because each individual solution and its positive smooth measure are smooth on the closure of the larger patch. No uniform bound for that individual supremum has been used to obtain (115).

We next lower the initial integrability exponent explicitly. By Lemma 30, \(\int_{U_R}Y\,d\nu\le e^{L_N}\) with \(L_N\) polynomial. Put \(M(r)=\max(1,\sup_{U_r}Y)\) and \(\theta=1-1/(2s_0)\). The inequality \(Y^{2s_0}\le M(R)^{2s_0-1}Y\) and (115) yield \[ \log M(r)\le\theta\log M(R) +\frac{L_N}{2s_0}+\frac{D_N}{s_0} +\frac C{s_0}\log\frac1{R-r}. \tag{116}\] Choose increasing radii \(r_j\) whose successive gaps are \(c2^{-j-1}\) and whose limit lies strictly inside the larger patch. Iterating (116) \(m\) times gives \[\log M(r_0) \le\theta^m\log M(r_m) +\sum_{j=0}^{m-1}\theta^j \left(\frac{L_N/2+D_N+C}{s_0} +\frac{Cj}{s_0}\right).\] The first term tends to zero: the supremum on that fixed larger patch is finite for the particular smooth solution. Moreover, \[\sum_{j\ge0}\theta^j=2s_0, \qquad \sum_{j\ge0}j\theta^j =\frac{\theta}{(1-\theta)^2}\le4s_0^2.\] Consequently \[ \log M(r_0)\le C(L_N+D_N+1+s_0)\le P_N. \tag{117}\] This computation is the required control of the parameter dependence: both iteration steps enlarge only a polynomial in the logarithm of the supremum. Uniform patches cover the entire enlarged core and the support of \(K\), so (117) bounds \(Z\) there with one polynomial independent of the number of patches.

Outside this covered region \(K=0\). If \(Z\) attained a larger maximum, exceeding also \(Z_0\), the maximum would be interior since \(Z=0\) on the outer sphere. At such a point \(V=2uA\nabla Z\) and \[\operatorname{div}V=2u\operatorname{tr}_A\nabla^2Z\le0, \qquad u\Xi\ge u\rho>0\] by (105), a contradiction. This proves the global upper bound on the first segment.

On the second segment \(a=0\), the trace equation again gives \(f=0\) and \(t=Z\). If \(b>0\), a positive interior maximum with \(Z>\max(0,-\epsilon+1/N)\) has \(m_0=0\), \(dt=0\), and hence \(\operatorname{div}V\le0<b u\rho\). Together with zero boundary values this is already a uniform upper bound. If \(b=0\), the transformed harmonic equation used in Lemma 29 gives \(Z=0\).

Finally Lemma 29 makes the lower bound strict, (104) gives the warp bounds, and \(D=e^{4(Z-t)}\le\exp(P_N)\) gives \(|df|_g=\sqrt{D-1}/l\le\exp(P_N)\). Lemma 28 and \(\eta^{-1}=e^{3N\epsilon/2}\) give \(|f|\le\exp(P_N)\). This proves (113). ◻

Proof. We give the compactness and improvement arguments, including the treatment of the face. The identities involving third derivatives are distributional. For a \(C^2\) function and metric their trace-reversed form follows by \(C^2\) smoothing: the fluxes converge locally uniformly, the Hessian quadratics converge in \(L^1\), and the Ricci terms converge locally uniformly. The original axial equation is substituted only after this identity has been established; neither \(\chi\) nor \(G\) is differentiated or mollified. The same argument on each side of a face retains the normal flux jump. Thus the proof applies in particular to the \(C^{2,\alpha}\) functions arising in continuation below.

A Hessian estimate above exponent two. Normalize the metric at the center of a small coordinate ball. The coordinate principal matrix \(a\) in Equation 118 satisfies \[|I-a|_{\mathrm F}\le1-c/2\] after shrinking the ball by an amount determined by the metric bounds. For a compactly supported function \(Y\) on Euclidean space, the Fourier transform gives \(\|D^2Y\|_2=\|\Delta Y\|_2\), where the matrix norm is Frobenius. The Calderón–Zygmund estimate at any fixed exponent larger than two (Gilbarg and Trudinger 2001, Theorem 9.9), interpolated with this identity, implies that the norm of \(D^2\Delta^{-1}:L^s\longrightarrow L^s\) tends to one as \(s\downarrow2\). Choose \(s_0>2\) sufficiently close to two that this norm times \(1-c/2\) is less than one. Writing \(\Delta Y=(I-a):D^2Y+a:D^2Y\) and absorbing gives the global estimate. Apply it to \(\zeta Y\), and use \[\|DY\|_{L^s(B_r)} \le \varepsilon r\|D^2Y\|_{L^s(B_r)} +C_{\varepsilon}r^{-1}\|Y\|_{L^s(B_r)}.\] Using intermediate balls and absorbing successively gives, for \(2\le s\le s_0\), \[ \|D^2Y\|_{L^s(B_{r/2})} \le C\bigl(\|a:D^2Y\|_{L^s(B_r)} +r^{-2}\|Y\|_{L^s(B_r)}\bigr). \tag{120}\] For clarity, the successive absorption uses radii tending geometrically from \(r/2\) to \(r\): choose the interpolation coefficient smaller than the inverse of the geometric loss to the power \(2s\). The resulting series converges. This proves Equation 120 without a derivative of \(a\).

The trace-reversed identity. In Euclidean coordinates put \(H=D^2z\), \(q=1-\chi\), and \(W=|Dz|^2/2\). Since \(He= D W/|Dz|\) wherever \(Dz\ne0\), direct multiplication gives the identity, also valid when \(Dz=0\), \[ A DW-GDz=(H-(\operatorname{tr}H)I)Dz, \qquad \operatorname{div}\bigl((H-(\operatorname{tr}H)I)Dz\bigr) =|H|^2-(\operatorname{tr}H)^2. \tag{121}\] Moreover \(\operatorname{tr}H=qH(e,e)+G\). Choose \((1-c)^2<\theta<1\). Young’s inequality then gives \[(\operatorname{tr}H)^2\le\theta |H|^2+C_cG^2.\] If \(|Dz|\le1\) and \(S=1-|Dz|^2\), we consequently have \[ \operatorname{div}(A DS+2GDz) \le-\kappa |D^2z|^2+C_cG^2,\qquad \kappa>0. \tag{122}\] This is a divergence inequality with measurable coefficients; it has no \(D\chi\) term. Intrinsically the last member of Equation 121 gains \(\operatorname{Ric}_g(\nabla z,\nabla z)\). Passing to coordinates with the volume density and replacing covariant by ordinary derivatives adds bounded flux and scalar errors. On rescaled balls whose metric tends to the Euclidean metric in \(C^1\), whose smooth-side curvature tends to zero, and whose normalized forcing tends to zero, these errors have the form \[ \operatorname{div}(a_1 DS+B) \le-\kappa_1|\nabla^2z|^2+\varepsilon, \qquad |B|\le\varepsilon, \tag{123}\] with fixed ellipticity constants and \(\varepsilon\to0\).

A gradient drop or entry into an affine regime. Fix \(0<r_*<1/32\). For every \(b_0>0\) there exist \(k_*\in(r_*,1)\) and an error tolerance \(\varepsilon_*>0\) such that a normalized solution on \(B_1\), with \(|\nabla z|\le1\) and errors at most \(\varepsilon_*\), satisfies one of \[ \sup_{B_{r_*}}|\nabla z|\le k_*; \qquad \sup_{B_{r_*}}|z-L|\le b_0r_*,\quad \tfrac12\le|DL|\le2 \tag{124}\] for an affine \(L\). To prove this, suppose the assertion fails, let the errors tend to zero, and let the gradient supremum on \(B_{r_*}\) tend to one. The scalar weak Harnack inequality (Gilbarg and Trudinger 2001, Theorem 8.18), applied to the nonnegative supersolution \(S\) in Equation 123, gives \[\left(\frac{1}{|B_{1/3}|}\int_{B_{1/3}} S^p\right)^{1/p} \le C\bigl(\inf_{B_{r_*}}S+\varepsilon\bigr)\longrightarrow0\] for a fixed \(p>0\). Equation 120 gives a uniform local \(L^2\) bound for \(DS\). Here is the extra truncation step needed to retain the negative Hessian term. Testing the supersolution inequality by \(\zeta^2(b-S)_+\) yields \[ \int_{\{S<b\}}\zeta^2|DS|^2 \le Cb^2\int|D\zeta|^2+C\|B\|_\infty^2 +C\varepsilon b. \tag{125}\] Indeed the derivative of \((b-S)_+\) contributes the negative of the elliptic energy on \(\{S<b\}\); the cutoff and \(B\) terms are absorbed by Young’s inequality. Since \(S\to0\) in measure, split \(DS\) into the sets \(\{S<b\}\) and \(\{S\ge b\}\). The first has limiting \(L^1\) norm at most \(Cb\) by Equation 125; the second has vanishing \(L^1\) norm by the uniform \(L^2\) bound and its vanishing measure. Thus \(DS\to0\) locally in \(L^1\). Testing Equation 123 with a fixed nonnegative cutoff now gives \(D^2z\to0\) locally in \(L^2\). After subtraction of a constant, the gradient bound gives uniform compactness, and the limit is affine. Its slope has length one because \(S\to0\) in measure. This contradicts failure of the second alternative in Equation 124.

Improvement near a nonzero affine function. Suppose on \(B_1\) that \(|z-L|\le b\le b_0\), \(1/4\le|DL|\le4\), and the gradient is bounded by a fixed constant. Suppose also that metric oscillation, connection errors, and forcing are at most \(b\delta\). For \(Y=(z-L)/b\), Equation 120 gives a uniform \(W^{2,s_0}\) bound on smaller balls; its source is bounded by \(C\delta\). Put \(e_0=DL/|DL|\). The directions satisfy \[|e\otimes e-e_0\otimes e_0| \le C\min\{1,b|DY|\}+o(1).\] Hence they converge in measure to the fixed axis as \(b_0,\delta\to0\). Hölder’s inequality, with exponent \(2s_0/(s_0-2)\) for this bounded coefficient error, proves \[ \|R\|_{L^2(B_{3/4})}\longrightarrow0, \qquad R=(\Delta_{e_0^\perp}+\chi\partial_{e_0}^2)Y. \tag{126}\] Only the axis is frozen; \(\chi\) remains an arbitrary measurable function of position.

There is a zero-Dirichlet correction \(U\in W^{2,2}(B_{3/4})\cap W^{1,2}_0(B_{3/4})\) with \[(\Delta_{e_0^\perp}+\chi\partial_{e_0}^2)U=R, \qquad \|U\|_{W^{2,2}}\le C_c\|R\|_2.\] Indeed solve \(U=\Delta_D^{-1}((1-\chi)U_{e_0e_0}+R)\) by contraction in the Hessian norm. On a convex ball integration by parts gives \[\int |D^2U|^2\le\int(\Delta U)^2;\] the difference is the boundary mean curvature times \((\partial_\nu U)^2\), which is nonnegative. Thus the contraction factor is at most \(1-c\). Poincaré’s inequality controls the lower derivatives. In dimension three the Sobolev embedding \(W^{2,2}\hookrightarrow C^{0,1/2}\) gives \(\|U\|_\infty\to0\). This is the dimension-specific uniform correction required here.

Write \(Y_0=Y-U\), and rotate so that \(e_0\) is the third coordinate axis. Then \(v=\partial_3Y_0\in W^{1,2}\) satisfies \[ \partial_1^2v+\partial_2^2v+\partial_3(\chi\partial_3v)=0 \tag{127}\] distributionally: it is exactly the distributional derivative of \(\Delta_{e_0^\perp}Y_0+\chi\partial_3^2Y_0=0\). The scalar divergence-form Hölder estimate (Gilbarg and Trudinger 2001, chap. 8) and its energy inequality therefore give, on smaller balls, \[[v]_{C^{\alpha_1}}\le C,\qquad \int_{B_r}|Dv|^2\le Cr^{1+2\alpha_1}\] for some \(\alpha_1\in(0,1)\). Since \(\Delta Y_0=(1-\chi)\partial_3v=:F_0\), Cauchy–Schwarz gives \[ \int_{B_r}|F_0|\le Cr^{2+\alpha_1}. \tag{128}\] This also controls all the first derivatives of \(Y_0\). To see the point explicitly, localize \(F_0\) to a slightly larger ball and take its Newton potential \(P\). In the difference \(DP(x)-DP(y)\), with \(h=|x-y|\), the two near regions have size bounded by \(C\sum_{j\ge0}(2^{-j}h)^{\alpha_1}\le Ch^{\alpha_1}\), by Equation 128. On the annuli of radius \(r\ge2h\), the derivative of the gradient kernel is bounded by \(Cr^{-3}\), giving \(Ch\sum_{r=2^jh}r^{\alpha_1-1}\le Ch^{\alpha_1}\). The remainder \(Y_0-P\) is harmonic. Its local norms are bounded by the already established \(W^{2,2}\) bound and the potential bound. Consequently \(DY_0\) is uniformly Hölder on an interior ball.

Choose \(0<\alpha_2<\alpha_1\), then a sufficiently small fixed \(\lambda\in(0,1/8)\) so that the Taylor error of \(Y_0\) is at most \(\frac12\lambda^{1+\alpha_2}\). Finally choose \(b_0\) and \(\delta\) so that the uniform correction is smaller than the other half. We obtain an affine \(L'\) with \[ \sup_{B_\lambda}|z-L'|\le b\lambda^{1+\alpha_2}, \qquad |DL'-DL|\le Cb. \tag{129}\]

Iteration and the boundary interface. In the improvement statement \(b\) is an allowed upper bound for the error, and need not equal the actual error. Fix \(b_0,\delta,\lambda\) as above, decreasing \(b_0\) so that \(Cb_0/(1-\lambda^{\alpha_2})<1/4\), and then choose \(k_*,\varepsilon_*\) in Equation 124. Initially shrink to a radius \(R_0\) and normalize the gradient by a fixed bound \(\Lambda_0\) so that all errors are at most \(\min(\varepsilon_*,b_0\delta)\). After \(j\) drops the physical radius and gradient normalization are \(R_j=R_0r_*^j\) and \(\Lambda_j=\Lambda_0k_*^j\). In particular the normalized forcing is \(R_jG/\Lambda_j\), and is multiplied by \(r_*/k_*<1\) at each drop. Metric and connection errors decrease at least by \(r_*\) and curvature errors by \(r_*^2\). At the first affine entry the physical radius is \(R_e=r_*R_j\), with the same \(\Lambda_j\). Rescale that ball and apply Equation 129 with the prescribed upper bound \(b=b_0\), even if the actual affine error is smaller. On iteration \(i\) use \(b_i=b_0\lambda^{i\alpha_2}\). The forcing divided by \(b_i\) is at most \(\delta\lambda^{i(1-\alpha_2)}\); the same comparison controls the connection errors. The slope increments are summable by the choice of \(b_0\), so the axis remains separated from zero. Only constants are subtracted in the rescaled original equation, so its axis is always the actual gradient axis. These two regimes give uniform affine approximation of order \(r^{1+\alpha}\), including the transition scale, for any sufficiently small \[0<\alpha\le \min\{\alpha_2,\log k_*/\log r_*\}.\] The affine approximation criterion for Hölder gradients can also be seen directly: affine slopes at consecutive dyadic scales differ by \(Cr^\alpha\), hence converge; comparing overlapping balls gives the same modulus between two centers. Thus the first estimate in Equation 119 follows. Subtract the first jet in Equation 120. Its error is \(O(r^{1+\alpha})\); connection terms are bounded. Squaring the resulting \(L^2\) estimate proves the second estimate.

For a face use Gaussian coordinates \((x',x_3)\), subtract its constant value, reflect \(z\) oddly and \(g\) evenly. The reflected function is \(C^1\cap W^{2,s}\), the metric is Lipschitz, and each smooth side has bounded curvature. The coordinate Hessian estimate applies unchanged. In the trace-reversed identity the only additional term is the jump of the normal flux at \(x_3=0\). Since the tangential gradient vanishes there, \[\bigl((\nabla^2z-(\Delta z)g)\nabla z\bigr)\cdot\nu =-(\partial_\nu z)\operatorname{tr}_{T}\nabla^2z.\] The constant boundary value gives \(|\operatorname{tr}_{T}\nabla^2z|\le C|\mathrm{II}|\,|dz|\). Thus the jump of the density-weighted flux is a bounded plane density \(J(x')\), of size \(C|\mathrm{II}|\,|dz|^2\). It can be cancelled exactly by adding the coordinate vector \(-J(x')\mathbf1_{\{x_3>0\}}\partial_3\) to the flux; its divergence is \(-J\delta_{\{x_3=0\}}\), and no tangential derivative of \(J\) is used. Under a spatial scale \(R_j\) and gradient normalization \(\Lambda_j\), the normal flux jump becomes \(R_jJ/\Lambda_j^2\). Since the current gradient bound is \(|dz|\le\Lambda_j\), its size is at most \(CR_j|\mathrm{II}|\), also after any number of gradient drops. The smooth-side Ricci error is at most \(CR_j^2|\operatorname{Ric}|\). Both therefore tend to zero independently of the shrinking gradient normalization. Hence Equation 123, the compactness alternative, and the affine improvement all remain valid across the face. This proves the boundary assertion. ◻

Proof. For the bounded flux part, scale to the unit ball. Testing by \((v-k)_+\) gives \(\int_{\{v>k\}}|Dv|^2\le C\|B\|_\infty^2|\{v>k\}|\). Sobolev’s inequality and the level-set iteration give \(\|v\|_\infty\le C\|B\|_\infty\) on the unit ball; the same argument for \(-v\) gives the other sign. Rescaling gives \(CrC_0\). For the measure part the scalar Dirichlet Green function has the bound \(0\le G(x,y)\le C|x-y|^{-1}\) for bounded measurable symmetric uniformly elliptic coefficients. This follows from the whole-space bound in (Littman et al. 1963, Theorem 7.1 and Equation (7.9)) by domain comparison. The coefficients may be extended elliptically outside the ball; symmetry is preserved. Integrating on dyadic balls centered at \(x\) gives \[\int G(x,y)\,d\mu(y) \le C\sum_{j\ge0}(2^{-j}r)^{-1} (2^{-j}r)^{1+\gamma} \le Cr^\gamma.\] For completeness the measure belongs to \(H^{-1}\): its Newton energy is bounded by the displayed potential bound times \(\mu(B_r)\), and the Newton energy identity bounds its action on \(H^1_0\). Mollifications preserve the mass-growth bound uniformly. Indeed, at scales \(s<\varepsilon\) their density is bounded by \(C\varepsilon^{\gamma-2}\), so their mass is at most \(Cs^3\varepsilon^{\gamma-2}\le Cs^{1+\gamma}\); at larger scales use the original bound on a ball enlarged by \(\varepsilon\). The bounded divergence-data estimate can first be applied to smooth matrices and then passed to the weak solution by its energy bound. The measure approximations have uniformly bounded \(H^{-1}\) norm and uniformly bounded potentials, so weak \(H^1_0\) compactness and weak-star \(L^\infty\) compactness give the same estimate for their limit. This proves the lemma. ◻

Proof. The bounds in Proposition 33 give uniform ellipticity of \(A_\chi\) and boundedness of every smooth function of \(x,Z,df\) that occurs in the equations. The trace equation, divided by \(lD^{-1/2}\), is \[ A_\chi:\nabla^2 f =\frac{\sqrt D}{l}\bigl(\eta f- \operatorname{tr}_{A_\chi}K_a\bigr). \tag{130}\] Its right side is bounded. Lemma 34 therefore bounds \(df\) in \(C^\alpha\) and gives \(\int_{B_r}|\nabla^2f|^2\le Cr^{1+2\alpha}\). The exact differential identity in Equation 80 gives \[ |dt|\le C(|dZ|+|\nabla^2 f|+1). \tag{131}\] Write the second equation as \[\operatorname{div}(M dZ+J)=F_1,\qquad M=2uA_\chi,\quad J=uA_\chi K_a(w,\cdot).\] Here \(M\) is symmetric uniformly elliptic and \(J\) is bounded. The quadratic formula for \(\mathcal T\) and Equation 131 imply \[ |F_1|\le C(1+|dZ|^2)+C|\nabla^2f|^2. \tag{132}\] All volume densities and connection terms can be incorporated into \(M,J,F_1\) without changing these assertions. Set \(d\mu=(1+|\nabla^2f|^2)\,dx\). For a fixed \(0<\gamma<\min(1,2\alpha)\) this measure satisfies \(\mu(B_r)\le Cr^{1+\gamma}\).

Choose a sufficiently large fixed \(L\) and put \(Q_\pm=e^{\pm LZ}\). For either sign the chain rule gives \[-\operatorname{div}(M dQ_\pm) =\operatorname{div}(Q_\pm'J)-Q_\pm'F_1 -Q_\pm''\bigl(M(dZ,dZ)+J\cdot dZ\bigr).\] The range of \(Z\) is bounded. Ellipticity and Young’s inequality show that the last term absorbs the quadratic term in Equation 132 if \(L\) is large. Thus \[ -\operatorname{div}(M dQ_\pm) \le\operatorname{div}B_\pm+C\mu,\qquad |B_\pm|\le C. \tag{133}\] By Lemma 35, subtracting the zero-Dirichlet potential of the right side on \(B_r\) changes \(Q_\pm\) by at most \(C(r+r^\gamma)\) and makes it a subsolution. The corrected supremum deficit is nonnegative and is a supersolution. Apply weak Harnack to that deficit. On at least half of \(B_{r/2}\), either \(Z\) is below the midpoint of its range on \(B_r\), or it is above that midpoint. In the first case the \(Q_+\) deficit, and in the second the \(Q_-\) deficit, is at least a fixed multiple of \(\operatorname{osc}_{B_r}Z\), up to the potential error. Weak Harnack then decreases one end of the range on \(B_{r/4}\) by a fixed fraction. More precisely, \[\operatorname{osc}_{B_{r/4}}Z \le\theta\operatorname{osc}_{B_r}Z+C(r+r^\gamma), \qquad 0<\theta<1.\] Iteration proves a uniform Hölder modulus for \(Z\). At an outer face both unknowns have zero trace. In flattened coordinates let \(P=\operatorname{diag}(1,1,-1)\). After incorporating the volume density, reflect by \(Z^-=-Z^+\), \(M^-=PM^+P\), \(J^-=-PJ^+\), and \(F_1^-=-F_1^+\), where opposite-side values are evaluated at reflected points. The reflected total flux is \(-P(M^+dZ^++J^+)\), so its normal component has matching traces and creates no plane source. For the principal part this also follows from the vanishing tangential gradient on the constant-data face. In the actual application \(K_a=0\) near the outer face, hence \(J=0\) there. The equation and its growth estimate hold across the face, and the same oscillation argument applies there.

The coefficients and right side of Equation 130 are now Hölder. Interior and Dirichlet Schauder estimates give \(C^{2,\alpha'}\) bounds for \(f\), with some \(\alpha'>0\). Expanding the second equation, all derivatives of its coefficients are derivatives of smooth functions of \(x,Z,df\). Consequently it has the form \[ M^{ij}\partial_{ij}Z=H(x,Z,dZ),\qquad |H(x,Z,dZ)|\le C(1+|dZ|^2), \tag{134}\] with a uniformly Hölder principal matrix. No bound for \(dZ\) has been used to reach this point.

Here is the absorption which supplies that bound. For \(s>3\) the local \(W^{2,s}\) estimate for a Hölder principal matrix has a uniformly bounded leading constant on sufficiently small patches. On balls or constant-Dirichlet half-balls the scaled derivative interpolation inequality is \[ \|dZ\|_{L^{2s}(Q)}^2 \le C\operatorname{osc}_{Q'}Z\, \|D^2Z\|_{L^s(Q')} +C r^{3/s-2}(\operatorname{osc}_{Q'}Z)^2, \tag{135}\] where \(Q\Subset Q'\) are concentric patches of comparable radius \(r\). It follows by applying the \(L^\infty\)–\(W^{2,s}\) interpolation inequality to a cutoff times \(Z-c\); take \(c\) between its maximum and minimum. For half-balls subtract the zero boundary value and reflect, or use a boundary cutoff in flattened coordinates. The Hölder modulus makes the oscillation coefficient as small as needed by fixing a sufficiently small patch radius. To justify absorption locally without any bound depending on the number of patches, let \(X\) be the supremum of their \(L^s\) Hessian norms. Each enlarged patch is covered by a fixed number of the smaller patches. The local estimate and Equation 135 give \(X\le\frac12X+C\). This supremum is individually finite on every smooth compact truncation. Thus \(X\le2C\), independently of its radius. Sobolev embedding gives a Hölder gradient for \(Z\). Schauder estimates applied to both equations now give second derivative bounds; differentiating gives bounds of every fixed order. The background geometry, end symbols, and outer sphere charts have uniform bounds at the chosen patch scale. The constants are therefore independent of the sufficiently large truncation radius. ◻

Proof. All constants here may depend on a bound for the trial’s \(C^{1,\alpha}\) norm and on this fixed domain. The global implicit relation in Equation 66 makes \(t\), \(l\), \(D\), and \(A_\chi\) smooth functions of \((x,Z,df)\), even at \(df=0\). The normalized equation is \[A_\chi:\nabla^2f-qf+\frac{\sqrt D}{l} \operatorname{tr}_{A_\chi}K_a=0, \qquad q=\eta\frac{\sqrt D}{l}>0.\] Continue it from \(\Delta_gf=f\) by using, for \(\sigma\in[0,1]\), \[ A_\sigma:\nabla^2f-q_\sigma f +\sigma\frac{\sqrt D}{l} \operatorname{tr}_{A_\chi}K_a=0, \quad A_\sigma=(1-\sigma)I+\sigma A_\chi, \quad q_\sigma=1-\sigma+\sigma q. \tag{136}\] The coefficient of \(f\) is strictly negative in this convention. At a positive maximum \(df=0\), so \(t=Z\), \(D=1\), and \(A_\chi=I\). The maximum principle gives \[|f|\le H:=\eta^{-1}\|\operatorname{tr}_gK\|_\infty\] for the entire continuation. If \(K=0\) it gives \(f=0\).

We prove a gradient bound before using uniform ellipticity. Write \(s=|df|\). When \(s\to\infty\) with \(Z\) in a bounded interval, the implicit equation forces \(t\to-\infty\). There \(p=N\) and \(l=e^{Nt}\), and the defining equation gives \[ t=-\frac{\log s}{N+2}+O(1),\qquad d\asymp s^{-4/(N+2)},\qquad \chi\ge\frac{c}{1+s},\qquad \frac{\sqrt D}{l}\le C(1+s). \tag{137}\] These estimates are uniform for bounded trial values; on a bounded \(s\) interval they follow from smoothness and positivity. The lower bound for \(\chi\) uses \(N>2\). Notice that it does not use the floor. The axial eigenvalue \(\chi_\sigma=1-\sigma+\sigma\chi\) obeys the same lower bound, while both transverse eigenvalues equal one. Since \(|f|\le H\), Equation 136 has normalized source of absolute value at most \(C(1+s)\).

Take a short fixed collar of the boundary and a smooth extension of the background metric across it. Choose a concave increasing modulus \[\psi(d)=K_0^{-1}\log(1+K_0Md),\qquad \psi''=-K_0(\psi')^2.\] First choose \(K_0\) sufficiently large depending on the source and geometry bounds. Then choose a sufficiently short \(d_0\) and a sufficiently large \(M\) so that \(\psi'\ge1\) on \([0,d_0]\) and \(\psi(d_0)>2H\). Such choices are compatible: one can first make \(K_0d_0<1/2\) and then let \(M\to\infty\). For the distance \(d\) to the boundary, the Hessian in its gradient axis is \(\psi''\); the transverse trace is bounded by \(C\psi'\). Equation 137 gives \[\chi_\sigma\psi''+C\psi' \le-cK_0\frac{(\psi')^2}{1+\psi'}+C\psi' \le(-cK_0/2+C)\psi'.\] For large \(K_0\) this is strictly more negative than the possible source. The negative barrier has the opposite strict inequality. At a first contact the gradients agree, and hence all the coefficients depending on the gradient agree. The strict sign and positivity of \(q_\sigma\) therefore exclude contact. The boundary values and the choice \(\psi(d_0)>H\) give \[ |f(x)|\le\psi(\operatorname{dist}(x,\partial\Omega_{N,R})) \quad\hbox{in the collar}. \tag{138}\]

For the interior consider \(f(x)-f(y)-\psi(\operatorname{dist}(x,y))\) for distances at most \(d_0\), using the short geodesic distance in the smooth extended metric. The value is nonpositive on the diagonal, at distance \(d_0\), and whenever one endpoint lies on the boundary, the last assertion by Equation 138 and monotonicity of \(\psi\). At a hypothetical positive interior maximum the endpoint gradients are parallel along the short geodesic and have the same magnitude \(\psi'\). Write \(e_x,e_y\) for that common transported direction. Varying just the first endpoint in its axial direction and then just the second endpoint gives, respectively, \[\nabla^2 f_x(e_x,e_x)\le\psi'',\qquad \nabla^2 f_y(e_y,e_y)\ge-\psi''.\] Paired parallel transverse endpoint variations, summed over the two transverse directions, give \[\operatorname{tr}_{e_x^\perp}\nabla^2f_x -\operatorname{tr}_{e_y^\perp}\nabla^2f_y \le C\operatorname{dist}(x,y)\psi'.\] This last bound is the second variation of the short geodesic length; the endpoint Jacobi fields have equal parallel initial and final values, and their index forms are bounded by curvature times its length. Thus the difference of the principal operators is at most \[ (\chi_\sigma(x)+\chi_\sigma(y))\psi''+Cd_0\psi' \le(-cK_0+Cd_0)\psi'. \tag{139}\] The right sides of the two equations have difference bounded below by \(-2C(1+\psi')\). Increasing \(K_0\) gives a contradiction. Crucially, Equation 139 uses the two axial coefficients separately; no modulus of \(\chi(x)-\chi(y)\) is assumed. We conclude \(|f(x)-f(y)|\le\psi(\operatorname{dist}(x,y))\) for short distances, and hence \(|df|\le M\).

Uniform ellipticity now follows from this gradient bound. The matrix \(A_\sigma\) still has precisely the structure of Lemma 34, and its right side is bounded. That lemma gives a Hölder gradient. Schauder estimates then give \(C^{2,\beta}\) bounds for some \(\beta>0\). At this stage \(df\) is Lipschitz, and the prescribed \(Z\) is \(C^{1,\alpha}\), so the coefficients and source of Equation 136 are \(C^\alpha\). Schauder estimates give \(C^{2,\alpha}\) bounds. Differentiating the equation and applying the same estimates, with the boundary estimates for tangential derivatives followed by the equation for the normal second derivative, gives \(C^{3,\alpha}\) bounds.

The linearization in \(f\) is uniformly elliptic with Hölder coefficients, bounded first-order terms, and zero-order term \(-q_\sigma<0\). Derivatives of the gradient-dependent coefficients produce only first-order terms, because those coefficients do not depend on the value of \(f\). The Dirichlet maximum principle and linear Schauder solvability make this linearization an isomorphism \(C^{2,\alpha}_0\to C^{0,\alpha}\). The implicit function theorem gives openness of the set of continuation parameters. The uniform \(C^{3,\alpha}\) bounds give compactness in \(C^{2,\alpha}\) and hence closedness. At \(\sigma=0\) the unique solution is zero, so the set is all of \([0,1]\). If two solutions of the final equation differed, a positive maximum of their difference would have matching gradients and an ordered Hessian, while their common \(q\) is positive. This contradicts the equations. Thus the solution is unique. Finally the same implicit function theorem, now with parameters \((a,Z)\), gives the asserted continuous dependence. Its local branches agree globally by uniqueness. ◻

Proof. Fix \(R\ge R_{\min}(N)\) as in Lemma 28. Parametrize the homotopy by \(\lambda\in[0,2]\): \[(a(\lambda),b(\lambda))= \begin{cases}(1-\lambda,1),&0\le\lambda\le1,\\ (0,2-\lambda),&1\le\lambda\le2. \end{cases}\] All quantities in the trace and scalar identities use \(K_a=aK\); the divergence equation is \(\operatorname{div}V=b\,u\Xi\). Let \(X=C^{1,\alpha}_0(\overline{\Omega_{N,R}})\) for a fixed \(\alpha\in(0,1)\). For a trial \(Z\in X\) solve the trace equation by Proposition 37, and evaluate \(t,w,u,A_\chi,\mathcal T\) and \(\Xi\) at this pair. Define \(T_\lambda Z=Y\) by the linear zero-Dirichlet equation \[ \operatorname{div}(2uA_\chi dY) =b\,u\Xi-\operatorname{div}(uA_\chi K_a(w,\cdot)). \tag{141}\] Only the gradient of the new unknown \(Y\) is left unfrozen. On a bounded set of trials, Proposition 37 bounds \(f\) in \(C^{2,\alpha}\), and all implicit coefficients have bounded \(C^{1,\alpha}\) norms. The scalar right side has bounded \(C^{0,\alpha}\) norm, since \(dt\) is computed from Equation 80. Ellipticity is uniform on that bounded set by the trial gradient estimate, even when \(t<-\epsilon\). Linear Dirichlet Schauder theory gives a uniform \(C^{2,\alpha}\) bound for \(Y\). Thus \((\lambda,Z)\mapsto T_\lambda Z\) is continuous and compact into \(X\). At a fixed point the linear solve first gives \(Z\in C^{2,\alpha}\), whereas the trace solution is already \(f\in C^{3,\alpha}\). Thus \(\Xi\in C^{1,\alpha}\) and the principal coefficient and drift flux are \(C^{2,\alpha}\). Schauder estimates give \(Z\in C^{3,\alpha}\), and the trace equation gives \(f\in C^{4,\alpha}\). Repeating gives smoothness. In particular the coupled a priori estimates apply to individually smooth fixed points; their eventual bounds for all fixed derivative orders control the chosen \(X\) norm, regardless of its initially prescribed exponent \(\alpha\).

Choose a radius \(M_X\) larger than the \(X\) norm of every above-floor fixed point, which is possible by Proposition 36 and the fixed truncation. Consider the relatively open bounded set \[ \mathcal U=\{(\lambda,Z):\|Z\|_X<M_X, \ \min_{\overline{\Omega_{N,R}}}t[Z,df[a(\lambda),Z]] >-\epsilon\}. \tag{142}\] Its openness follows from the continuous dependence in Proposition 37. A fixed point cannot occur at its norm boundary by the choice of \(M_X\). At its other boundary it would have minimum \(t=-\epsilon\) and would be above the closed floor. Lemma 28 excludes such a minimum on the outer boundary; Lemma 29 excludes it in the interior. There is no inner boundary in the filled manifold.

For completeness the variation of the domain in Equation 142 presents no degree obstruction. The images of the bounded parameter-trial cylinder under \(T\) have compact closure. Therefore any convergent sequence of parameters and associated boundary fixed points would converge to a boundary fixed point, already excluded. Around any fixed parameter the compact set of its fixed points has an open neighborhood whose closure lies inside its slice of \(\mathcal U\). By compactness the same neighborhood contains every fixed point in the relevant slices for all sufficiently nearby parameters; otherwise a contrary sequence converges to an excluded fixed point. Excision and the usual fixed-domain homotopy invariance of Leray–Schauder degree show that the degrees on the slices are locally constant. A finite interval covering makes them constant along the full path. At \(\lambda=2\) we have \(K_a=0\), \(f=0\), and \(b=0\). The linear solve in Equation 141 is therefore identically zero for every trial. Its sole fixed point is \(Z=0\), with \(t=0>-\epsilon\), and the degree of \(I-T_2\) on the terminal slice is one. Hence the initial slice has degree one and contains a fixed point. This proves existence on every sufficiently large truncation, with the estimates already established.

We next verify the end before taking the exhaustion limit. All bounds in the remainder of the proof are uniform in the truncation radius at this fixed \(N\). Outside a fixed sphere \(K=0\), so Equation 130 is the homogeneous proper equation \[ A_\chi:\nabla^2 f-q(x)f=0,\qquad q(x)=\eta\frac{\sqrt D}{l}\ge\eta/e>0. \tag{143}\] The eigenvalues of \(A_\chi\) stay in a fixed positive interval. For a sufficiently small fixed \(c>0\), the functions \(C e^{-c(r-r_0)}\) have \((A_\chi:\nabla^2-q)C e^{-c(r-r_0)}<0\) for \(r_0\) sufficiently large. Indeed their Hessian is bounded by \(C(c^2+c/r)e^{-c(r-r_0)}\), whereas the zero-order term is at least \((\eta/e)C e^{-c(r-r_0)}\). Choose \(c\) first, then \(r_0\). The negatives give the opposite inequality. Choose \(C\) to dominate the uniformly bounded values on \(r=r_0\); the outer Dirichlet data are zero. Comparison gives \(|f|\le Ce^{-cr}\) on every truncated end. The local estimates in Proposition 36 give uniform smooth coefficients for Equation 143; its interior and boundary estimates then give exponential decay of every fixed derivative of \(f\). Alternatively, interpolation between this exponential height bound and the uniform higher derivative bounds gives the same conclusion, with a possibly smaller positive exponent for each order. The implicit relation implies that \(t-Z\) and its fixed derivatives are exponentially small.

On the bounded fixed-\(N\) range of \(Z\), define the smooth increasing function \(\Phi\) by \[ \Phi(0)=0,\qquad \Phi'(0)=1,\qquad \frac{\Phi''(z)}{\Phi'(z)} =1+p(z)-\delta_0s_{p(z)},\qquad s_{p}=p+\tfrac12. \tag{144}\] Its derivative is bounded above and away from zero on that range. Putting \(f=K=0\) in Lemma 27 gives exactly \[2\Delta_g Z+2\bigl(1+p(Z)-\delta_0s_{p(Z)}\bigr)|dZ|^2 =\rho-\vartheta(N(Z+\epsilon))C_N\rho_0.\] The exponential bounds just proved and the uniform local derivative bounds show that the actual end equation differs from this one by \(O_j(e^{-c_jr})\) for every fixed order. Thus \[ |\Delta_g\Phi(Z)|\le C r^{-3-\beta},\qquad \Phi(Z)|_{r=R}=0. \tag{145}\] This scalar Poisson bound is independent of the radius \(R\).

Here are explicit barriers fixing the end constant. Put \(\gamma=\beta/2\) and, after increasing \(r_0\), let \(W(r)=r^{-1}-r^{-1-\gamma}>0\). The prepared metric has \(g-\delta=O_j(r^{-1})\). Since \(\Delta_\delta r^{-1}=0\) and \(\Delta_\delta r^{-1-\gamma} =\gamma(1+\gamma)r^{-3-\gamma}\), \[\Delta_g W\le-c_\gamma r^{-3-\gamma}\] for large \(r_0\). Choose a constant multiple of \(W\) to dominate the inner boundary values and the right side of Equation 145; this is possible because \(\gamma<\beta\). The zero outer boundary values lie between the positive and negative barriers. The maximum principle therefore gives \(|\Phi(Z)|\le Cr^{-1}\), and the bounds for \(\Phi'\) give \[ |Z|+|t|\le Cr^{-1} \tag{146}\] on every truncated end. In particular no undetermined additive constant is introduced in the exhaustion.

We record the derivative decay rather than infer it from an unscaled estimate. Rescale an annulus of radius \(r\) to unit size. The bounded right side of Equation 145 and Equation 146 give, by local Poisson \(W^{2,s}\) estimates for \(s>3\), \(|d\Phi(Z)|\le Cr^{-2}\) and a scaled Hölder bound for its gradient. The exponentially small error has all unscaled derivatives exponentially small, so it remains negligible after this rescaling. The other right-side term is a smooth function of \(Z\) times the prescribed weight \(\rho_0\); its scaled Hölder norms are bounded by \(Cr^{-3-\beta}\). Schauder estimates give \(D^2Z=O(r^{-3})\). Differentiating the transformed equation and repeating the scaled estimates inductively gives \(Z=O_j(r^{-1})\) for every fixed \(j\). Since \(t-Z\) is exponential, the same holds for \(t\). Finally the untransformed equation gives \[\Delta_g t=O(r^{-3-\beta})+O(|dZ|^2)+O(e^{-cr}) =O(r^{-3-\beta}),\] because \(\beta<1\) and \(|dZ|^2=O(r^{-4})\).

Take any sequence \(R\to\infty\). The compact-patch estimates and a diagonal subsequence give a smooth solution on \(\Omega_N\), with all bounds above and the stated end estimates. Initially the limit has \(t\ge-\epsilon\). If equality held at a point, it would be an interior minimum of a smooth solution and the same strict contradiction in Lemma 29 would apply. Hence \(t>-\epsilon\). No limit in \(N\) has been taken. Since \(Z\to0\) and \(N\epsilon>2\), the cutoff \(\vartheta(N(t+\epsilon))\) vanishes sufficiently far out. The exact zero-trace expression for \(\mathcal T\) and the exponential trace error give \(\mathcal T=O(r^{-4})+O(e^{-cr})\) there. Also \(\eta a_w|df|\) is exponential, \(\rho=O(r^{-3-\beta})\), and \(u=e^Zl(Z)+O(e^{-cr})\to1\). These functions are integrable over the three-dimensional end, and smoothness handles the compact part. Thus \(u\Xi\in L^1(\Omega_N)\) and the final assertions follow. ◻

Proof. The end conclusion of Theorem 38 gives, at fixed \(N\), exponential decay of \(f\) and each of its fixed-order derivatives, and \[t,Z=O_j(r^{-1}),\qquad t-Z=O_j(e^{-c_jr}),\qquad \Delta_g t=O(r^{-3-\beta}).\] Here the constants and the positive \(c_j\) may depend on \(N\). Since \(K=0\) sufficiently far out, \(\mathcal T=O(r^{-4})\) up to exponentially decaying terms, \(u=1+O(r^{-1})\), and \(\eta a_w|df|\) decays exponentially. For \(N\epsilon>2\), the penalty cutoff \(m_0=\vartheta(N(t+\epsilon))\) vanishes sufficiently far out. Consequently \(u\Xi\) is absolutely integrable. The divergence equation and the absence of any boundary in \(\Omega_N\) prove the flux identity in Equation (150).

The definition of \(V\), \(A_\chi=\mathop{\mathrm{Id}}+O_j(e^{-c_jr})\), and the same end estimates show that \[V=2\nabla t+O(r^{-3})+O(e^{-cr}).\] Thus its limiting flux is twice the limiting flux of \(dt\). The graph term \(l^2df^2\) makes no contribution to ADM energy. Direct substitution of \(e^{2t}g\) into the ADM integral gives \[E(e^{2t}g)-E(g) =-\frac1{4\pi}\lim_{R\to\infty} \int_{r=R}\partial_{\nu_R}t\,dA_g.\] The terms containing \((g-\delta)dt\), \(t\,\partial g\), or \(t\,dt\) have integrals \(O(R^{-1})\); changing between Euclidean and \(g\) normals and measures has the same vanishing cost. This proves the coefficient and sign in Equation (150) without requiring a separately chosen monopole coefficient for \(t\).

It remains to estimate the possibly negative part of the integral. Write \(\sigma=|df|_{g_N}\). On the support of \(m_0\) the floor gives \(-\epsilon<t<-\epsilon+N^{-1}\), and therefore \(\ell\le l\le e\ell\) for large \(N\). Since \(u=e^t l\sqrt D\) and \(D=1+l^2\sigma^2\), direct calculation gives \[ u\le C(\ell+\ell^2\sigma),\qquad uv\le C\ell^2\sigma,\qquad u a_w\sigma=e^tl^2\sigma^2\ge c\ell^2\sigma^2. \tag{152}\] The negative term in \(u\Xi\) is consequently bounded in absolute value by \[\mathcal P_N\ell\rho_0 +\mathcal P_N\ell^2\sigma(\rho_0+\rho_1).\] Young’s inequality, using a fixed fraction of the positive \(\delta_0\eta u a_w\sigma\), bounds the second summand by \[\frac{\delta_0\eta}{2}u a_w\sigma +\mathcal P_N\frac{\ell^2}{\eta} (\rho_0^2+\rho_1^2).\] Here \(\rho_0=r^{-3-\beta}\) and \(\rho_1=r^{-2b_1}\). Their required integrals have polynomial bounds: the compact contribution is \(O(1+N)\), while on the end \(\rho_0\), \(\rho_0^2\), and \(\rho_1^2\) are integrable, the last because \(4b_1>3\). In particular, no integral of \(\rho_1\) is being assumed finite. Positivity of \(\mathcal T\), proved in Proposition 25, now gives \[\mathfrak F_N\ge -\mathcal P_N\left(\ell+\frac{\ell^2}{\eta}\right) =-\mathcal P_N(\ell+\ell^{1/2}).\] Equation (151) follows. ◻

Proof. We give the construction when \(\sigma_i=1\). Use the signed collar coordinate \(s\) increasing toward the original end. Choose a small fixed number \(d_0>0\). On the cap put the comparison height equal to \(-d_0\). There \(K=-L_0g_N\), so the trace of \(K\) in \(A_\chi\) is \(-L_0(2+\chi)\) and this constant height is a supersolution once \(L_0\) is large.

On the adjoining product cylinder choose a nonnegative slope \(a(s)\) which starts at zero, becomes a truncated increasing exponential, and ends at a small fixed positive value \(a_0\). It can be arranged that \[|a'(s)|\le C(a(s)+\eta),\qquad \int a(s)\,ds<d_0/8.\] Indeed, grow exponentially from a value of order \(\eta\) to \(a_0\), smoothly cutting off near the first value and making the last value constant. The required length is \(O(1+|\log\eta|)=O(1+N)\), and the integral is bounded by a fixed multiple of \(a_0\). Let \(H_0(s)\) be the resulting height, starting at \(-d_0\).

For a height test \(H_0\), the Hessian tensor in the trace equation is \[\frac{\nabla^2H_0} {\sqrt{(\eta/l)^2+|dH_0|^2}}.\] On a product cylinder only its axial entry can be nonzero. The bound on \(a'\) gives \[\frac{|a'|}{\sqrt{(\eta/l)^2+a^2}} \le C(1+l)\le C(1+e).\] Multiplication by \(\chi\le1\) can only improve this estimate. A fixed choice of \(L_0\) therefore makes \(H_0\) a supersolution on the entire product, including its small-slope starting portion.

Continue with a small fixed positive slope through the fixed transition and up to \(S\), keeping the total increase below \(d_0/4\). On these leaves the prepared extension satisfies \(H_s+\operatorname{tr}_{S_s}K<0\). Increase the slope within the fixed strict collar on the exterior side until the test height exceeds the uniform bound \(C\) in Equation (149) at its outer face. All slopes on these fixed transitions are bounded below by a fixed positive number, and all their derivatives are fixed. At a putative comparison contact the gradients of the solution and the test agree, so the solution’s floor gives \(\eta/l\le\eta/\ell=\ell^{1/2}\). At such contacts the transverse Hessian trace converges uniformly to \(H_s\), while \(\chi\to0\) uniformly and the axial Hessian contribution vanishes. Thus the full test trace converges uniformly to \(H_s+\operatorname{tr}_{S_s}K\). First choose \(d_0\) smaller than the absolute strict expansion bound on these fixed collars, and then choose \(N\) large. The trace is then below \(-d_0\), hence below the test height, at every possible contact in the remaining ramp.

For completeness, the comparison uses the actual solution \(Z\) as a fixed function. At a positive maximum of \(h-H_0\), their gradients agree, so the defining equation for \(t\) gives the same \(t\) and the same positive matrix \(A_\chi\) for both functions. Their Hessians are ordered. Consequently \(F[h]\le F[H_0]\le H_0<h\), contradicting \(F[h]=h\). There is no inner boundary, and the test dominates on the outer collar face. It follows that \(h\le H_0\), which is less than \(-d_0/2\) on the cap, neck and original boundary. A slightly smaller bound holds on a neighborhood by continuity. For \(\sigma_i=-1\) the same construction applies to \(-h\) and \(-K\). There are finitely many components, so one may take the minimum of their positive height gaps as \(b_2\). ◻

Proof. Since \(F=h=\eta f\), differentiation gives \(w(F)=\eta a_w|df|\). Substitute the equations into the scalar identity of Proposition 24. On \(|h|\le b_3\), using Equation (154) and dropping the nonnegative penalty contribution, one obtains \[ \frac{e^{2t}}2R_{\widehat g} \ge I_g(w)+\rho+(1-\delta_0)\mathcal T +(1-\delta_0)\eta a_w|df|. \tag{159}\] We estimate the cost of \(\psi\) directly through the trace equation. The graph inverse is \(A_d\), and its volume density is \(\sqrt D\). Differentiation gives the identities \[\begin{align*} |dh|_{A_d}&\le\eta/l,\tag{160}\\ e^{2t}\Delta_{\widehat g}h &=\frac{\eta}{l\sqrt D} \left(\operatorname{div}_{g_N}w+(1-p)w(t)\right), \tag{161}\\ \operatorname{div}_{g_N}w &=\operatorname{tr}_{A_d}H^f+p d\,w(t). \tag{162}\end{align*}\] For example, \(\sqrt D\,A_d\nabla h=(\eta/l)w\); the divergence formula for \(\bar g\), followed by the three-dimensional conformal Laplacian formula, proves Equation (161). Differentiating \(w=lD^{-1/2}\nabla f\) proves Equation (162).

In the axial decomposition of Section 5, \[\operatorname{tr}_{A_d}H^f =h+(d-\chi)j-\operatorname{tr}_{A_d}K.\] Since \(|d-\chi|\le d\) and \(0<d\le1\), the coercivity estimate controls \((d-\chi)j\), \(d\,w(t)\), and \(|dt|_{A_d}\) by \(\mathcal P_N\sqrt{\mathcal T}\). On the compact original region where \(\zeta'\) or \(\zeta''\) can be nonzero, Equations (161)–(162) therefore give \[ \left|e^{2t}\Delta_{\widehat g}h\right| \le\mathcal P_N\frac\eta l(1+\sqrt{\mathcal T}). \tag{163}\] This estimate uses no quantitative bound on second derivatives of the solution beyond the structural coercivity inequality.

In dimension three the conformal curvature identity reads \[ \frac{\lambda^2}{2}R_{\widetilde g} =\frac{e^{2t}}2R_{\widehat g} -2e^{2t}\Delta_{\widehat g}\psi -e^{2t}|d\psi|_{\widehat g}^2. \tag{164}\] The absolute value of its two error terms is at most \[\mathcal P_N A_N\frac\eta l(1+\sqrt{\mathcal T}) +C(A_N+A_N^2)\left(\frac\eta l\right)^2.\] Here \[A_N\frac\eta l\le\ell^{1/4},\qquad A_N^2\left(\frac\eta l\right)^2\le\ell^{1/2}.\] Young’s inequality absorbs the \(\sqrt{\mathcal T}\) term into half of \((1-\delta_0)\mathcal T\), with remainder tending to zero even after the polynomial factors. The remaining errors are absorbed by the positive minimum of \(\rho\) on this fixed transition region. Off that region \(\psi\) is constant, so there is no error. This proves Equation (157).

Compact support of \(\psi\) follows from the height bound. It leaves the end energy unchanged. The estimates in Theorem 38, the strong prepared end, and the exponential graph error imply \(e^{2t}\bar g-\delta=O_2(r^{-1})\). Its scalar curvature is integrable: on the end the conformal formula involves the integrable prepared \(R_g\), \(\Delta_g t=O(r^{-3-\beta})\), \(|dt|^2=O(r^{-4})\), and exponentially decaying graph errors. The compact part is smooth. Finally, constant scaling by \(e^{2\epsilon}\) followed by \(y'=e^\epsilon y\) multiplies ADM mass by \(e^\epsilon\). Lemma 39 proves Equation (158). ◻

Proof. Choose a constant matrix \(\mathsf R\in SO(2)\) which sends the charge vector \((Q_E,Q_B)\) to \((Q,0)\). For \(Q>0\) one can take \[\mathsf R=\frac1Q \begin{pmatrix}Q_E&Q_B\\-Q_B&Q_E\end{pmatrix};\] for \(Q=0\) take the identity. Put \((X_e,X_o)^T=\mathsf R(\mathcal E,\mathcal B)^T\) and \(\alpha=\iota_{X_e}dV_g\). A determinant-one rotation preserves both the sum of squared norms and the cross product, so \[\begin{align*} I_g(w) &=|X_e|_g^2+|X_o|_g^2+2X_o\cdot(w\times X_e)\\ &=|X_e|_g^2-|w\times X_e|_g^2 +|X_o+w\times X_e|_g^2. \tag{166}\end{align*}\] To identify the first two terms, choose an oriented \(g\)-orthonormal frame in which \(w=a_we_1\). Then \[\bar g=\operatorname{diag}(d^{-1},1,1),\qquad \bar g^{-1}=\operatorname{diag}(d,1,1),\] and the components of \(\alpha=\iota_{X_e}dV_g\) give \[ |\alpha|_{\bar g}^2 =X_{e,1}^2+d(X_{e,2}^2+X_{e,3}^2) =|X_e|_g^2-|w\times X_e|_g^2. \tag{167}\] The identity remains valid at \(w=0\) by continuity.

Combine Equations (157), (166), and (167), retaining the different scaling powers of scalar curvature and two-form norm: \[ \frac12R_{\widetilde g} \ge\lambda^{-2}I_g(w) \ge\lambda^{-2}|\alpha|_{\bar g}^2 \ge\lambda^{-4}|\alpha|_{\bar g}^2 =|\alpha|_{\widetilde g}^2. \tag{168}\] The third inequality uses \(\lambda\ge1\) from Equation (156).

Define \[(\widetilde\alpha_1,\widetilde\alpha_2)^T =\mathsf R^{-1}(\alpha,0)^T, \qquad \iota_{\widetilde{\mathcal E}}dV_{\widetilde g} =\widetilde\alpha_1, \quad \iota_{\widetilde{\mathcal B}}dV_{\widetilde g} =\widetilde\alpha_2.\] Both forms are closed. Orthogonality of \(\mathsf R\) and the isometry between vector and contracted volume-form norms in dimension three identify the sum of squared vector norms with \(|\alpha|_{\widetilde g}^2\). This proves Equation (165). Closedness is precisely divergence freedom in \(\widetilde g\). The total flux vector is \(\mathsf R^{-1}(4\pi Q,0)=(4\pi Q_E,4\pi Q_B)\).

The field associated with \(\alpha\) itself is explicitly \[\widetilde X_e=\lambda^{-3}D^{-1/2}X_e.\] At fixed \(N\), \(D-1\) and its derivatives decay exponentially, \(t=O_j(r^{-1})\), and \(\psi=0\) near infinity. Hence the original \(O_1(r^{-2})\) field decay gives the asserted decay of both new fields; the constant coordinate dilation has no effect on its order. Flux integrals of two-forms are invariant under this coordinate change.

This construction preserves the two total charges. It imposes no claim about the individual charges of the boundary components. All forms have been used only on the original exterior; their extension through a cap is unnecessary. ◻

Proof. Truncate beyond the confining radius and equip the compact region with the auxiliary second fundamental form zero. The inner boundary has expansion zero with respect to the normal pointing into the region; the outer sphere has positive expansion with respect to the normal pointing out. These are the weak inner and strict outer barriers in (Andersson and Metzger 2009, Theorem 5.1). In particular neither strict stability of the inner boundary nor an energy condition is required. Definitions 7.1–7.2 of that paper prescribe the entire inner boundary in the bounding class of trapped sets. Theorem 7.3 and Remark 7.4 give the smooth outer boundary of their union and its outermostness. The weak-barrier version permits coincidence with inner components. Since the auxiliary second form vanishes, the resulting marginal surface is minimal. Keeping the full frontier of the component incident on the end, or equivalently filling bounded complementary pockets, gives the asserted full cut. The outer mean-convex spheres exclude a further compact minimal enclosure outside the truncation. This proves outermostness in the complete exterior as well as in the compact barrier problem. ◻

Proof. Choose a fixed original coordinate sphere beyond the radius containing \(\{|h|\ge b_3/4\}\) and beyond a fixed further neighborhood. It encloses \(O_N\) for all large \(N\) and lies where \(\psi=0\). Its area in \(e^{2(t+\epsilon)}(g+l^2df^2)\) is at most \(e^{\mathcal P_N}\) by Equation (149). This proves the proposed upper bound for the infimum with a radius independent of \(N\).

We justify compactness separately at fixed \(N\). Strong asymptotic flatness gives, sufficiently far out, a foliation by spheres with positive definite outward second fundamental form. Follow their unit-normal trajectories, parametrized by the radius. Tangential metrics increase along these trajectories. Projection from an outer leaf to an inner leaf consequently contracts two-dimensional area. For almost every sufficiently large radius \(R\), clipping a bounded competitor at that leaf does not increase its perimeter: along each normal trajectory where clipping creates boundary, the bounded competitor must exit farther out; projection of that portion of its old boundary covers the new trace, with area counted with multiplicity. BV slicing and the area formula give the same inequality for finite-perimeter competitors. Thus the infimum over all bounded competitors equals the infimum in one fixed sufficiently large compact truncation. BV compactness and lower semicontinuity (Ambrosio et al. 2000, Theorem 3.23 and Proposition 3.6), and the closed containment condition \(B\supset O_N\) almost everywhere give a minimizer. Since it minimizes among all bounded competitors, the outer truncation imposes no local constraint. Large mean-convex spheres also exclude a free minimal contact with that outer truncation.

We prove that the perimeter support has no contact with \(O_N\). Write \[\gamma_N=e^{2(t+\epsilon)}\bar g.\] On the \(2s_N\)-neighborhood of \(O_N\), the plateau is constant, and \[ \widetilde g=e^{2A_N}\gamma_N, \qquad g_N\le\gamma_N\le L_N^2 g_N, \qquad 1\le L_N\le e^{\mathcal P_N}. \tag{171}\] Only pointwise metric comparison is asserted here.

Suppose \(q\in O_N\cap\operatorname{supp}|D\mathbf1_{B_N}|\). Because \(B_N\) contains \(O_N\) almost everywhere, such a point is in \(\partial O_N\). Take a shortest background geodesic of length \(s_N\) from \(q\) to \(H_N\). The ball of radius \(s_N/4\) about its midpoint lies in \(O_N\cap B_{s_N}(q)\). Uniform local background geometry gives \[ |B_N\cap B_{s_N}(q)|_{g_N}\ge c s_N^3. \tag{172}\]

We obtain a lower bound for the complementary volume without using regularity of either frontier. Put \(v(r)=|B_r(q)\setminus B_N|_{g_N}\) for \(0<r<s_N\). Full minimality against \(B_N\cup B_r(q)\), cancellation outside the ball, and removal of the constant factor \(e^{2A_N}\) imply, for almost every \(r\), \[ P_{g_N}(B_N;B_r(q)) \le P_{\gamma_N}(B_N;B_r(q)) \le L_N^2v'(r). \tag{173}\] The last derivative is the background area of the complementary trace on the sphere, by coarea. In the uniform normal charts, background metric balls correspond to Euclidean balls and volume and perimeter are uniformly comparable. Thus the Euclidean absolute and relative isoperimetric inequalities (Ambrosio et al. 2000, Theorem 3.46 and Remark 3.50) hold here with uniform constants. Local absolute isoperimetry for \(B_r(q)\setminus B_N\), including its boundary on the sphere, gives \[v(r)^{2/3} \le C\bigl(P_{g_N}(B_N;B_r(q))+v'(r)\bigr) \le C(L_N^2+1)v'(r).\] Every ball at a point of perimeter support has positive complementary volume. Integrating the resulting inequality for \(v^{1/3}\) from \(\delta\) to \(r\) and then letting \(\delta\downarrow0\) yields \[ v(r)\ge c(L_N^2+1)^{-3}r^3. \tag{174}\] Relative isoperimetry in \(B_{s_N}(q)\), together with Equations (172) and (174), therefore forces \[P_{g_N}(B_N;B_{s_N}(q)) \ge c(L_N^2+1)^{-2}s_N^2.\] Restoring the constant area multiplier in Equation (171) gives \[ P_{\widetilde g}(B_N) \ge e^{2A_N}c(L_N^2+1)^{-2}s_N^2 \ge\exp(2e^{N\epsilon/4}-\mathcal P_N). \tag{175}\] For fixed \(\epsilon>0\), this contradicts Equation (170) when \(N\) is large.

The minimizing frontier is now disjoint from the obstacle. It is locally perimeter minimizing without a constraint and is therefore smooth and embedded by three-dimensional perimeter regularity (Huisken and Ilmanen 2001, Regularity Theorem 1.3(iii)). Filling every bounded complementary pocket cannot increase perimeter and preserves the obstacle condition. The frontier of the remaining exterior is the whole smooth minimal boundary incident on the end. It is nonempty because a bounded set containing the nonempty open obstacle cannot have zero perimeter in the connected filled manifold. This proves all assertions. ◻

Proof. Apply Lemma 43 to the exterior of \(\Gamma_N^0\) from Lemma 44. The comparison end is strongly asymptotically flat, so its large coordinate spheres have positive outward mean curvature and confine compact minimal surfaces by the maximum principle at a largest-radius point. We obtain an outermost full minimal enclosure \(\Gamma_N\) and take its exterior \(\widetilde\Omega_N\).

The obstacle contains every cap and a neighborhood of the entire original boundary. Both \(\Gamma_N^0\) and its outer enclosure are therefore full cuts in the original exterior, with every component counted. Their surviving exterior is disjoint from \(H_N=\{|h|\ge b_3\}\). Lemmas 41 and 42 consequently apply everywhere on \(\widetilde\Omega_N\), up to its boundary. No part of a filling is retained. The construction supplies smoothness, completeness with boundary, one end and the stated asymptotics. Finally, metric dominance on each tangent two-plane and the full-cut definition give \[\mathop{\mathrm{Area}}_{\widetilde g}(\Gamma_N) \ge\mathop{\mathrm{Area}}_g(\Gamma_N)\ge a_g(S),\] and Equation (158) gives the mass bound. ◻

Proof. Put \(r_*=(a_g(S)/(4\pi))^{1/2}\). We prove the family of bounds \[ E_*\ge sQ+\frac{1-s^2}{2}r_*\qquad(0<s<1). \tag{179}\] The numerical input is the neutral enclosing-area theorem (OpenAI 2026, Definitions 1.1–1.2 and Theorem 1.3). In dimension three it states that a smooth connected oriented complete one-ended exterior with nonempty compact boundary, integrable neutral constraint densities satisfying \(\mu\ge|J|\), finite future-timelike ADM vector, decay \(g-\delta=O_6(r^{-q_0})\), \(K=O_5(r^{-1-q_0})\) for some \(1/2<q_0<1\), and weakly future-trapped boundary on every component obeys \(m\ge\tfrac12(a/(4\pi))^{1/2}\) when its full-cut infimum \(a\) is positive. We verify this contract for the auxiliary data constructed below.

Proof of Theorem 3. First suppose the original ADM vector is future timelike, and write \(m=\sqrt{E^2-|P|^2}\). Proposition 22 supplies strict prepared rest exteriors indexed by \(j\) with \[E_{*,j}\longrightarrow m,\qquad (Q_{E,j},Q_{B,j})=(Q_E,Q_B),\qquad a_{g_j}(S)\longrightarrow a_g(S).\] The area convergence is for the full original cut class. More explicitly, the lower metric comparison \(g_j\ge(1-\delta_j)g\), with \(\delta_j\to0\), gives \(a_{g_j}(S)\ge(1-\delta_j)a_g(S)\); local smooth convergence on any fixed full cut gives the reverse upper limit after taking its area arbitrarily close to the infimum. Thus no original minimizing cut is needed. The preparation keeps the fixed sign on each boundary component.

For each \(j\), Proposition 46 has already taken the limits in the order: outer Dirichlet exhaustion at fixed \(N\), then \(N\to\infty\), then \(\epsilon\downarrow0\). Apply its conclusion and only now let \(j\to\infty\). It follows that \[m\ge Q,\qquad r_A\le m+\sqrt{m^2-Q^2}.\] Since \(E>0\), the first inequality is equivalent to \(E\ge\sqrt{|P|^2+Q^2}\).

Finally, Proposition 8 gives \(a_g(S)>0\) for every original exterior in the theorem. We recall explicitly how Lemma 19 eliminates the remaining ADM endpoint using the timelike conclusion just proved. Its conformal energy-raising family preserves \(P,Q_E,Q_B\), the matter condition and the componentwise boundary signs, increases every full-cut area, and changes the energy to \(E_s=E+2s\). If \(E\le|P|\), take \(s>(|P|-E)/2\). The family then has a future-timelike ADM vector, so the preceding result applies and in particular gives \[\sqrt{(E+2s)^2-|P|^2} \ge\frac12\sqrt{\frac{a_{g_s}(S)}{4\pi}} \ge\sqrt{\frac{a_g(S)}{16\pi}}>0.\] Here we used \(x+\sqrt{x^2-Q^2}\le2x\) for \(x\ge Q\). Letting \(s\downarrow(|P|-E)/2\) makes the left side tend to zero, a contradiction. Hence \(E>|P|\) in the original data. The already proved timelike case completes the theorem, with no boost of speed one and no additional area or timelikeness hypothesis. ◻

Proof. This is the envelope formula of Proposition 14; the following two inequalities specify how it is used here. On the common compact truncation, the area density has the expansion \[dA_{g_t}|_T=\left(1+\tfrac t2\operatorname{tr}_T h+O(t^2)\right)dA_g|_T,\] where the remainder is uniform in the point and the two-plane \(T\). Every fixed \(M\in\mathcal M_B\) is a competitor at positive \(t\), giving the upper derivative bound. Conversely choose minimizing sets \(M_t\) at positive \(t\). Metric comparison and the same expansion show \(P_g(M_t)\to A\). A subsequence converges in measure to a member \(M_0\in\mathcal M_B\), and its perimeter convergence is strict. Continuity of Equation (189) gives \[a(t)-A\ge t\,a'_{M_t}(h)+O(t^2), \qquad \liminf_{t\downarrow0}\frac{a(t)-A}{t}\ge a'_{M_0}(h).\] The uniform remainder is bounded by a constant times the uniformly bounded perimeter. Taking a subsequence realizing the lower limit proves Equation (190). ◻

Proof. One-point compactify the active-ray space at infinity. If it is already compact, adjoin an isolated point instead. Assign the limiting values just calculated to the direction functions. The boundary and minimizing family are compact. Finite linear combinations \(V\) of all the directions give continuous functions on their disjoint union through \[ V\longmapsto\left( \frac{\mathfrak C'(V)}{\rho_*},\ -\theta_\varepsilon'(V),\ \left[c\,a'_M(h)-16\pi\mathcal L'\right]_{M\in\mathcal M_B} \right). \tag{196}\] We claim its linear image misses the open cone of functions that are strictly positive everywhere.

Indeed, a direction in that cone has positive \(V_*\)-coefficient, by its value at infinity. Add its compact and prototype changes linearly and apply the indicated positive conformal change. The forms stay closed and retain their end fluxes. On a compact region, strict positivity on the closed active set makes the first constraint derivative uniformly positive there. Continuity gives this conclusion in a neighborhood of that set; outside the neighborhood the unperturbed constraint has a positive minimum. A uniform Taylor remainder therefore gives the DEC for all sufficiently small positive parameters. The same argument gives the strict chosen boundary expansion.

Here the far end requires a uniform estimate, rather than compact continuity. The conformal identities retain a positive multiple of the original nonnegative ball constraint. Their additional term is a positive constant times \(t\rho_*\). The nonconformal linear errors are \(tO(r^{-3-q}+r^{-5})\). The quadratic metric and tensor prototype errors are \(O(t^2r^{-4})\); the remaining conformal and mixed errors are \(O(t^2\rho_*)\), after the background nonnegative term has been retained. Since \(\beta<q\) and \(\beta<1\), first choose the end radius large enough for the linear errors to be absorbed, and then choose \(t>0\) small enough for the quadratic errors to be absorbed. This proves feasibility simultaneously on every end ray. The estimates also show integrability of the changed constraints and their Taylor remainders. The conformal gradient flux and the explicit prototype fluxes give finite, differentiable ADM limits. The weak decay, completeness, and cut hypotheses are preserved. Thus these paths belong to the numerical class; no preservation of equality-subclass outermostness is needed.

Equation (190) and positivity of the third component in Equation (196) imply \(c\,a'_+(0)>16\pi\mathcal L'\). This contradicts the numerical bound and Equation (188). The claim follows.

Separation of a linear subspace from a disjoint open convex cone gives a nonzero continuous functional that annihilates the subspace and is nonnegative on positive functions. The representation of positive functionals on continuous functions gives positive measures on the three compact test spaces. If the measure on \(\mathcal M_B\) had zero mass, evaluation on \(V_*\), using Equation (194) also at the adjoined point, would be strictly positive. This contradicts annihilation. Normalize that mass to one. On the finite active rays divide the resulting measure by \(\rho_*\), obtaining \(\Lambda_B\). The possible atom at infinity contributes zero to compact tests and the four prototypes. Rearranging the annihilation identity proves Equation (195) and the weighted finiteness assertion. ◻

Proof. On a neighborhood of the support prescribe a Gaussian metric \(\mathbf G=-dt^2+g_t\) with \(g_0=g\), \(\partial_tg_t|_0=2K\), and future normal \(n=\partial_t\). Prescribe a two-form \(\mathbf F\) whose initial electric and magnetic fields are the given ones. The spatial pullbacks are \(\mathbf F|_{t=0}=\alpha_2\) and \((*_{\mathbf G}\mathbf F)|_{t=0}=-\alpha_1\). The spatial closure equations are precisely \(d\alpha_1=d\alpha_2=0\). The components of \(d\mathbf F=0\) and \(d(*_{\mathbf G}\mathbf F)=0\) with one normal slot determine all six normal derivatives of the two-form from its spatial derivatives and the already prescribed first metric jets. Thus both Maxwell equations hold at the slice. For the graph variation with velocity \(sn\), Cartan’s formula gives explicitly \[ \alpha'_1=-d(s\mathcal B^\flat),\qquad \alpha'_2=d(s\mathcal E^\flat). \tag{198}\] These are compact closed variations.

Let \[ \mathsf S_{ab}=2\left(\mathbf F_{ad}\mathbf F_b{}^d -\tfrac14\mathbf G_{ab}\mathbf F_{cd}\mathbf F^{cd}\right), \qquad T_\delta=\mathbf{Ein}-\mathsf S. \tag{199}\] The constraint equations give \(T_\delta(n,n)=\mu_m\) and \(T_\delta(n,\cdot)|_{T\Omega}=J_m\). Choose the second normal metric jets so that \[ (T_\delta)_{ij}=\frac{(J_m)_i(J_m)_j}{\mu_m+\delta}. \tag{200}\] This is a free prescription: the dependence of the spatial Einstein tensor on \(H=\partial_t^2g_t|_0\) is \(\tfrac12(H-\operatorname{tr}_g(H)g)\). Its inverse sends a desired coefficient \(A\) to \(2A-(\operatorname{tr}_gA)g\) in dimension three. Smooth time collars realizing these finite jets can be obtained by a quadratic Taylor polynomial and a cutoff. The contracted Bianchi identity and the Maxwell equations at the slice give \(\nabla^a(T_\delta)_{ab}=0\) there. In particular they determine the normal derivatives of its normal contractions from spatial derivatives; no further evolution equation is imposed.

We justify a limit used in this calculation. If smooth \(\mu\ge|J|\) on a compact set, then \[Q_\delta=\frac{J\otimes J}{\mu+\delta} \longrightarrow Q=\begin{cases}J\otimes J/\mu,&\mu>0,\\0,&\mu=0\end{cases} \quad\text{in }C^1.\] At a zero of \(\mu\), nonnegativity and domination give \(d\mu=dJ=0\). In a fixed smooth frame, \[|dQ_\delta|\le2|dJ|+|d\mu|,\qquad |Q_\delta|\le\mu.\] These bounds are uniform in \(\delta\). On a sufficiently small neighborhood of the compact zero set the right sides, including the first derivatives, are uniformly small. On its complement the denominator has a positive lower bound and convergence is smooth. This proves the claim, including differentiability with derivative zero on the zero set.

Apply it to Equation (200). Where \(\mu_m>0\), the limiting tensor is \(T=\gamma\otimes\gamma/\mu_m\), where \(\gamma(n)=\mu_m\) and \(\gamma|_{T\Omega}=J_m\). The covector \(\gamma\) is causal. At an active ray with positive \(\mu_m\), one has \[|v|=1,\qquad J_m=-\mu_m v^\flat,\qquad \ell=n+v\text{ null},\qquad \gamma(\ell)=0.\] Parallel transport \(\ell\) along any spatial curve. Its pairing with the causal covector field \(\gamma\) is nonnegative and vanishes at the point under consideration. Its derivative there is therefore zero. It follows that \((\nabla_eT)(Z,\ell)=0\) for spatial \(e\) and arbitrary \(Z\). The divergence identity, in an orthonormal frame, now gives \[ (\nabla_nT_\delta)(n,\ell) =\sum_i(\nabla_{e_i}T_\delta)(e_i,\ell)\longrightarrow0. \tag{201}\] At \(\mu_m=0\), both the limiting tensor and its spatial covariant derivatives vanish, so the same conclusion holds for every active \(|v|\le1\).

For completeness the varying arguments in this contraction also matter. Under the graph variation and metric-square-root transport, \[D_tn=\nabla s,\qquad D_tv=v(s)n,\qquad D_t\ell=\nabla s+v(s)n.\] The tangential term \(sK^\sharp v\) in the derivative of a graph tangent vector is canceled by its square-root transport. Differentiating the constraint therefore gives \[\begin{align*} \tfrac12\mathfrak C'_\delta &=s(\nabla_nT_\delta)(n,\ell) +T_\delta(\nabla s,\ell) +T_\delta(n,\nabla s+v(s)n). \end{align*}\] At a positive-density active ray, the last term is \(J_m(\nabla s)+\mu_m v(s)=0\), and the middle term is \(\delta J_m(\nabla s)/(\mu_m+\delta)\), which tends uniformly to zero. At a zero-density ray both terms vanish. Together with the proved \(C^1\) convergence and Equation (201), this proves uniform convergence on the compact active set. The variations have no boundary or ADM contribution. Since \(\Lambda_B\) is locally finite, Equation (195) and dominated convergence give Equation (197). ◻

Proof. There is no cancellation between different tangent planes in Equation (197). Indeed, perimeter submodularity and the common lower bound \(A\) show that unions and intersections of two minimizers are minimizers. Their regularity excludes a transverse crossing. Lemma 15 further shows that the plane incidence set is closed: strict convergence, uniform local density, and graphical regularity prevent a sheet from disappearing or its tangent plane from changing at a limiting point. Thus the average of the tangent-plane area measures has the form \(d\eta(x)\,\delta_{T(x)}\), with a single plane over each relevant spatial point. Equation (197) reads \[\int s(x)\operatorname{tr}_{T(x)}K\,d\eta(x)=0\] for every compact smooth \(s\). Hence its integrand vanishes \(\eta\)-almost everywhere. Fubini’s Theorem gives the claim on \(\pi_B\)-almost every frontier. On each free smooth sheet it then holds everywhere by continuity. ◻

Proof. Every minimizing set for an enlarged obstacle is also a minimizing set for the original obstacle, because both minima equal \(A\). Choose one of the frontiers supplied by Lemma 50. Away from \(S\) this is a free original minimizer, so it is smooth and minimal, including at contact with an interior component of \(B\). On the coincidence set with that component, its second graph derivatives agree almost everywhere with those of \(B\). Thus \(H_B=0\) there. Its chosen marginality gives \(\operatorname{tr}_{T_M}K=0\), independently of the chosen sign. Off the coincidence set the same assertion follows from Lemma 50. Continuity consequently gives \(H=\operatorname{tr}_{T_M}K=0\) throughout the frontier in \(\operatorname{int}\Omega\).

At contact with \(S\), the obstacle regularity gives a \(C^{1,\alpha}\cap W^{2,2}\) graph. On its coincidence set with \(S\) the second derivatives agree almost everywhere, and the original condition \(H_S+\operatorname{tr}_S K=0\) applies. Off that set the same equation has just been proved. The frontier therefore satisfies the uniformly elliptic expansion graph equation almost everywhere through the contact patch. Writing its principal part in divergence form, local elliptic regularity first gives \(C^{2,\alpha}\) and then smoothness, since the coefficients and the original data are smooth. The strong comparison principle for two touching ordered solutions of this graph equation gives local coincidence with \(S\). The contact set is consequently both open and closed in connected \(S\). If it is nonempty it is all of \(S\), whose area already equals \(A\); no additional frontier component remains.

A frontier disjoint from \(S\) would instead be a smooth full enclosing surface in the interior with \(\theta_+=0\), contrary to the original outermostness hypothesis. Thus the chosen frontier must be exactly \(S\). Its regular bounded filled side is the original obstacle, so it cannot contain the additional open set of an enlarged obstacle. This proves the exclusion. With \(B=S\), the same reasoning applies to almost every member selected by \(\pi_S\), showing that their area functionals all equal \(a'_S\). ◻

Proof. For compactly supported interior tensor variations \(k\), the part of \(\mathfrak C'\) depending on \(k\) integrates to \[2\int\left[u(\tau\operatorname{tr}k-K:k) +X^i\nabla^j(k_{ij}-\operatorname{tr}k\,g_{ij})\right]dV.\] There is no interior area term by Proposition 51. Integration by parts in distributions and invertibility of spatial trace reversal give Equation (203).

For a compact interior metric variation \(h\), the second-order principal term is the scalar curvature adjoint \(\nabla^2u-(\Delta u)g\). Connection variation in the momentum constraint has at most one derivative of \(h\); after integration by parts it has at most one derivative of \(u,X\). The variations of the fixed flux forms’ norms and of the transported ray are algebraic. All other terms in the metric equation are therefore smooth linear expressions in \(u,X,\nabla u,\nabla X\). Taking the trace solves for \(\Delta u\), since in three dimensions \(\operatorname{tr}(\nabla^2u-(\Delta u)g)=-2\Delta u\), and hence solves for the full Hessian of \(u\).

Taking a divergence of Equation (203) and using its trace \(\operatorname{div}X=-u\tau\) gives a vector Laplace equation for \(X\) with only first derivatives of the moments on the right. Together with the traced metric equation this is a linear elliptic system with diagonal Laplace principal symbol. Local distributional elliptic regularity makes the moment measures smooth. To see that no derivatives remain unaccounted for, differentiate the symmetrized equation and combine its three index permutations. For \(C_{ij}=-2uK_{ij}\) this expresses \(2\nabla_i\nabla_jX_k\) as \(\nabla_iC_{jk}+\nabla_jC_{ik}-\nabla_kC_{ij}\) plus curvature contractions with \(X\). Thus every second derivative of \(X\), as well as every second derivative of \(u\), is a smooth linear function of the first jets. This is the asserted closed system.

It also proves extension through the boundary. In a smooth collar, restrict the system for the first jets to each inward normal segment from a fixed interior collar slice. It is a linear ordinary differential equation with coefficients smooth up to \(S\), so its solution extends smoothly to \(S\). Smooth dependence on the tangential initial point gives a smooth extension of all the moments. Uniqueness of the normal equations preserves the identities between these extended jets and derivatives of the extended fields.

It remains possible a priori that \(\Lambda\) has a boundary moment not represented by these densities. Take a metric test with \(h=\nabla h=0\) on \(S\) and arbitrary normal second derivatives of its tangential block. In collar coordinates the principal term \(R'_g=\nabla^i\nabla^jh_{ij}-\Delta\operatorname{tr}h-\langle \operatorname{Ric},h\rangle\) then has arbitrary boundary value, through \(-\partial_\nu^2\operatorname{tr}_S h\). The momentum, electromagnetic, transported-ray, area, and expansion variations vanish on \(S\). After integrating the smooth interior moments by parts, their boundary terms also vanish because \(h\) has zero first jet. Equation (195) consequently annihilates every boundary test against the scalar moment mass. That mass is zero. Its vector moment is dominated by the scalar mass and is zero as well. ◻

Proof. Write \(Y=(u,X)\) in the end coordinates. Inspection of the preceding adjoint and its prolongation gives \[ |\partial^2Y|\le C r^{-1-q_0}|\partial Y| +C r^{-2-q_0}|Y|. \tag{205}\] The derivative coefficients contain \(\partial g,K\); the value coefficients contain \(\partial^2g,\partial K\), their quadratic products, and squared electromagnetic fields. These are controlled by the original \(O_2/O_1\) hypotheses; no extra derivative bound on the original data is being used.

Here is the growth argument. On each radial segment, first bound \(|Y(r)|\) by its value at a fixed sphere plus \(\int_R^r|\partial Y(t)|dt\). Substitution in the integrated form of Equation (205) and Gronwall’s inequality with the integrable kernel \(t^{-1-q_0}\) give uniformly bounded first derivatives and \(Y=O(r)\). The same equation then gives \(\partial^2Y=O(r^{-1-q_0})\), so \(\partial Y\) has a limit on each ray with error \(O(r^{-q_0})\). Two ray values on a sphere can be joined by an arc of length at most \(\pi r\); their derivative difference is \(O(r^{-q_0})\). Thus the limiting derivative matrix \(L\) is independent of the ray, and \(Y=Lx+O(r^{1-q_0})\).

The inequality \(u\ge|X|_g\) holds in every direction at infinity. Applying it on opposite rays first forces the linear part of \(u\) to vanish, and then forces every linear part of \(X\) to vanish. Consequently \(\partial Y=O(r^{-q_0})\) and \(Y=O(r^{1-q_0})\). Reinsert these bounds in Equation (205) to obtain \(\partial^2Y=O(r^{-1-2q_0})\). Its radial integral from infinity gives \(\partial Y=O(r^{-2q_0})\). Since \(2q_0>1\), \(Y\) is bounded. One more insertion gives \(\partial^2Y=O(r^{-2-q_0})\), \(\partial Y=O(r^{-1-q_0})\), and a common constant limit \(Y_\infty\) with error \(O(r^{-q_0})\).

Integrate the smooth adjoint identity against the four prototypes, truncated at radius \(R\), and let \(R\to\infty\). The limiting scalar boundary term is \(16\pi u_\infty E'\), and the vector boundary term is \(16\pi X_\infty^iP'_i\). The errors involve a decaying coefficient times a prototype or its derivative, or a derivative of \(Y\) times a prototype, and their sphere integrals are \(O(R^{-q_0})\). The volume integrals converge by the weighted finiteness in Lemma 48. The prototypes vanish near \(S\), so Equation (195) identifies these limits with \(16\pi(b^0E'+b^iP'_i)\). Their four independent flux values give \(Y_\infty=(b^0,b^i)\), proving Equation (204).

If \(u\) vanished at an interior point, nonnegativity would give \(du=0\) there and domination would give \(X=0\) and \(\nabla X=0\) there. The entire first jet would be zero. Uniqueness along curves for the closed first-order system in Lemma 52 would make \(u,X\) identically zero on the connected exterior. This contradicts \(u_\infty=E/m>0\). ◻

Proof. Define \(\mathbf F\) uniquely by stationarity and its electric and magnetic contractions with the future slice normal \(n=u^{-1}(\partial_z-X)\). With the given orientation its spatial parts are \(\alpha_2\) and \(-\alpha_1\) for \(\mathbf F\) and \(*_{\mathbf G}\mathbf F\). Contraction with \(\xi=un+X\) gives Equation (209), including both cross-product signs.

In Equation (195), vary \(\alpha_1\) by any smooth compactly supported closed two-form \(\eta\) in the interior. The only contribution is \(-4\int\beta_E\wedge\eta\), hence it is zero. Varying \(\alpha_2\) gives the same assertion for \(\beta_B\). We explain why this test class implies exactness on arbitrary topology. Testing first with exact \(\eta=d\gamma\) proves \(d\beta=0\). For any embedded oriented loop choose a trivialized oriented normal disk bundle and a compact disk two-form of integral one, extended constantly along the loop and by zero through the tube sides. It is a compactly supported closed two-form. Since \(\beta\) is closed, its pairing with this form equals the period of \(\beta\) around the central loop: all parallel loops in the tube have the same period. Thus every such period is zero. Smooth one-cycles can be represented by finite sums of embedded loops in a three-manifold, so every period is zero. Path integration defines a global potential. This argument uses all compact closed two-form variations, not only the exact ones.

The forms in Equation (209) are smooth up to \(S\). In a collar, integration in the normal direction extends their potentials smoothly there; local potentials on overlapping charts differ by the same constants as in the interior, so the extensions agree. Finally, stationarity and Cartan’s identity give \(i_\xi d\mathbf F= -d(i_\xi\mathbf F)=0\) and the corresponding identity for \(*\mathbf F\). Their spatial exterior derivatives vanish because \(d\alpha_1=d\alpha_2=0\). These two conclusions give \(d\mathbf F=d(*\mathbf F)=0\) on the product. ◻

Proof of Proposition 54. The slice metric is \(g\), and stationarity of Equation (206) gives its second fundamental form \(-u^{-1}\operatorname{sym}\nabla X=K\), by Equation (203). The definition of \(\mathbf F\) recovers the two original fields. Lemma 55 proves Maxwell closure. Causality and the end normalization follow from Equations (202) and (204); \(\xi\) is nonzero because \(u>0\).

We derive the stress assertion to fix its normalization. Write \(\mathsf B(L,M)=L:M-\operatorname{tr}L\operatorname{tr}M\). In a compact metric variation keep the lapse and shift fixed, and set \(b=-u^{-1}\operatorname{sym}\nabla X\), so \(b=K\) at the base point. Integrating the momentum term by parts gives the gravitational functional \[\int 2\{u\mu+J(X)\}\,dV =\int u\{R-\mathsf B(K,K)+2\mathsf B(K,b)\}\,dV.\] Its first variation at \(b=K\) is that of \(\int u\{R+\mathsf B(b,b)\}\,dV\), since their difference is \(-\int u\mathsf B(K-b,K-b)dV\), whose first derivative is zero. The Gauss scalar-curvature identity identifies the latter functional, up to a divergence, with the spacetime scalar action per unit Killing time of Equation (206). Integrating the scalar-curvature variation twice by parts gives its coefficient in a covariant spatial metric variation as \(-u\mathbf{Ein}_{ij}\). This calculation uses the coframe \(dx^i+X^i dz\): its dual spatial vectors are tangent to the slices, so the indicated Einstein slots are exactly the spatial ones.

With a fixed flux form, the vector is its contravariant density divided by \(\sqrt{\det g}\). Consequently \[\bigl((|\mathcal E|^2+|\mathcal B|^2)dV\bigr)' =\left(\mathcal E_i\mathcal E_j+\mathcal B_i\mathcal B_j -\tfrac12(|\mathcal E|^2+|\mathcal B|^2)g_{ij}\right)h^{ij}dV.\] The coefficient for twice the lapse times this expression is \(-u\mathsf S_{ij}\). Also \(\langle\mathcal E\times\mathcal B,X\rangle dV\) is independent of the metric when the two vector densities and \(X\) are fixed. The transported-ray derivative in Equation (191) adds \(-J_{m(i}X_{j)}h^{ij}\). Including a volume variation costs nothing because \(u\mu_m+J_m(X)=0\). The zero interior metric variation in Equation (195) therefore yields Equation (207).

The normal-normal and normal-spatial components of \(\mathbf{Ein}-\mathsf S\) are \(\mu_m,J_m\) by Gauss–Codazzi and the original constraints. If \(\mu_m=0\), the DEC gives \(J_m=0\), and Equation (207) makes all components zero. If \(\mu_m>0\), complementarity and the two inequalities \(|J_m|\le\mu_m\), \(|X|\le u\) imply \[|X|=u,\qquad J_m=-\frac{\mu_m}{u}X^\flat.\] Since \(\xi^\flat(n)=-u\) and its spatial restriction is \(X^\flat\), these normal components and Equation (207) give Equation (208). This also proves its stated support property. In these formulas \(\mu_m,J_m\) are geometric densities; there is no additional factor of \(8\pi\). ◻

Proof. The integration normal of the exterior at its boundary is \(-\nu\). An arbitrary compact \(K\)-variation \(k\) therefore leaves the boundary expression \[-2\int_S\{k(X,\nu)-\operatorname{tr}k\,X_\nu\}\,dA -\int_S\operatorname{tr}_S k\,d\zeta=0.\] The mixed normal-tangential components first give \(X_{\mathrm{tan}}=0\), and the tangential trace gives \(d\zeta=2X_\nu dA\).

Next use a compact scalar test \(s\) and the variation \((h,k)=(2sg,sK)\), with the flux forms fixed. The differential part of \(\mathfrak C'\) is \(4(-\Delta s+K(\nabla s,v))\), by Lemma 16. The expansion derivative at a marginal boundary is \(2\partial_\nu s\). After the already proved interior adjoint cancels the interior terms, Equation (195) becomes \[\begin{align*} -2c\int_Ss\,dA &=4\int_S\left[u\partial_\nu s -(\partial_\nu u+K(X,\nu))s\right]dA -2\int_S\partial_\nu s\,d\zeta. \end{align*}\] The independent boundary value and normal derivative of \(s\) give \(X_\nu=u\) and \(\partial_\nu u+K(X,\nu)=c/2\). This proves Equation (210).

Let \(L_+\) be the expansion linearization for outward normal motions of \(S\); its principal part is \(-\Delta_S\). Apply the multiplier identity to Lie derivatives of all four original data under a compact vector field that equals \(s\nu\) on \(S\). These are legitimate jets up to the boundary: extend the data smoothly into a collar solely to take the derivative. The flux-form Lie derivatives are exact. At an interior active ray, the transported DEC quantity is a nonnegative function with value zero, so its Lie derivative vanishes. There is no boundary constraint atom by Lemma 52. The area and expansion derivatives are \(\int_SH_Ss\,dA\) and \(L_+s\), respectively, while the ADM derivative is zero. Since \(d\zeta=2u\,dA\), we obtain \[ cH_S=2L_+^*u. \tag{213}\] Outer area minimization gives \(H_S\ge0\), by nonnegative outward normal area variations. Thus \(u\ge0\) is a supersolution of a smooth uniformly elliptic operator on connected compact \(S\). The strong minimum principle, applied locally with the bounded zeroth-order coefficient, implies either strict positivity everywhere or identical vanishing. No positive stability eigenvalue is assumed.

If \(u>0\) on \(S\), then \(W=0\) there. Tangential derivatives of \(W\) vanish, and the normal component of Equation (203) gives \[\partial_\nu W =2u\partial_\nu u-2\langle\nabla_\nu X,X\rangle =2u(\partial_\nu u+uK_{\nu\nu})=2\kappa u>0.\] This is Equation (211) and gives the positive deleted collar. If \(u=0\) throughout \(S\), then \(X=0\) there and all its tangential covariant derivatives vanish. The mixed and normal components of Equation (203) force its normal derivatives to vanish as well. Equation (210) gives \(\partial_\nu u=\kappa\), proving Equation (212) and again \(W>0\) for sufficiently small positive \(s\).

Finally Equation (204) gives \(W\to1\) at infinity. Equation (208) makes \(\mu_m=0\) wherever \(W>0\), so \(h_*\) vanishes on both the boundary collar and a far end neighborhood. It is smooth in the remaining compact interior region because \(u>0\) there. Extension by zero proves the last claim. ◻

Proof. Write \(\mathfrak A=I^-(\mathfrak E)\); the future case has the same proof with time orientation reversed. The end has timelike Killing orbits, so \(\mathfrak E\subset\mathfrak A\). The set \(\mathfrak A\) is open and invariant under every Killing translation. It is connected: a point of \(\mathfrak A\) can be joined to \(\mathfrak E\) by a timelike curve entirely in \(\mathfrak A\), and \(\mathfrak E\) is connected. Hence \(\mathfrak A=\mathbb R\times A\) for a connected open subset \(A\) of \(\operatorname{int}\Omega\) containing its distant end.

Suppose that its relative boundary \(\mathfrak H=\partial_{\mathfrak R} \mathfrak A\) is nonempty. A boundary of a chronological past is achronal and locally separates its past side from an excluded future side. To recall the local regularity used here, choose a small coordinate cylinder whose vertical lines are timelike. Achronicity permits at most one boundary point on each such line. The openness of the past set and local timelike cone bounds show that the boundary is a graph over a spatial ball, with a uniform Lipschitz bound. Shrinking the cylinder removes any edge of this graph. This description also shows that both of its local sides are nonempty.

The boundary is invariant. At every differentiability point its tangent hyperplane therefore contains \(\xi\). An achronal tangent hyperplane cannot contain a timelike vector. Since \(\xi\) is nonzero and causal, it is null there, and the hyperplane is precisely \(\xi^\perp\). Continuity gives \(W=0\) throughout \(\mathfrak H\). This proves smoothness as well. Indeed, in the preceding graph chart write the graph as \(x^0=f(x^1,x^2,x^3)\). The smooth plane field \(\xi^\perp\) is transverse to the vertical direction. Its graph slopes are smooth functions \(a_i(x^0,x)\); the weak derivatives of \(f\) satisfy \(\partial_i f=a_i(f(x),x)\) almost everywhere. The right sides are continuous. Mollification, or integration on coordinate lines, shows that these are the classical derivatives of a \(C^1\) function. This identity then gives successively \(C^2,C^3,\ldots\) regularity. In particular no regular-level-set assumption on \(W\) is involved.

Intersecting with \(z=0\) gives a smooth embedded two-sided surface \(C=\partial A\) in \(\operatorname{int}\Omega\). The intersection is transverse because the \(z\)-slice is spacelike while the null normal \(\xi\) is nonzero. Invariance identifies \(\mathfrak H\) with \(\mathbb R\times C\). Since \(W>0\) in a deleted collar of \(S\) and on the end, \(C\) lies in a compact subset of \(\operatorname{int}\Omega\). It is closed there and thus compact. A compact smooth embedded surface has finitely many components: finitely many connected local submanifold charts cover it, and each component contains one of these charts. In particular there is no accumulating family of extra frontier components.

Choose the normal \(\nu_C\) into the end-side exterior. On each connected component of \(C\), \[\xi=u(n+\varepsilon\nu_C),\qquad \varepsilon\in\{1,-1\},\] where \(n\) is the future normal to \(z=0\) and the sign is constant on that component. The null second form of \(\mathfrak H\) along its Killing normal is zero: for tangent vectors \(Y,Z\) to a spacelike section it is symmetric by hypersurface orthogonality and has symmetric part \(\tfrac12(\mathcal L_\xi\mathbf G)(Y,Z)=0\). Its trace therefore vanishes. Thus \(C\) is marginal for one of the two expansion signs on each component.

The connected collar of the connected surface \(S\) contains no frontier point. It is consequently wholly contained in \(A\) or wholly excluded from \(A\). The closure of \(A\) in \(\Omega\) is a connected closed smooth exterior with full boundary \(C\cup S\) in the first case and \(C\) in the second. It contains the distant end. The local excluded side at every point of \(C\) shows that its complementary obstacle contains an additional nonempty open subset outside the original obstacle. Its components have the marginal signs just determined, together with the original sign on \(S\) if that component is present. This contradicts Proposition 51. Hence \(\mathfrak H\) is empty. Connectedness of \(\mathfrak R\) proves the assertion. ◻

Proof. Choose a smooth spatial orthonormal frame \(e_i\) on an original collar patch and extend it, and \(n\), by stationarity. For spatial directions the spacetime connection is nonsingular: \[\boldsymbol\nabla_{e_i}n=K_i{}^j e_j, \qquad \boldsymbol\nabla_{e_i}e_j=\Gamma_{ij}^k e_k+K_{ij}n.\] In particular spatial covariant differentiation preserves smooth one-sided coefficients. The spatial derivatives of \(\xi=u n+X\) are \[ \boldsymbol\nabla_{e_i}\xi =(e_i u+K(e_i,X))n +(uK_i{}^j+D_iX^j)e_j. \tag{217}\] Skew symmetry of \(\boldsymbol\nabla\xi\) determines its remaining components without division by \(u\). The boundary identities \(u=0\), \(X=0\), \(DX=0\), and \(du=\kappa\nu^\flat\) give Equation (216). Also \([\xi,e_i]=[\xi,n]=0\), so the connection coefficients along \(\xi\) are the coefficients of \(\boldsymbol\nabla_{e_i}\xi\) and \(\boldsymbol\nabla_n\xi\) and are bounded and smooth in the same sense.

The order-zero Maxwell coefficients are the original electric and magnetic fields. Gauss–Codazzi determines \(\mathbf R_{ijkl}\) and \(\mathbf R_{nijk}\) from \(g,K\) and their spatial derivatives. In the deleted collar the Ricci tensor is \[\mathbf{Ric}_{ab} =2\left(\mathbf F_{ac}\mathbf F_b{}^c -\tfrac14\mathbf G_{ab}\mathbf F_{cd}\mathbf F^{cd}\right),\] because the Maxwell stress is trace free. The contraction identity \[ \mathbf R_{ninj}=\sum_k\mathbf R_{kikj}-\mathbf{Ric}_{ij} \tag{218}\] then supplies the remaining curvature block (with the same curvature convention in Gauss–Codazzi and this identity). All order-zero coefficients therefore have smooth limits.

Here is an induction which controls every normal derivative. Write \(F_j=\boldsymbol\nabla^j\mathbf F\) and \(R_j=\boldsymbol\nabla^j\mathbf{Rm}\), with derivative slots listed from outermost to innermost. Suppose all orders below \(j\) have smooth limits. Any component of order \(j\) having a spatial derivative slot has a smooth limit: commute that derivative to the outermost slot and use the displayed spatial connection formulas. The commutators are sums of contractions of lower derivatives of curvature with lower derivatives of the tensor being differentiated. Their total derivative order is \(j-2\), so the induction hypothesis controls every term.

It remains to treat derivative slots \(n,\ldots,n\). Apply \(j-1\) covariant derivatives to closure and co-closure of \(\mathbf F\), and evaluate their free derivative slots on \(n\). Metric compatibility allows these tensor identities to be contracted directly; derivatives of a chosen extension of the frame do not enter. For spatial \(i,l\) they read \[\begin{align*} F_j(n^j;i,l) &=-F_j(n^{j-1},i;l,n)-F_j(n^{j-1},l;n,i),\tag{219}\\ F_j(n^j;n,i) &=\sum_lF_j(n^{j-1},l;l,i). \tag{220}\end{align*}\] Here \(n^j\) means \(j\) derivative slots equal to \(n\). Every right-hand side has a spatial derivative slot and was already controlled. These identities give all of \(F_j\) first.

The differentiated differential Bianchi identity similarly gives \[ R_j(n^j;i,l,a,b) =-R_j(n^{j-1},i;l,n,a,b) -R_j(n^{j-1},l;n,i,a,b). \tag{221}\] By pair symmetry this covers every curvature component except the block with two normal tensor slots, \(R_j(n^j;n,i,n,l)\). Differentiating Equation (218) supplies that component. The differentiated Ricci tensor is a sum of products of the already controlled \(F_0,\ldots,F_j\). This closes the joint induction. Each right-hand side has smooth coefficients on the original closed collar, so the conclusion is smooth one-sided dependence, including all spatial derivatives, rather than merely bounded limits. No identity in the induction solves a time evolution equation by dividing by \(u\).

For \(T=\boldsymbol\nabla^j\mathbf F\) or \(\boldsymbol\nabla^j\mathbf{Rm}\), Killing invariance of the connection gives \(\mathcal L_\xi T=0\). Written covariantly this is \[0=\boldsymbol\nabla_\xi T+(\boldsymbol\nabla\xi)\cdot T.\] The first term tends to zero: it is \(u\boldsymbol\nabla_nT+X^i\boldsymbol\nabla_{e_i}T\), and the next-order jets have just been controlled. Thus \(B\cdot T=0\) at the edge. In the null basis \(n+\nu,n-\nu\) the two eigenvalues of \(B\) are \(\kappa,-\kappa\). A component of an invariant covariant tensor can be nonzero only when its total boost weight is zero. Counting its derivative slots as well as its tensor slots, it therefore contains equal numbers of the two normal null directions. Their simultaneous sign reversal has even total degree. This proves the \(I_b\) invariance. ◻

Proof. In the first case Taylor’s formula to arbitrarily high order gives \[a(r,y)=\sum_{j=0}^{L}a_j(y)r^{2j} +r^{2L+2}b_L(r,y),\] where \(b_L\) is smooth. Put \(p=r^2\). Repeatedly applying \(\partial_p=(2r)^{-1}\partial_r\) to the remainder, up to order \(h\leq L\), bounds the result by \(C p^{L+1-h}\); the same estimate holds after any prescribed tangential derivatives. Hence \(a(\sqrt p,y)\) has the derivatives at \(p=0\) prescribed by the polynomial, continuously to every order. In the second case, Hadamard’s formula writes \(a=r b\) with \(b\) smooth; the Taylor coefficients of \(b\) have the parity of the first case. This proof also treats a smooth flat remainder and makes no analyticity assumption. ◻

Proof. First suppose \(u|_S>0\). The equation \(dW=2\kappa X^\flat\) on \(S\) makes \(W\) a global positive defining function on a boundary collar. Use coordinates \((W,y)\) there. Division of a smooth tensor vanishing on the boundary by its defining function gives a smooth one-form \(\beta\) with \[X^\flat=\frac{dW}{2\kappa}+W\beta.\] Set \[ V=e^{\kappa z},\qquad U=W e^{-\kappa z},\qquad p=UV=W. \tag{222}\] Substitution into Equation (214) cancels the singular terms and gives \[ \mathbf G=\kappa^{-2}dU\,dV+\frac{2}{\kappa}U\beta\,dV+g, \tag{223}\] where \(\beta,g\) are pulled back from \((p,y)\); in such pullbacks \(dW=VdU+UdV\). This is smooth at both null faces and at their intersection. At the intersection its normal block is \(\kappa^{-2}dU\,dV\) and its tangential block is \(g|_{TS}\).

To extend the Maxwell tensor, write in the original regular chart \[\mathbf F=dz\wedge e+\alpha_2, \qquad e=\iota_\xi\mathbf F.\] At \(W=0\) the vector \(\xi\) is the null normal of a smooth Killing horizon. Its null second form vanishes by the Killing equation. The null Raychaudhuri equation, either with an affine generator or with its nonaffine term proportional to the zero expansion, gives \(\mathbf{Ric}(\xi,\xi)=0\). Electrovacuum holds up to this horizon by continuity from the deleted collar. Thus \[0=\mathbf{Ric}(\xi,\xi) =2|\iota_\xi\mathbf F|_{\mathbf G}^2.\] The one-form \(\iota_\xi\mathbf F\) annihilates \(\xi\). On the orthogonal space of a null vector the metric is positive semidefinite with precisely its null line as kernel. The last equality therefore implies that \(e\) is proportional to the metric dual of \(\xi\), hence to \(dW\), at the horizon. Smooth division gives \(e=b\,dW+W\gamma\) with \(b,\gamma\) smooth on the collar. The potentially singular terms become \[dz\wedge b\,dW=-\frac{b}{\kappa}dU\wedge dV, \qquad dz\wedge W\gamma=\frac{U}{\kappa}dV\wedge\gamma.\] All remaining terms are smooth pullbacks from \((p,y)\).

Now suppose \(u|_S=0\). In original normal distance \(s\) the boundary identities give \(u=\kappa s+O(s^2)\) and \(X=O(s^2)\) with smooth Taylor factors. Thus the tensors \[ A_s=W^{-1}X^\flat,\qquad h_s=g+W^{-1}X^\flat\otimes X^\flat \tag{224}\] are smooth up to \(S\), and \(h_s\) is positive definite and equals \(g\) there. The metric is \(\mathbf G=-W(dz-A_s)^2+h_s\). Let \((r,y)\) be the Gaussian normal coordinates of \(h_s\) at \(S\), and put \[H(r,y)=\int_0^r(A_s)_r(t,y)\,dt, \qquad z'=z-H(r,y).\] In these coordinates \(A_s-dH\) has no radial component. It follows that \(G_{rr}=1\) and \(G_{ra}=0\) for \(a\ne r\). Therefore the curves \(r\mapsto(z',r,y)\) are unit spacelike geodesics, and \(Y_A=\partial_{y^A}\) and \(\xi=\partial_{z'}\) along them are Jacobi fields. Notice that \(r\) is a smooth original defining function with \(r=s+O(s^2)\).

We prove the precise parity of their coefficients. The connection along these geodesics is smooth one-sided: their velocities are smooth linear combinations of original spatial directions and \(\xi\), whose connection coefficients were controlled in Lemma 58. A parallel orthonormal frame with a prescribed limiting basis at \(r=0\) is therefore obtained by a linear ODE with smooth coefficients on a closed half interval. The radial velocity has limiting value \(\nu\). The values \(Y_A(0)\) are tangent to \(S\), while \(\xi(0)=0\) and \(D_r\xi(0)=\kappa n\).

For completeness, \(D_rY_A(0)\) is normal to \(S\). The Killing identity \(\boldsymbol\nabla(\boldsymbol\nabla\xi)=\mathbf{Rm}*\xi\) shows that the tangential derivative of \(B=\boldsymbol\nabla\xi\) vanishes at the edge. Hence the projector \(P=B^2/\kappa^2\) onto \(\operatorname{span}(n,\nu)\) is parallel there in every tangent direction. The vector \((1-P)\partial_r\) vanishes at the edge, with its tangential derivatives. Differentiating this equality tangentially yields \((1-P)\boldsymbol\nabla_{Y_A}\partial_r=0\). Since coordinate fields commute, this is the assertion about \(D_rY_A(0)\).

Let \(I_b\) act in the parallel frame as in Lemma 58. The \(j\)th radial derivative of curvature at zero is its \(j\)th covariant derivative contracted with \(j\) copies of \(\nu\). Its transformation under \(I_b\) consequently acquires the factor \((-1)^j\). Differentiate the Jacobi equation \[D_r^2Y+\mathbf R(Y,\partial_r)\partial_r=0.\] The initial values are fixed by \(I_b\) and their initial derivatives change sign. Induction in this equation gives \(I_b D_r^jY(0)=(-1)^jD_r^jY(0)\) for both \(Y=Y_A\) and \(Y=\xi\). The contractions \(\mathbf G(Y_A,Y_B)\), \(\mathbf G(\xi,Y_A)\) and \(\mathbf G(\xi,\xi)\) therefore have only even Taylor coefficients. The initial values and Equation (216) give \[\mathbf G(\xi,Y_A)=O(r^2),\qquad \mathbf G(\xi,\xi)=-\kappa^2r^2+O(r^4).\] For \(\mathbf F\), the same differentiated contraction rule uses all its covariant jets. Contractions on two Jacobi fields have even parity, and contractions with one radial velocity and one Jacobi field have odd parity. Also \(\mathbf F(\xi,Y_A)=O(r^2)\) because \(\xi(0)=0\). Lemma 59 now gives the following actual smooth coefficient formulas, with \(p=r^2\): \[\begin{align*} \mathbf G={}&dr^2+(-\kappa^2p+p^2a)\,dz'^2 +2p b_A\,dz' dy^A+c_{AB}\,dy^A dy^B,\tag{225}\\ \mathbf F={}&\tfrac12 f_{AB}\,dy^A\wedge dy^B +p e_A\,dz'\wedge dy^A+r q_A\,dr\wedge dy^A +r d\,dr\wedge dz'. \tag{226}\end{align*}\] All coefficients in these displays are smooth in \((p,y)\) up to \(p=0\). Smooth dependence of the geodesic and parallel-frame ODEs on \(y\) makes these conclusions uniform with every tangential derivative. The statements are tensorial on overlapping charts of \(S\), so they define global tensors on its collar.

Set \[ V=r e^{\kappa z'},\qquad U=r e^{-\kappa z'}. \tag{227}\] Then \[dr^2-\kappa^2r^2dz'^2=dU\,dV, \qquad p\,dz'=\frac{U\,dV-V\,dU}{2\kappa},\] \[r\,dr=\frac{V\,dU+U\,dV}{2}, \qquad r\,dr\wedge dz'=\frac{dU\wedge dV}{2\kappa}.\] Equations (225)–(226) are thus smooth in \((U,V,y)\). Their metric at \(U=V=0\) has normal block \(dU\,dV\) and tangential block \(g|_{TS}\).

In each case extend the finitely many smooth coefficient tensors in \(p\geq0\) to \(|p|<p_0\), keeping the displayed invariant forms. A smooth extension exists by collar extension in each coordinate patch and a partition of unity on the compact \(S\). Lorentz signature and time orientation hold near \(B_0\) by the nondegenerate leading blocks. They hold on the entire sufficiently small band: the boost \(U\mapsto e^{-\kappa t}U\), \(V\mapsto e^{\kappa t}V\) takes any point with \(p\ne0\) to one with \(|U|=|V|=\sqrt{|p|}\), and takes a point on a punctured null face arbitrarily close to \(B_0\). The extended formulas are invariant under these boosts.

Glue this open band to the deleted collar of \(\mathfrak R\) by Equation (222) or Equation (227). This is a collar gluing of smooth manifolds. To check its separation property, a sequence in the overlap which tends to an added null face has \(p\to0\) or unbounded original time and cannot have a limit in \(\mathfrak R\); a sequence tending to the outer collar boundary has no limit in the open added band. No pair of distinct retained points is therefore identified by a limiting overlap sequence. Local collar charts and the old charts give a Hausdorff, second-countable manifold. The translations and boosts agree on the overlap, and both are defined for every real parameter. Their resulting Killing flow is complete.

In the positive-lapse case the original regular chart gives \(U=0,V=e^{\kappa z}\) on \(S\); in the zero-lapse case \(r=0\) gives \(B_0\) and the original \(z=0\) slice has a smooth finite \(z'\) offset. These observations also identify the induced metric on \(B_0\) and the indicated original boundary. The extension was constructed for this geometric purpose; only its restriction to \(\mathfrak R\) is used in the field equations below. ◻

Proof. Proposition 54 gives a constant limiting metric for Equation (214), with error \(O_2(\rho^{-q_0})\), and a unit timelike limiting \(\xi\), since \((b^0)^2-|b|^2=1\). A constant linear change of coordinates puts this metric in rest form \(\eta\) while keeping the new spatial coordinates constant along Killing orbits. Explicitly, in the limiting product coordinates one can take \(t_*=z-b_i x^i\) and a spatial linear map whose Euclidean quadratic form is \(\delta_{ij}+b_i b_j\). The old and new spatial radii are comparable. Maxwell components remain \(O_1(\rho^{-2})\) under this constant change.

We describe the coordinate improvement to account for the weak initial derivative bounds. For a stationary scalar the wave operator has the form \[L=\mathbf G^{ij}\partial_i\partial_j+d^i\partial_i, \qquad \mathbf G^{ij}=\delta^{ij}+O_1(\rho^{-q_0}),\quad d^i=O_0(\rho^{-1-q_0}).\] It is uniformly elliptic on a sufficiently distant end because \(W>0\) there. Solve \(L\chi^a=-\boldsymbol\Box x_*^a\), for the three linear spatial rest coordinates and for \(x_*^0=t_*\). The sources are \(O_{0,\gamma}(\rho^{-1-q_1})\); this Hölder estimate follows on scaled annuli from the available second metric derivatives, after decreasing \(q_1\) and, if necessary, \(\gamma\).

Here is a construction of the right inverse at this weight. Extend the scalar operator to the Euclidean Laplacian inside a large sphere with a smooth cutoff. This extension is only for the coordinate equation. For a source \(f=O_{0,\gamma}(\rho^{-1-q_1})\) the Newton kernel with its constant term subtracted on the far tail defines \[P f(x)=-\frac1{4\pi}\int_{\mathbb R^3} \left(\frac1{|x-y|}-\frac{\chi_\infty(y)}{|y|}\right)f(y)\,dy,\] where \(\chi_\infty=1\) outside a fixed ball and vanishes near zero. The tail is absolutely convergent because the difference of kernels is \(O(|x|\,|y|^{-2})\). Splitting the integral into \(|y|<|x|/2\), \(|y|\asymp |x|\), and \(|y|>2|x|\) gives \(|Pf(x)|\leq C(1+|x|)^{1-q_1}\). For example, the last region is bounded by \(C|x|\int_{2|x|}^{\infty}s^{-1-q_1}\,ds\); the other regions give the same power. Differentiation of the kernel for first derivatives and scaled interior Schauder estimates for second derivatives give the weighted \(C^{2,\gamma}\) bound. Thus \(P\) is a bounded right inverse from weight \(-1-q_1\) to weight \(1-q_1\).

After rescaling the cutoff radius to one, the coefficient norm of \(L-\Delta\) between these spaces tends to zero with that radius. The convergent Neumann series for \(1+P(L-\Delta)\) gives the required corrections \(\chi^a=O_{2,\gamma}(\rho^{1-q_1})\). They are smooth by local elliptic regularity. Their first derivatives tend to zero, so the new spatial coordinates are a diffeomorphism on a further tail. Since the corrections are stationary, \[\boldsymbol\Box t=\boldsymbol\Box x^i=0, \qquad \xi(t)=1,\qquad \xi(x^i)=0.\] In particular \(\xi\) remains exactly \(\partial_t\).

In these harmonic coordinates the reduced Ricci equation has scalar principal part \[ \mathbf G^{ij}\partial_i\partial_j\mathbf G_{ab} =\mathcal Q_{ab}(\mathbf G,\partial\mathbf G) -2\mathbf{Ric}_{ab}, \tag{229}\] where \(\mathcal Q\) is quadratic in first derivatives with smooth bounded coefficients near \(\eta\). The coordinate construction starts with scaled \(C^{1,\gamma}\) bounds for the metric error and scaled \(C^{0,\gamma}\) bounds for the Maxwell field. The Ricci source is quadratic in \(\mathbf F\), since the end is electrovacuum. Schauder estimates in Equation (229) therefore give scaled \(C^{2,\gamma}\) bounds for the metric. The Maxwell equations then give one additional field derivative. One way to see the latter step precisely is to square closure and co-closure: \[\boldsymbol\Box\mathbf F=\mathbf{Rm}*\mathbf F.\] By stationarity this is elliptic. Its coefficients have the metric regularity just obtained and its zeroth-order source has scaled \(C^{0,\gamma}\) bounds. Interior \(W^{2,p}\) estimates first improve the bounded first field derivatives to \(C^{0,\gamma'}\) bounds for any fixed \(\gamma'<1\) by taking \(p\) large; Schauder estimates then apply. Equivalently the stationary Hodge system is elliptic because the symbol of \(d+\delta\) squares to \(\mathbf G^{ij}\zeta_i\zeta_j>0\) for a nonzero spatial covector. Repeating these two estimates gives, on every scaled annulus, \[\partial^j(\mathbf G-\eta)=O(\rho^{-q_1-j}),\qquad \partial^j\mathbf F=O(\rho^{-2-j})\] for every fixed \(j\). The constants may depend on \(j\). This bootstrap uses only the original two metric derivatives and one field derivative to start, and only ellipticity on the distant timelike region.

It follows from Equation (229), replacing its principal coefficients by \(\delta^{ij}\) and using the just-proved second derivative estimate, that \[ \Delta_\delta(\mathbf G_{ab}-\eta_{ab}) =O_j(\rho^{-2-2q_1}+\rho^{-4}) \tag{230}\] for every \(j\). The right side is integrable in three dimensions because \(2q_1>1\). Multiply a component of the decaying perturbation by a cutoff on the end and call its Euclidean Laplacian \(f\). The unsubtracted Newton integral of \(f\) is now \(O(\rho^{-1})\): the inner region is bounded by \(C\rho^{-1}\|f\|_{L^1}\) and the comparable and outer regions are \(O(\rho^{-2q_1})\). Its difference from the cutoff perturbation is an entire harmonic function tending to zero, hence vanishes by the maximum principle. Scaled estimates for the Poisson equation and its derivatives prove the first estimate in Equation (228) to every order. The field estimate is already known. One further derivative implies each desired scaled Hölder estimate. Finally \(h_{s,ij}=\mathbf G_{ij}-\mathbf G_{0i}\mathbf G_{0j}/\mathbf G_{00}\) and \(W=-\mathbf G_{00}\) give the asserted orbit estimates. ◻

Proof. The covector in Equation (231) is timelike on a sufficiently small positive collar. This can be checked at \(U=V=\sqrt p\) in the smooth metrics of Proposition 60 and then everywhere on the positive collar by boost invariance. In the positive-lapse case it is \(dz-(2\kappa W)^{-1}dW\). Choose a smooth radial cutoff \(\chi(W)\) equal to one near zero and zero outside this collar and set \[s_0=z-\frac1{2\kappa}\int^W\frac{\chi(w)}{w}\,dw.\] The integration constant is chosen so that the displayed local formula holds exactly. Its differential is a convex combination of \(dz\) and \(dz-(2\kappa W)^{-1}dW\). Both covectors have value one on the future causal \(\xi\), so they lie in the same component of the timelike covector cone; that cone is convex. This proves timelikeness and extends \(s_0\) to \(z\) plus a constant off the collar.

In the zero-lapse case Equation (231) is \(z'=z-H(r,y)\). Cut off \(H\) in a sufficiently thin original collar. The additional spatial differential stays bounded because \(H=O(r)\) and a cutoff derivative of order \(r^{-1}\) is multiplied by \(H\). For any bounded spatial one-form \(\lambda\), \[\mathbf G^{-1}(dz+\lambda,dz+\lambda) =|\lambda|_g^2-u^{-2}(1-\lambda(X))^2.\] Here \(u=\kappa r+O(r^2)\) and \(X=O(r^2)\). Thus the negative term dominates uniformly on a sufficiently thin collar, including the cutoff annulus. This again continues the desired time as \(z\) plus a constant.

On the far end replace this time by the harmonic rest time \(t\) plus a constant. Both time covectors have value one on \(\xi\); their limiting Minkowski covectors, and the segment joining them, are uniformly timelike. Their difference of functions is a spatial linear function plus \(O(\rho^{1-q_1})\) and a constant. If the interpolation cutoff varies by one over a sufficiently long interval of \(\log\rho\), the extra differential is at most \(C/L\), where \(L\) is the length of that interval. Choose \(L\) large, and then its starting radius large. This error is smaller than the uniform timelike margin of the convex segment. The resulting smooth function has all the asserted local formulas and is temporal globally. Each modification was a globally defined stationary function, so \(\xi(s_0)=1\) is preserved.

We check the Cauchy assertion quantitatively. Write \(r=\sqrt{UV}\) in the positive collar and use \((s_0,r,y)\) there. At \(s_0=0\) the smooth extension has a positive radial coefficient, a positive tangential metric, and time coefficient a negative constant times \(r^2\). The remaining terms in the formulas of Proposition 60 can be absorbed into these leading quadratic forms for small \(r\). Uniformly for a causal tangent vector one obtains \[ |dr|^2+|dy|^2\leq C r^2|ds_0|^2. \tag{232}\] Here \(|dy|\) is measured in a fixed metric on the compact \(S\). Boost invariance extends the bound to every \(s_0\). In particular \(|d\log(UV)/ds_0|\leq C\), so a causal curve cannot reach the deleted inner end in a finite amount of \(s_0\) time. On the outer end, Equation (228) gives \(|dx/ds_0|\leq C\), excluding escape to spatial infinity in finite time. On the compact part between these two regions, temporality and stationary smooth coefficients give the same bounded-speed estimate in any fixed auxiliary spatial metric. This argument also covers compact loci where \(W=0\), since \(u\) and \(ds_0\) remain nondegenerate there.

If an inextendible future causal curve had a finite upper endpoint for its \(s_0\) range, these estimates would confine it to a compact spatial set away from both ends and make its spatial coordinates Cauchy as \(s_0\) approaches that endpoint. It would have a limit in \(\mathfrak R\) and could be extended by a short timelike curve, a contradiction. The past argument is identical. Thus its \(s_0\) range is all of \(\mathbb R\), and strict monotonicity gives exactly one intersection with every level. ◻

Proof. We retain only a sufficiently small open band \(|UV|<p_0\) on the added side. The invariant smooth metric formulas imply \[\mathbf G^{UU}=U^2a(p,y),\quad \mathbf G^{VV}=V^2b(p,y),\quad \mathbf G^{UV}=c(p,y),\qquad c\geq c_0>0,\] where \(a,b,c\) are smooth and bounded for \(|p|<p_0\). Choose a fixed large \(C\) and define \[ \mathcal U=U e^{-Cp},\qquad \mathcal V=V e^{-Cp}. \tag{235}\] A direct computation gives \[ e^{2Cp}|d\mathcal U|_{\mathbf G}^2 =U^2\{a(1-Cp)^2-2C(1-Cp)c+C^2p^2b\}. \tag{236}\] The analogous formula interchanges \(U,a\) with \(V,b\). Taking \(C\) large and then \(p_0\) small makes the braces strictly negative. The two gradients are nonzero even on their respective null faces. With the right-wedge future orientation, \(\mathcal U\) is nonincreasing and \(\mathcal V\) nondecreasing along future causal curves in the band. Their difference \(\mathcal T=\mathcal V-\mathcal U\) has timelike differential: the two causal covectors have opposite orientations and are linearly independent. In particular \(\mathcal T\) is temporal through the intersection of the faces.

These monotonicities also control curves which leave a coordinate neighborhood. First the right wedge is causally convex in the full attachment. A future curve leaving it across a null face must cross \(U=0\) into \(U\leq0,V>0\). It cannot return across that face because \(\mathcal U\) cannot increase to positive values. It cannot leave the added band elsewhere except through the right wedge, the only old region continuing beyond the band. The corresponding past statement uses \(V=0\). The same sign argument excludes a departure and return through the right wedge for endpoints in any one of the other open quadrants. Thus the temporal functions \(s_0\) and \(\mathcal T\) establish strong causality at points off the faces.

There is one additional global issue at a face: a curve could otherwise travel far into the right wedge and return near that face. Fix a small \(p_1>0\) at which the exact logarithmic formula (231) still holds. Suppose a future causal curve with endpoints \(a,b\) sufficiently close to a face leaves \(p<p_1\) through \(e\), and subsequently returns through \(f\). Both crossing points have \(p=p_1\) in the right wedge. The segment between them has nondecreasing \(s_0\), by causal convexity and Lemma 63. Hence \[\frac{V_f}{U_f}\geq\frac{V_e}{U_e},\qquad V_f\geq V_e,\quad U_f\leq U_e.\] Monotonicity on the initial and final band segments then gives \[ \mathcal U(a)\mathcal V(b) \geq \mathcal U(e)\mathcal V(f) \geq p_1 e^{-2Cp_1}>0. \tag{237}\] If either factor has the wrong sign, such a curve is already impossible. Otherwise the left side tends to zero as both endpoints approach any fixed point on a null face, contradicting the fixed positive lower bound. This excludes the excursion.

For a curve that stays in the band, both corrected coordinates stay between their endpoint values. The map \((U,V)\mapsto(\mathcal U,\mathcal V)\) is a local diffeomorphism throughout the smaller band, since its Jacobian is \(e^{-2Cp}(1-2Cp)>0\). Thus endpoints near a fixed face point confine \(U,V\) to a fixed small coordinate box. In that box and for all \(y\in S\), compactness and the timelike covector \(d\mathcal T\) give a uniform causal speed bound \(|dy|\leq C\,d\mathcal T\). A sufficiently small difference of endpoint \(\mathcal T\) values prevents angular escape from any prescribed neighborhood of the point. Combined with Equation (237), this is the defining local no-departure-and-return property of strong causality at every face point, including \(B_0\).

The open part of \(\Sigma\) is acausal by its temporal level-set description and causal convexity of the right wedge. No causal curve can join it to \(B_0\). A future curve from \(B_0\) starts with \(\mathcal U\leq0,\mathcal V\geq0\) and cannot enter the right wedge; a past curve has the opposite obstruction from \(\mathcal V\leq0\). A hypothetical connection in the other direction is the same statement with endpoints reversed. A causal curve between two points of \(B_0\) would have both corrected coordinates constant zero. It would then be tangent to \(B_0\), whose metric is positive definite, and must be constant. Hence the closed hypersurface \(\Sigma\) is acausal.

Every inextendible causal curve in the full spacetime through a point of the right wedge has a maximal portion in that wedge which is inextendible as a curve of the wedge. By Lemma 63 this portion intersects \(s_0=0\). Thus \(\mathfrak R\subset D(\Sigma)\), and openness gives the inclusion into its interior.

For the reverse inclusion, in the left wedge \(U,V<0\) the complete Killing orbits are timelike and avoid \(\Sigma\). In the future quadrant \(U<0,V>0\), a timelike curve can be continued to the past into the left wedge while staying in the band. To verify this directly, boost a point to \((U,V)=(-d,d)\), where \(d=\sqrt{|UV|}\). At fixed \(y\) follow the past-directed segment \[U(\lambda)=-d+\tfrac12d\lambda, \qquad V(\lambda)=d-2d\lambda, \qquad 0\leq\lambda\leq1.\] Its leading normal metric norm is a strictly negative constant times \(d^2\), and the perturbation is smaller uniformly when the band is small. It is therefore timelike. Along it \(|UV|\leq d^2<p_0\), and it ends in the left wedge. Continue there on a complete timelike Killing orbit in the past direction; in the future direction remain in the future quadrant until an inextendible end of the curve. Both portions avoid \(\Sigma\). The past quadrant has the time-reversed construction. Corners can be rounded inside the open timelike cones. Thus every point of the three other open quadrants lies on an inextendible timelike curve avoiding \(\Sigma\). None belongs to \(D(\Sigma)\) even for the timelike-curve convention. Finally each null face has empty interior and borders an excluded quadrant. It cannot contribute to \(\operatorname{int}D(\Sigma)\). This proves Equation (234) with either convention. ◻

Proof. We apply the class-(b) existence theorem of Chruściel and Wald (Chruściel and Wald 1994, Definitions 2.1, 2.3 and 2.4, Equation (4.1), Theorem 4.2). We verify its geometric conditions on the particular spacetime just constructed.

The spacetime is smooth, time oriented and strongly causal by Proposition 60 and Lemma 64. The hypersurface \(\Sigma\) in Equation (233) is connected, closed, acausal, spacelike and smooth up to the compact boundary \(B_0\). It is the union of a compact core and one Euclidean end. The smooth Killing field has a complete flow on the full spacetime and fixes \(B_0\) pointwise. The end lies in a stationary chart with uniformly negative \(G_{00}\), uniformly positive spatial metric, and \(\xi=\partial_t\). Equation (228) supplies all derivatives and the weighted Hölder bounds of their asymptotic definition. In particular we may use differentiability order two and the fixed exponent \(q_2\in(1/2,1)\).

Let \(\mathfrak M_{\rm ext}\) denote the Killing development of this end, as in their definition of black and white hole regions. It is exactly a stationary end of the original right wedge. Equation (234) identifies the open domain of dependence as \(\mathfrak R\), and Proposition 57 gives \[\mathfrak R\subset I^-(\mathfrak M_{\rm ext}) \cap I^+(\mathfrak M_{\rm ext}).\] Thus neither black nor white hole points meet that open domain; this is their Equation (4.1). All hypotheses of (Chruściel and Wald 1994, Theorem 4.2) are now satisfied. We record the analytic details that give regularity at the fixed boundary and the derivative estimates in this application. Let \(\tau_0\) be the smooth time function furnished by (Chruściel and Wald 1994, Proposition 4.2), and let \(\Sigma_* = \{\tau_0=0\}\cap D(\Sigma)\) be its zero hypersurface in the closed domain, with boundary \(B_0\). Its interior is Cauchy in \(\mathfrak R\). On the end \(\tau_0\) is Killing time; choose the additive constant of the harmonic rest coordinate so that \(\tau_0=t\) there. On this strong end we first make the reference graph maximal outside a compact set. Here is a direct construction. Rescale a sufficiently distant end by \(x=R y\), \(t=R\theta\), and extend its stationary metric coefficients to Minkowski coefficients inside \(|y|=1\). The extended coefficients satisfy \[\partial^j(\mathbf G-\eta) =O(\varepsilon\langle y\rangle^{-1-j}),\qquad \varepsilon=O(R^{-1}),\qquad \langle y\rangle=1+|y|,\] with the same scaled Hölder bounds. This auxiliary extension is used only to solve the end graph equation. For a stationary graph \(\theta=a(y)\) its normalized maximal equation is \(\Delta a+\mathcal Q(a)=0\). It depends on \(Da,D^2a\), not on \(a\). Use the norm controlling \[\frac{|a|}{\log(2+|y|)},\qquad \langle y\rangle|Da|,\qquad \langle y\rangle^2|D^2a|\] and the corresponding scaled Hölder seminorm, with \(a(0)=0\). The Euclidean Newton operator with its constant tail subtracted, followed by subtraction of its value at zero, is a bounded right inverse from \(C^{0,\gamma}_{-2}\) to this space. Indeed, splitting the integral into \(|z|<|y|/2\), \(|z|\asymp|y|\), and \(|z|>2|y|\) gives the logarithmic bound for its value and \(O(\langle y\rangle^{-1})\) for its first derivative; local scaled Schauder estimates give the second derivative and its seminorm. The subtracted kernel is \(O(|y||z|^{-2})\) in the last region, so that integral converges absolutely.

On a ball of radius \(C\varepsilon\) in this norm, \(\mathcal Q(0)=O_{0,\gamma}(\varepsilon\langle y\rangle^{-2})\) and the Lipschitz norm of \(\mathcal Q\) into \(C^{0,\gamma}_{-2}\) is \(O(\varepsilon)\). These estimates follow by writing the principal coefficients as their Euclidean values plus \(O(\mathbf G-\eta)+O(|Da|^2)\); all remaining terms contain a metric derivative. The contraction map \(a\mapsto-P\mathcal Q(a)\) therefore gives the required graph. Differentiating its uniformly elliptic equation on scaled annuli gives all higher derivatives. Returning to \(x\) and cutting off the graph on a fixed annulus gives a smooth spacelike reference hypersurface \(\Sigma_e\) equal to \(\Sigma_*\) on the compact core, and maximal on the distant end, where its height \(a_e\) satisfies \[a_e=O(\log\rho),\qquad \partial^j a_e=O(\rho^{-j}) \quad(j\geq1).\] The cutoff is spacelike because its gradient is small on that annulus. The end Killing time \(\tau_e=t-a_e\) still has \(\mathbf G-\eta=O_j(\rho^{-1})\) in its coordinates.

Subtract the same cut-off end correction from \(\tau_0\) and call the resulting time function \(\tau\). Its differential remains timelike: it is unchanged on the compact core, and the correction has uniformly small gradient in the transition annulus and the end. Thus \(\tau\) is smooth through \(B_0\), has \(\Sigma_e\) as its zero hypersurface in \(D(\Sigma)\), and equals \(\tau_e\) on the distant end. Its zero level is Cauchy in \(\mathfrak R\): the compact and inner ends are unchanged, and the end correction is sublinear with vanishing derivative, so the end causal speed bound and no-escape argument, or (Chruściel and Wald 1994, Lemma 4.2), apply. Exhaust \(\Sigma_e\) by its compact truncations, whose boundaries are \(B_0\cup C_i\) with \(C_i\) an end coordinate sphere. Compactness of their domains of dependence is supplied by (Chruściel and Wald 1994, Theorem 3.2), using the communication condition already verified. It is this compactness, not merely compactness of the truncations, that permits the smooth Dirichlet construction of (Bartnik 1984, Theorem 4.2). As in the cited proof, its domain and edge conditions hold for these acausal truncations. We obtain smooth maximal spans \(M_i\) with the exact boundary \(B_0\cup C_i\).

For clarity, the uniform height argument permits this fixed inner boundary. Choose a fixed sufficiently distant cylinder \(\rho=R_0\), and translate \(M_i\) by a Killing parameter \(b_i\) so that its least \(\tau\) value on the cylinder is \(1\). Spacelikeness bounds the oscillation on that cylinder uniformly. The compactness statement (Chruściel and Wald 1994, Corollary 3.6) then confines the translated compact portions to one compact set. The geometric constants in Bartnik’s tilt estimates are uniform on these surfaces: on the compact portion this follows from the smooth regular time \(\tau\), and on the end from its stationary identification with \(\tau_e\) and the strong coefficient bounds. In particular the curvature, the lapse and its first logarithmic derivative, and the derivative of the reference unit normal are bounded. If \(b_i\leq0\), the translated outer boundary has time \(b_i\), while \(B_0\) is fixed and has time zero. Comparison on the end with the maximal reference translates also bounds the translated height above: its interface height is bounded, and its outer height is nonpositive. Together with the compact-core bound this gives a uniform upper bound for the translated height. Thus the cylinder is separated by at least one unit of time from both boundary values, and its distance below the supremum of the translated height is uniformly bounded. The interior tilt estimate (Bartnik 1984, Theorem 3.1(iv), Equation (3.14)) gives a uniform tilt bound there. The exterior test-function argument in (Bartnik 1984, proof of Theorem 5.3, Equations (5.13)–(5.18)) now bounds \(-b_i\). That argument integrates only over the end, after subtracting an exterior barrier and the outer boundary time. Its cylinder trace has bounded oscillation and bounded tilt; these are precisely the two inner-boundary estimates used in its boundary term. The fixed surface \(B_0\) is outside this integration region and contributes no additional term. The test function gives \(\log(2+|b_i|)\leq C\), with \(C\) independent of \(i\). For large positive \(b_i\), an additional bounded translation puts the greatest cylinder time at \(-1\); applying the same argument with time reversed bounds \(b_i\). This is the translation argument in (Chruściel and Wald 1994, proof of Theorem 4.2, Equations (4.22)–(4.26)). Consequently \(|b_i|\) is uniformly bounded. Compactness on the interior and comparison with the maximal reference translates on the end give \(|\tau|\leq C\) on the original \(M_i\).

The ensuing tilt estimate includes the actual boundary. Both components of \(\partial M_i\) lie in \(\{\tau=0\}\); their ambient mean-curvature vectors have a uniform bound, since \(B_0\) is fixed and smooth and \(C_i\) are end spheres. The boundary maximum argument in (Bartnik 1984, Theorem 3.1(iii), Equations (3.10)–(3.12)) therefore gives a uniform tilt bound on all of \(M_i\), including \(B_0\). Its hypotheses bound the geometry of the prescribed edge; they do not assume the unknown boundary gradient. In smooth coordinates across \(B_0\) this makes the graph equation uniformly elliptic with fixed smooth boundary data. Boundary gradient Hölder estimates and Schauder estimates, followed by bootstrapping, give uniform bounds of every order on each fixed closed collar. The interior estimates give the same conclusion on every compact subset. A diagonal subsequence consequently converges smoothly, up to the same embedded \(B_0\), to a spacelike maximal hypersurface.

One can also read off its end estimates directly. Relative to \(\Sigma_e\), the graphs \(M_i\) have uniformly bounded heights. For any fixed \(0<\beta<1\), the graphs \(a_e\pm C\rho^{-\beta}\) are upper and lower barriers outside a fixed large sphere. The principal term has the strict sign of \(\Delta\rho^{-\beta}=\beta(\beta-1)\rho^{-\beta-2}\); the coefficient and drift errors are lower order, since \(Da_e=O(\rho^{-1})\), \(D^2a_e=O(\rho^{-2})\) and the stationary metric has the strong decay above. Choose \(C\) to dominate the bounded inner trace. The outer trace relative to \(\Sigma_e\) is zero, so comparison gives the same bound uniformly in \(i\). Scaled elliptic estimates give all derivative and Hölder bounds for the limiting difference. Thus its full end graph has \(f=O(\log\rho)\) and \(\partial^j f=O(\rho^{-j})\) for \(j\geq1\), which imply (238) for every \(q_2<1\). In particular its induced metric has two controlled asymptotic derivatives and its second form has one, as required below. The Cauchy assertion and covering translates follow from (Chruściel and Wald 1994, Lemma 4.2 and Theorem 4.2).

Finally, the complete nonzero causal orbit of \(\xi\) through any point of \(\mathfrak R\) is an inextendible causal curve there, so it meets the Cauchy hypersurface exactly once. It is transverse to that hypersurface: a nonzero causal vector cannot be tangent to a spacelike hyperplane. The intersection map is smooth by the implicit function theorem, giving the stated global graph and covering property directly. Its future lapse is strictly positive in the interior. This application uses the existence part of the cited theorem; the additional alternatives stated there for a foliation are not needed. ◻

Proof. Only the boundary assertion requires more than the global graph already proved. In a smooth collar of \(\widehat\Sigma\) choose a defining function \(q\geq0\). Its boundary is the actual embedded surface \(B_0\), and its inward spacelike tangent has a positive spacelike component in the \((U,V)\) normal plane. Because its interior is in \(U,V>0\), the two components of this normal-plane vector are both strictly positive. Subtracting its component tangent to \(B_0\) does not change that conclusion, since the normal and tangential blocks are orthogonal at \(B_0\). Hadamard’s formula consequently gives \[U=q\,a(q,y),\qquad V=q\,b(q,y), \qquad a(0,y)>0,\quad b(0,y)>0.\] It follows that \(r=q\sqrt{ab}\) is a smooth defining function, and the orbit projection \((q,y)\mapsto(r,y)\) has an invertible boundary derivative. Its boundary map on \(S\) is the identity in these natural labels. Also the graph time \(s_0=(2\kappa)^{-1}\log(b/a)\) is smooth to the boundary. This proves the collar diffeomorphism. In the positive-lapse case \(r=\sqrt W\); it is not the original defining function \(W\). In the zero-lapse case it is the Gaussian coordinate of Equation (227).

The Killing contractions \(\iota_\xi\mathbf F\) and \(\iota_\xi(*_{\mathbf G}\mathbf F)\) are invariant and annihilate \(\xi\). Their pullbacks under any two sections of the orbit projection are therefore the same one-forms on the orbit space. Their global exactness on the original interior implies exactness on \(\widehat\Sigma\setminus B_0\). Both contractions are smooth on the attachment and vanish on \(B_0\), since \(\xi=0\) there. Their restrictions to the closed hypersurface are closed by continuity from its interior. Local collar primitives consequently extend the global primitives smoothly: the difference of two primitives has zero differential on an overlapping connected interior patch and is a constant. A finite collar cover, or integration along its radial curves, matches these constants. The resulting boundary differential is zero, and \(B_0\) is connected, so each potential has one constant value on the whole boundary. This reasoning uses no simple-connectivity assumption on \(\Omega\). ◻

Proof. Each Killing orbit in the right wedge is a complete, nonzero causal curve. It is inextendible there: in the original product its spatial label is fixed and its \(z\) coordinate traverses all of \(\mathbb R\). The Cauchy property of \(\operatorname{int}\Sigma'\) therefore gives exactly one intersection with each orbit. A nonzero causal vector cannot be tangent to a spacelike hypersurface. Consequently the restriction of orbit projection is a local diffeomorphism, and its bijectivity makes it a global diffeomorphism. Flowing the section thus gives a diffeomorphism \[\mathbb R\times\operatorname{int}\Sigma' \longrightarrow \mathbb R\times\operatorname{int}\Omega.\] For a compact cut its two sections bound the compact chain swept out by the intervening finite orbit segments, which proves the homology assertion without a filling behind either boundary.

For clarity, the identification of boundary labels does not assert that the two hypersurfaces have the same transverse defining function. In the attachment coordinates of Proposition 60, \(B_0=\{U=V=0\}\) and the right wedge has \(U,V>0\). A spacelike hypersurface through \(B_0\) is transverse to both null faces. If \(s\geq0\) is a smooth defining function on that hypersurface, then \[U=s\,a(s,y),\qquad V=s\,b(s,y),\qquad a(0,y),b(0,y)>0.\] Thus \(\sqrt{UV}=s\sqrt{ab}\) is a smooth defining function, and the restriction of \(y\) to \(B_0\) is its original boundary label. This also proves that the graph approaches precisely that boundary and no other limiting orbit. The end assertion follows from the rest-chart graph estimates in Proposition 65. ◻

Proof. The exact potentials of Lemma 55 remain exact after rotation and pullback to the global graph. In particular, \[ b'=\iota_\xi(*_{\mathbf G}\widehat{\mathbf F})|_{T\Sigma'} =u'(\mathcal B')^\flat+(X'\times\mathcal E')^\flat=d\psi. \tag{241}\] The primitive extends smoothly to \(B_0\). To see this without an assumption on its periods at the boundary, take a smooth collar primitive locally, subtract its constant difference from the interior primitive, and glue; smoothness of the one-form supplies the local primitives. Since \(\xi=0\) on \(B_0\), its differential vanishes there. Connectedness of \(B_0\), inherited from \(S\), makes \(\psi\) one constant \(\psi_0\) on the whole boundary.

At infinity \(b'=O(r^{-2})\). Radial integration gives a finite limit along each ray, and joining two rays along a sphere gives oscillation at most \(Cr^{-1}\). Hence there is a single constant \(\psi_\infty\) with \(\psi-\psi_\infty=O(r^{-1})\). Since \(\operatorname{div}_{g'}\mathcal B'=0\) and its total flux is zero, the divergence theorem on the portion of \(\Sigma'\) inside a large coordinate sphere gives \[\begin{split} \int_{\Sigma'}b'(\mathcal B')\,dV_{g'} &=\lim_{R\to\infty}\int_{S_R} (\psi-\psi_\infty)g'(\mathcal B',\nu_R)\,dA_{g'} -\psi_0\int_{B_0}g'(\mathcal B',\nu)\,dA_{g'}\\ &=0. \end{split}\] The outer error is \(O(R^{-1})\) and the inner total flux is zero. This is precisely where a single connected boundary is needed: zero total magnetic charge would not cancel independently chosen constants on several components. Equation (241) therefore yields \[ \int_{\Sigma'}X'\cdot(\mathcal E'\times\mathcal B')\,dV_{g'} =-\int_{\Sigma'}u'|\mathcal B'|^2\,dV_{g'}. \tag{242}\]

We check the normalization and sign of the other integral. Proposition 54 gives \[\operatorname{Ein}_{\mathbf G}-\mathsf S =h_*\xi^\flat\otimes\xi^\flat, \qquad h_*\geq0,\qquad\operatorname{supp}h_*\subset\{W=0\}.\] Here \(h_*\) is supported away from the inner and outer collars. For the geometric constraint densities used throughout this paper, \(\mathsf S(n',e_i)=2(\mathcal E'\times\mathcal B')_i\). Also \(\mathbf G(\xi,n')=-u'\) and \(\mathbf G(\xi,e_i)=X'_i\). The mixed Einstein constraint on the maximal slice is consequently \[ \nabla^jK'_{ij} =2(\mathcal E'\times\mathcal B')_i-h_*u'X'_i. \tag{243}\] There is no \(8\pi\) in this identity. The \(8\pi\) factor belongs to the ADM momentum flux and to conversion from physical matter densities, not to these geometric densities.

Multiply Equation (243) by \(X'^i\) and integrate. Integration by parts and the Killing equation give \[\int_{\Sigma'} X'^i\nabla^jK'_{ij}\,dV_{g'} =-\int_{\Sigma'}K'_{ij}\nabla^jX'^i\,dV_{g'} =\int_{\Sigma'}u'|K'|^2\,dV_{g'}.\] The inner boundary term vanishes because \(X'=0\) there. At infinity \(X'=O(r^{-q_2})\) and \(K'=O(r^{-1-q_2})\) for some \(q_2>1/2\), so its absolute value is \(O(R^{1-2q_2})\to0\). All volume integrals converge: the gravitational integrand is \(O(r^{-2-2q_2})\), the Maxwell squared norms are \(O(r^{-4})\), and \(h_*\) has compact support. Substituting Equation (242) gives the exact nonnegative identity \[ \int_{\Sigma'}u' \bigl(|K'|^2+2|\mathcal B'|^2+h_*|X'|^2\bigr)\,dV_{g'}=0. \tag{244}\] It follows that \(K'=\mathcal B'=0\) in the interior and hence at its boundary by smoothness.

Now \(X'\) is a spatial Killing field and vanishes on the whole boundary. Its covariant derivatives in boundary tangent directions are zero. Skew symmetry of \(\nabla X'\) then forces all remaining components of \(\nabla X'\) to vanish there as well. The Killing identities form the first-order linear system \(\nabla X'=A\), \(\nabla A=\operatorname{Rm}_{g'}*X'\) along any curve. Its zero initial data at a boundary point give \(X'=0\) along every such curve, and connectedness gives \(X'=0\) everywhere. Thus \(W=(u')^2>0\) in the interior. Every orbit meets this slice, and \(W\) is invariant under the flow, so the same conclusion holds throughout the right wedge. The support condition now forces \(h_*=0\). Flowing the orthogonal slice gives Equation (240); invariance and the vanishing of the tangential restriction of \(\widehat{\mathbf F}\) give its stated form. Restriction back to \(z=0\) proves \(\mu_m=J_m=0\) in the original open exterior. Smoothness of the original data extends these equalities to \(S\). ◻

Proof. Smoothness through \(B_0\), compactness of the core and completeness of the end follow from Proposition 65 and Lemma 67. The symbol bounds for the stationary end in Proposition 62, together with Equation (245), give the stated strong asymptotic flatness and field bounds. Fluxes were checked in the proof of Proposition 68.

We spell out why the mass is that of the original data. In the unimproved stationary coordinates the spacetime perturbation of its limiting constant Minkowski metric has differentiated decay \(O_2(r^{-q_0})\), \(q_0>1/2\). It is electrovacuum near infinity. The skew-slot flux in Lemma 17 therefore has spacetime divergence \[O(r^{-2-2q_0})+O(r^{-4}).\] Stokes’ Theorem between cuts in the original asymptotic plane and cuts in a plane orthogonal to the limiting Killing direction has error \(O(R^{1-2q_0})+O(R^{-1})\to0\): the connecting chains have volume \(O(R^3)\) and remain at radii comparable to \(R\). The same estimate permits replacing comparable large cuts by coordinate spheres. With the positive-\(K\) convention, Lemma 17 pairs a translation \((b^0,b^i)\) with \(b^0E+\sum_i b^iP_i\). Here the limiting unit Killing translation furnished by Proposition 54 is \[b^0=\frac E m,\qquad b^i=-\frac{P_i}{m}.\] Its flux on the original section is therefore \[b^0E+\sum_i b^iP_i =\frac{E^2-|P|^2}{m}=m.\] On the orthogonal rest plane that same flux is its ADM energy. Its spatial metric and \(h_s\) differ by the quadratic shift term, which is \(O_1(r^{-2q_0})\) in this chart and contributes \(O(R^{1-2q_0})\) to the ADM flux.

Finally the stationary harmonic correction has spatial coordinate displacement \(O_2(r^{1-q_1})\), \(q_1>1/2\), as in Proposition 62. Its leading metric change is a symmetric flat gradient \(2\partial_{(i}a_{j)}\). The ADM divergence of this term is \(\Delta a_i-\partial_i\operatorname{div}a\), whose divergence vanishes identically. Extend \(a\) smoothly across the inside of a coordinate sphere; the flux of this leading term is exactly zero by the Euclidean divergence theorem. Products of coordinate and metric errors have flux \(O(R^{1-2q_1})\to0\), with a smaller decay exponent chosen if necessary. The energy is consequently still \(m\) in the strong asymptotic coordinates. This argument uses the original invariant ADM flux, not the mass of a deformation. ◻

Proof. Put \(N=u'\). We first verify the electrostatic system, including its potential and asymptotic normalization. By Lemma 55 and the constant duality rotation, the one-form \(\iota_\xi\widehat{\mathbf F}=N(\mathcal E')^\flat\) has a global primitive. Choose its negative, denoted by \(\Psi\), and add a constant so that \(\Psi\to0\) at infinity. This normalization is possible because \(d\Psi=O(r^{-2})\): radial integration gives a limit along each ray, and the oscillation on a sphere is \(O(r^{-1})\). Thus \[d\Psi=-N(\mathcal E')^\flat,\qquad \widehat{\mathbf F}=d\Psi\wedge dT,\qquad \Psi=O(r^{-1}).\] The primitive extends smoothly to \(B_0\), as in the proof of Proposition 68. Its differential vanishes there because \(\xi=0\), and connectedness makes its boundary value a single constant \(\Psi_0\). This is a scalar Killing potential; no global vector potential or simple connectivity is needed.

The spatial Ricci tensor of \(-N^2dT^2+g'\) is \(\operatorname{Ric}_{g'}-N^{-1}\nabla^2N\). The spatial component of the purely electric Maxwell Ricci tensor is \(|\mathcal E'|^2g'-2(\mathcal E')^\flat\otimes(\mathcal E')^\flat\). Together with Equation (246) and \(\operatorname{div}_{g'}\mathcal E'=0\), this gives \[\begin{aligned} N\operatorname{Ric}_{g'} &=\nabla^2N-\frac2N\,d\Psi\otimes d\Psi +\frac1N|d\Psi|^2g',\\ \Delta_{g'}N&=\frac1N|d\Psi|^2,\qquad \Delta_{g'}\Psi=\frac1N\langle dN,d\Psi\rangle. \end{aligned}\] These signs agree with the convention \(\widehat{\mathbf F}=d\Psi\wedge dT\).

We next establish the lapse expansion required by electrostatic uniqueness. Proposition 62 gives \(N=1+O_j(r^{-1})\), \(g'-\delta=O_j(r^{-1})\), and \(\mathcal E'=O_j(r^{-2})\) for every fixed \(j\), with scaled Hölder bounds. Hence the lapse equation implies \(\Delta_\delta(N-1)=O_j(r^{-4})\). Multiply \(N-1\) by an end cutoff and extend it by zero to \(\mathbb R^3\). Its Euclidean Laplacian \(b\) is integrable and is \(O_j(r^{-4})\) off a compact set. The Newton potential \(-(4\pi)^{-1}\int b(y)|x-y|^{-1}\,dy\) equals this cutoff function, since their difference is an entire harmonic function tending to zero. Subtracting the monopole term and splitting the integral into \(|y|<r/2\), \(|y|\asymp r\), and \(|y|>2r\) gives an \(O(r^{-2}\log r)\) remainder. Scaled Poisson estimates give the corresponding first two derivative bounds. Consequently, for a constant \(\mu\), \[N=1-\frac\mu r+O_2(r^{-2}\log r) =1-\frac\mu r+o_2(r^{-1}).\]

For completeness, this coefficient is the original invariant mass. Let \(\mathcal G_{ij}=\operatorname{Ric}_{g',ij} -\tfrac12R_{g'}g'_{ij}\). Expanding about the Euclidean metric, the linearized contracted Bianchi identity gives \[\partial_j\left(\mathcal G^{\rm lin}_{ij}x^i +\tfrac12(\partial_i g'_{ij}-\partial_j g'_{ii})\right)=0.\] Extend the metric perturbation smoothly through a fixed ball. The flux of this divergence-free vector field is zero. The nonlinear terms have size \(O(r^{-4})\) and give an \(O(r^{-1})\) error in the resulting flux identity. Thus, with \(n_\delta\) and \(dA_\delta\) Euclidean, \[E_{\mathrm{ADM}}(g') =-\frac1{8\pi}\lim_{R\to\infty} \int_{S_R}\mathcal G_{ij}x^i n_\delta^j\,dA_\delta.\] The electrostatic equations give \(\mathcal G=N^{-1}\nabla^2N -2(\mathcal E')^\flat\otimes(\mathcal E')^\flat =\partial^2N+O(r^{-4})\) in these coordinates. The last flux is therefore \(-8\pi\mu+o(1)\), proving \(\mu=E_{\mathrm{ADM}}(g')=m\) by Lemma 69. Also, with \(\nu\) pointing from \(B_0\) toward the end, integration of the lapse equation gives \[4\pi\mu=\int_{B_0}\partial_\nu N\,dA +\int_{\Sigma'}N|\mathcal E'|^2\,dV>0.\] Here \(\partial_\nu N=\kappa>0\) follows from the bifurcation attachment and the normalization of \(\xi\).

We can now apply the connected-horizon electrostatic uniqueness theorem of Borghini, Cederbaum, and Cogo (Borghini et al. 2025, Theorem 3.1 and Proposition 6.1). The spatial manifold is connected, oriented, one-ended and complete up to its compact connected boundary by Lemma 69. The metric, lapse and potential are smooth up to that actual boundary, \(N>0\) inside, \(N=0\) and \(|dN|=\kappa>0\) on \(B_0\), and \(\Psi\) is constant there. The displayed electrostatic system and asymptotics are exactly the required ones. The additional charge expansion imposed in that theorem when \(\mu=0\) is irrelevant because \(\mu=m>0\). It follows that the entire system is the subextremal Reissner–Nordström system with mass \(m\) and some signed charge \(q\), including its normalized lapse and potential \(N=\sqrt{1-2m/r+q^2/r^2}\) and \(\Psi=q/r\). Its isometry extends smoothly to the nondegenerate boundary.

The sign is fixed by the actual flux, not by a choice of potential convention: \(\mathcal E'=-N^{-1}\nabla\Psi\) for our convention, so the charge of this model is \(q\). Lemma 69 therefore gives \(q=Q\). This also covers \(Q=0\). More directly, in that case integration of \(\operatorname{div}(\Psi N^{-1}\nabla\Psi) =N^{-1}|d\Psi|^2\) gives \[\int_{\Sigma'}N^{-1}|d\Psi|^2\,dV=4\pi\Psi_0 Q=0.\] The boundary expression is regular because \(N^{-1}d\Psi=-(\mathcal E')^\flat\); the term at infinity tends to zero. Thus \(\Psi=0\), and the zero-charge case of the same uniqueness theorem is Schwarzschild.

The identified horizon radius is \(m+\sqrt{m^2-Q^2}=r_h\). The boundary isometry is smooth in proper-distance collars, where \(N\) vanishes simply. If the isometry initially reverses orientation, compose it with a reflection of the round sphere. That reflection fixes the radial lapse, potential and electric field, and makes the resulting identification orientation preserving. This proves all three identifications in Equation (247) on the closed exterior. ◻

Proof. The global product in Equation (240) and Proposition 70 identify the right wedge with the static Reissner–Nordström exterior, with the form in Equation (248). The original section is a global graph there by Lemma 67. All original tensors are already induced by \(\mathbf G,\mathbf F\) on that section; in particular its second fundamental form is the original \(K\), not the zero second fundamental form of \(\Sigma'\). Its unit normal need not be the static normal. The contraction identities defining the original fields consequently recover both full fields even when they are nonradial or nonparallel on that section.

It remains to verify smoothness and injectivity at the original boundary. On the open original orbit base write \[ \mathbf G=-W(dz-A_s)^2+h_s,\qquad A_s=W^{-1}X_g^\flat,\qquad dT=dz-A_s. \tag{249}\] The last equality is global: \(T\) is the Killing-flow coordinate based on the orthogonal global section \(\Sigma'\), so \(T-z\) is a single-valued function of the orbit label. On the original section it says \(dT=-A_s\). Let \(r_*\) be the Reissner–Nordström tortoise coordinate, \(dr_*/dr=f^{-1}\). Since \(f'(r_h)=2\kappa\) and \(W=f(r)\), \[ r_*=(2\kappa)^{-1}\log W+a(W) \tag{250}\] for a smooth function \(a\) near \(W=0\). To verify this expansion, use the smooth inverse \(r=r(W)\) of \(f\) near \(r_h\) and subtract \((2\kappa W)^{-1}\) from \(dr_*/dW\); the difference is smooth. The normalization of \(\kappa\) is the one from Proposition 56, equivalently the surface gravity of the now normalized static Killing field.

If \(u|_S=0\), Proposition 60 gives smooth \(A_s\) and a smooth defining function \(\sqrt W\) in the original collar. Hence \(T|_{z=0}\) extends smoothly there. The orbit metric is a smooth nondegenerate metric in that collar, and the base isometry extends smoothly by its normal geodesic collars; its angular boundary label is a diffeomorphism. The exterior Kruskal coordinates \[\mathsf U=-e^{\kappa(r_*-T)},\qquad \mathsf V=e^{\kappa(r_*+T)}\] are, by Equation (250), smooth multiples of \(\sqrt W\) on this section. Both vanish simply in its inward normal parameter, while the angular variables parametrize \(S\). Their derivative vector is a nonzero spacelike direction in the normal two-plane, since the two nonzero factors have opposite signs. This gives a smooth immersion through the bifurcation sphere.

The exterior in these Kruskal coordinates has \(\mathsf U<0<\mathsf V\); the attachment coordinates of Section 10 have \(U,V>0\).

If \(u|_S>0\), the other part of Proposition 60 gives \(W\) as a smooth original boundary defining function and \[X_g^\flat=(2\kappa)^{-1}dW+W\beta, \qquad A_s=(2\kappa)^{-1}d\log W+\beta,\] where \(\beta\) is smooth in \((W,y)\). Equations (249) and (250) show that the advanced time \[v=T+r_*\] has smooth differential \(dv=-\beta+a'(W)dW\) on the original section, and hence has a finite smooth extension to \(S\). In these coordinates the spacetime metric is \[-f\,dv^2+2\,dv\,dr+r^2\sigma_2.\] The sign of the logarithmic cancellation therefore selects the future horizon, where \(r=r_h\) and \(v\) is finite.

Here the angular map also is smooth in the original defining function \(W\), a fact that requires more than smoothness in \(\sqrt W\). Put \(s=\sqrt W\). The formula for \(X_g^\flat\) shows that \[h_s=g+W^{-1}\bigl((2\kappa)^{-1}dW+W\beta\bigr)^2\] extends as a smooth nondegenerate tensor in \((s,y)\) invariant under \(s\mapsto-s\). At \(s=0\) its normal coefficient is \(\kappa^{-2}\) and its tangential metric is \(g|_{TS}\). Reflection is consequently a local isometry fixing the boundary. In normal geodesic collars the angular labels are constant along the normal geodesics and invariant under this reflection. The angular components of the base isometry are thus smooth even functions of \(s\). Taylor’s formula with parameters makes every such function smooth in \(s^2=W\) on the one-sided collar. The radial coordinate \(r=r(W)\) is smooth as well. We have obtained a smooth map in the regular coordinates \((v,r,\text{angles})\), and \(dr/dW=(2\kappa)^{-1}\) at \(S\) is nonzero. Together with the boundary angular diffeomorphism this proves immersion at the future horizon and shows that \(S\) maps onto a full horizon cross-section.

In either case the limiting pullback of the ambient metric is the original smooth positive definite \(g\), so the immersion is spacelike up to \(S\). The induced future normal is a smooth algebraic function of its first derivatives and the ambient metric; it therefore extends smoothly as well. The second fundamental form and the two field contractions agree in the open exterior and extend by smoothness, proving their equality with the original tensors on \(S\).

The open graph is injective, its boundary angular map is injective, and the boundary image is disjoint from the open exterior. Finally its end escapes properly: the base radius tends to infinity and the rest-chart graph has the original strictly spacelike asymptotic plane slope, up to sublinear errors. Its image approaches spatial infinity in that end. Compactness of the remaining core and the boundary collar then make the injective immersion proper, hence an embedding. The spatial orientation was fixed in Proposition 70; flowing the future normal preserves the corresponding spacetime orientation. This proves the stated orientation and all the required original-data matching. ◻

Proof of Theorem 5. The equality variation and stationary construction in Propositions 54, 56, and 65, together with Lemma 55 apply to the original equality data. Proposition 68 proves static electrovacuum in their entire right wedge. Lemma 69 preserves the original invariant mass, and Proposition 70 identifies its orbit metric, lapse, and rotated electric field. The inverse duality formula (248) recovers the prescribed two charge signs. Proposition 71 then realizes the entire original data, with smooth future normal and fields at its horizon boundary. The vanishing of original non-electromagnetic matter was proved in Proposition 68. ◻

Proof of Theorem 6. Write \(Q_*^2=q_E^2+q_B^2\) and \(R_+=M+\sqrt{M^2-Q_*^2}\). The degenerate induced metric on the future horizon is \(R_+^2\sigma_2\), with its null direction along the generators. Every full smooth cross-section, including the bifurcation sphere, therefore has area \(4\pi R_+^2\). The assumed original cut condition implies \[A_{\min}(S)=|S|_g=4\pi R_+^2.\] The converse assumes that the actual invariant ADM mass is \(M\) and that the actual outward fluxes are \(q_E,q_B\). Substitution now gives the required equality. The examples below verify those assumptions for three explicit types of slicing. ◻

Proof. Use the converse parameters \(M,q_E,q_B\) and set \(Q_*^2=q_E^2+q_B^2\). Let \[f(r)=1-\frac{2M}{r}+\frac{Q_*^2}{r^2},\qquad g_0=f^{-1}dr^2+r^2\sigma_2, \qquad r\geq R_+.\] Start with the static exterior down to the bifurcation sphere. It is smooth there in proper distance or Kruskal coordinates; its future normal and both induced Maxwell fields are smooth, \(K=0\), and its boundary is a connected minimal sphere with \(\theta_+=0\). Its metric is complete with that boundary included. Consider also time graphs \(T=h(y)\) in areal Cartesian variables \(y=r\omega\) of the following two types:

1. \(h=\varepsilon h_0(r)\), where \(h_0\) is smooth with compact support in \((R_+,\infty)\), and has a critical point \(r_0\) with \(h_0''(r_0)\ne0\);

2. \(h=\chi(r/L)\,v\cdot y\), where \(\chi=0\) for \(r/L\leq1\), \(\chi=1\) for \(r/L\geq2\), \(L\) is sufficiently large, and \(|v|\) is sufficiently small and nonzero.

All graphs agree with the static slice near \(S\). Their induced metric is \[ g_h=g_0-f\,dh\otimes dh. \tag{251}\] For the radial graph it is \((f^{-1}-f(h')^2)dr^2+r^2\sigma_2\) and is positive for small \(\varepsilon\). For the cutoff graph, bounded derivatives of \(\chi\) give \(f|dh|_{g_0}^2\leq C|v|^2\). Choose \(|v|\) such that this is at most a fixed \(\eta<1\). Then \[ g_h\geq(1-\eta)g_0. \tag{252}\] These estimates prove spacelikeness and completeness, and the unchanged inner collar supplies smoothness of the graph and its future normal at the bifurcation sphere.

These are electrovacuum initial data with exactly the normalizations of Definition 1. One can check this directly in the static coordinates: the mixed diagonal components of the Einstein tensor are \[\frac{rf'+f-1}{r^2},\quad \frac{rf'+f-1}{r^2},\quad \frac{f''}{2}+\frac{f'}r,\quad \frac{f''}{2}+\frac{f'}r,\] which equal \(r^{-4}(-Q_*^2,-Q_*^2,Q_*^2,Q_*^2)\), the components of \(2(F_{ac}F_b{}^c-\frac14\mathbf G_{ab}F_{cd}F^{cd})\). The forms \[F=\frac{q_E}{r^2}dT\wedge dr+q_BdA_{\sigma_2},\qquad *F=\frac{q_B}{r^2}dT\wedge dr-q_EdA_{\sigma_2}\] are closed. Gauss–Codazzi with \(K(U,V)=\mathbf G(\nabla_U n,V)\) gives \(\mu=|\mathcal E|^2+|\mathcal B|^2\) and \(J=2(\mathcal E\times\mathcal B)^\flat\) on every graph; the spatial pullbacks of the closed forms give both divergence constraints. Thus \(\mu_m=J_m=0\) and the matter DEC holds. For every graph sphere \(r=R\), \(dr\) vanishes on its tangent space, so directly \[ \frac1{4\pi}\int_{r=R}F=q_B,\qquad -\frac1{4\pi}\int_{r=R}*F=q_E. \tag{253}\] The orientation is toward increasing \(r\). These are precisely the induced magnetic and electric fluxes, respectively. Closure identifies them with the fluxes on the actual end coordinate spheres, even when those coordinate spheres are not the surfaces \(r=R\).

We next check the full cut condition in the original graph metric. On the product \([R_+,\infty)\times\mathbb S^2\) let \[\alpha=R_+^2dA_{\sigma_2}.\] It is closed and has comass \(R_+^2/r^2\leq1\) in the static and radial graph metrics, since their radial and angular blocks are orthogonal and the angular metric is \(r^2\sigma_2\). For the cutoff graph, Equation (252) shows that its comass is at most \(R_+^2/((1-\eta)r^2)\) in the changed region. Choose \(L>R_+/\sqrt{1-\eta}\). On the unchanged inner region the previous exact bound applies, so the comass is at most one everywhere on this graph too.

If \(\Gamma=\partial D\) is any full cut as in Definition 2, truncate \(D\) by a large sphere outside \(\Gamma\). Stokes’ Theorem for this compact region, with all its inner boundary components and coincident obstacle portions counted, gives \[\int_\Gamma\alpha=\int_{r=R}\alpha=4\pi R_+^2.\] The comass bound yields \(|\Gamma|_{g_h}\geq4\pi R_+^2=|S|_{g_h}\). It covers disconnected cuts and cuts meeting the obstacle. Taking \(D=\Omega\) shows the infimum equals this value. No extension or filling behind the horizon has entered the calculation.

The boundary expansion remains zero on the unchanged collar. Every coordinate sphere with \(r>R_+\) has strictly positive \(\theta_+\) on these examples. On the static slice its value is \(2\sqrt f/r\). For the compact radial change, the value varies continuously with the height in \(C^2\) on its fixed compact support, so sufficiently small \(\varepsilon\) preserves strict positivity; outside that support it has the static value. For the cutoff graph choose \(L\) large enough that \(2\sqrt f\geq1\) for \(r\geq L\). On any annulus with radii comparable to \(R\geq L\), rescale spatial coordinates and time by \(R\). The background metric and its derivatives are uniformly bounded and converge to the Minkowski ones as \(R\to\infty\), and the rescaled height has \(C^2\) norm at most \(C|v|\), with a constant independent of \(R\) and of whether the annulus meets the cutoff. The induced normal, mean curvature, and tangential \(K\) trace depend smoothly on these derivatives as long as the graph stays uniformly spacelike. Consequently \[\left|r\theta_+(r)-2\sqrt{f(r)}\right|\leq C'|v| \qquad(r\geq L)\] pointwise on each sphere. Taking \(C'|v|<1/2\) proves the claim.

To deduce the required outermostness, suppose a compact smooth enclosing surface entirely in the open graph had \(\theta_+\leq0\). At a point of largest \(r\) it is tangent to the coordinate sphere, with the same outward normal: the radially outward side is in the component containing infinity. The inner surface’s local graph lies on the inner side of the coordinate sphere. In coordinates on their common tangent plane, the second derivative test and the mean-curvature principal part \(-\Delta\) give \(H_{\mathrm{surface}}\geq H_{\mathrm{sphere}}\) at contact. Their tangential \(K\) traces are equal there because their tangent planes coincide. Hence its expansion is at least the strictly positive sphere expansion, a contradiction. The argument applies to a largest-radius point among all components, so it also excludes disconnected enclosing weakly outer-trapped surfaces.

The radial example has nonzero original second fundamental form. At its critical radius the future normal is the static normal, and the graph Hessian formula gives \[K_{rr}(r_0)=\sqrt{f(r_0)}\,h''(r_0)\ne0.\] Thus these examples are not time-symmetric.

For completeness we calculate the ADM fluxes, including the momentum sign of the boosted examples, directly. The static metric in areal Cartesian coordinates is \[(g_0)_{ij}=\delta_{ij}+\frac{2M}{r}\frac{y_i y_j}{r^2} +O_2(r^{-2}).\] Its energy flux is \(M\) and its momentum flux is zero, as also follows from the calculation below at \(v=0\). The compact radial graph has exactly the same end data.

Rotate the spatial axes for the cutoff graph so \(v=\beta e_3\), where \(\beta\) may have either sign. On its uncut tail set \[T=\beta\gamma x^3,\qquad y=(x^1,x^2,\gamma x^3),\qquad \gamma=(1-\beta^2)^{-1/2},\qquad d=\gamma^2-1.\] The induced Minkowski metric in \(x\) is Euclidean. Let \(A_0=\operatorname{diag}(1,1,\gamma^2)\), \(w=A_0x\), and \(r^2=x\cdot A_0x\). The leading induced metric is \[ (g_h)_{ij}=\delta_{ij}+H_{ij}+O_2(|x|^{-2}),\qquad H_{ij}=2M\left(\frac{d\,\delta_{i3}\delta_{j3}}r +\frac{w_iw_j}{r^3}\right). \tag{254}\] Because \(\partial_i r=w_i/r\), differentiation gives \[\partial_jH_{ij}-\partial_iH_{jj} =\frac{2M}{r^3} \bigl[(\operatorname{tr}A_0+d)w_i-(A_0w)_i -d\delta_{i3}w_3\bigr] =\frac{4M\gamma^2x_i}{r^3}.\] On \(|x|=R\) put \(\mu=x^3/R\) and \(\rho^2=1+d\mu^2\), so \(r=R\rho\). The \(16\pi\) flux therefore is \[ E=\frac{M\gamma^2}{2} \int_{-1}^1(1+d\mu^2)^{-3/2}\,d\mu =\gamma M, \tag{255}\] where the primitive is \(\mu/\sqrt{1+d\mu^2}\).

To compute \(K\) with its sign, the leading spacetime perturbation in \((T,y)\) is \(h_{00}=2M/r\), \(h_{0a}=0\), \(h_{ab}=2My_ay_b/r^3\). Its required connection coefficients are \[\Gamma^0_{0a}=\frac{My_a}{r^3},\qquad \Gamma^a_{00}=\frac{My_a}{r^3},\qquad \Gamma^a_{bc} =M\left(\frac{2\delta_{bc}y_a}{r^3} -\frac{3y_ay_by_c}{r^5}\right),\] up to \(O_1(r^{-3})\); the remaining mixed time coefficients at this order are zero. Put \(F_0=T-\beta y^3\) and \(\sigma=\sqrt{-\mathbf G^{-1}(dF_0,dF_0)} =\gamma^{-1}+O(r^{-1})\). The future normal is \(n=-\sigma^{-1}\nabla F_0\). For the constant tangent vectors \(e_i\) to the plane this gives \[K_{ij}=-\sigma^{-1}\operatorname{Hess}F_0(e_i,e_j) =\gamma(\Gamma^0_{\alpha\lambda}-\beta\Gamma^3_{\alpha\lambda})e_i^\alpha e_j^\lambda +O_1(R^{-3}).\] Writing \(z=x^3\), substitution yields \[ K_{ij}=M\beta\gamma^2 \left[ \frac{x_i\delta_{j3}+x_j\delta_{i3} -2z\delta_{ij}-dz\delta_{i3}\delta_{j3}}{r^3} +\frac{3zw_iw_j}{r^5} \right]+O_1(R^{-3}). \tag{256}\] The trace at this order is \[\tau=M\beta\gamma^2 \left[-\frac{(4+d)z}{r^3}+\frac{3z|w|^2}{r^5}\right] +O(R^{-3}).\] Contracting with the Euclidean outward normal \(x/R\) gives \[R^2(K_{3j}-\tau g_{3j})\frac{x^j}{R} =M\beta\gamma^2 \frac{1+(3+4d)\mu^2}{(1+d\mu^2)^{5/2}}+O(R^{-1}).\] Consequently the actual \(8\pi\) momentum flux is \[ P_3=\frac{M\beta\gamma^2}{4} \int_{-1}^1\frac{1+(3+4d)\mu^2}{(1+d\mu^2)^{5/2}}\,d\mu =\gamma M\beta. \tag{257}\] Indeed an antiderivative is \(\mu[1+(1+2d)\mu^2]/(1+d\mu^2)^{3/2}\), whose endpoint difference is \(4/\gamma\). The two transverse momentum components vanish by reflection in the corresponding coordinate. Terms quadratic in asymptotic errors and the charge-squared terms have integrated error \(O(R^{-1})\) and do not alter either flux. Rotating back gives \(P=\gamma Mv\). Thus the actual invariant mass is \(\sqrt{E^2-|P|^2}=M\), and the momentum is nonzero when \(v\ne0\).

Equations (254) and (256) also verify \(g_h-\delta=O_2(R^{-1})\) and \(K=O_1(R^{-2})\) in the actual Euclidean end chart. Pulling back the smooth Maxwell forms and contracting with the uniformly timelike graph normal gives both fields \(O_1(R^{-2})\). Their stress, and hence \(\mu\) and \(|J|\), are \(O(R^{-4})\) and integrable. The compact core causes no integrability issue. Together with Equation (253), the cut calibration, and the expansion barrier, these checks establish every initial-data and connected rigidity-subclass hypothesis for all three families. The converse just proved supplies equality. ◻