Proof. The scalar-curvature transformation in dimension three is \(e^{4f}\mathop{\mathrm{Scal}}_{g_f}=\mathop{\mathrm{Scal}}_g-8\Delta_g f-8|df|^2\). Both quadratic terms in \(K\) scale by \(e^{-4f}\), which proves Equation (10). Put \(\pi=K-(\mathop{\mathrm{tr}}_gK)g\). Then \(\pi_f=e^{2f}\pi\). For a covariant symmetric tensor \(T\), the connection difference for \(\widetilde g=\Omega^2g\) gives, in dimension three, \[\widetilde\nabla^{,i}(\Omega T)_{ij} =\Omega^{-1}\bigl(\nabla^iT_{ij} +2T_{ij}\nabla^i\log\Omega -(\mathop{\mathrm{tr}}_gT)\partial_j\log\Omega\bigr).\] Take \(\Omega=e^{2f}\) and use \(\mathop{\mathrm{tr}}_g\pi=-2\mathop{\mathrm{tr}}_gK\). The terms containing \(\mathop{\mathrm{tr}}_gK\) cancel, leaving Equation (11). Taking its norm introduces another factor \(e^{-2f}\). The triangle inequality, with the factor two between the constraint normalizations \(16\pi\) and \(8\pi\), yields Equation (12). Finally, \(\nu_f=e^{-2f}\nu\), \(H_f=e^{-2f}(H+4\partial_\nu f)\), and \(\mathop{\mathrm{tr}}_{B,g_f}K_f=e^{-2f}\mathop{\mathrm{tr}}_{B,g}K\). This proves Equation (13). The last assertion follows from \(\Delta\log u+|d\log u|^2=u^{-1}\Delta u\). ◻

Proof. Choose smooth \(b\ge|K|_g\), positive everywhere and equal to \(b_0r^{-1-\beta}\) sufficiently far out, where \(b_0>0\) is chosen large enough. This is possible because \(\beta<q\). Choose a smooth positive regularizer \(\zeta\), equal to \(r^{-2}\) on the end, and extend \(w\) smoothly and positively to \(N\). We solve \[ -\Delta_g\phi=b\sqrt{|d\phi|_g^2+\zeta^2}+w, \qquad \partial_\nu\phi=-1, \qquad \phi\longrightarrow0. \tag{19}\] The regularizer makes this equation smooth even at critical points.

First truncate at a large coordinate sphere \(S_R\) and prescribe \(\phi=0\) there. The Neumann and Dirichlet conditions occur on disjoint boundary components. For \(0\le t\le1\), replace \(b\) in Equation (19) by \(tb\). At \(t=0\), the mixed Laplacian is invertible: its Dirichlet face gives the Poincare inequality, and the weak solution obtained from its coercive quadratic form is smooth by boundary regularity. At any solution, the linearization is a Laplacian with a smooth drift of norm at most \(b\) and with the homogeneous mixed boundary conditions. The strong maximum principle and the Hopf boundary principle make its kernel zero. It is a Fredholm perturbation of the mixed Laplacian of index zero, and is therefore invertible. The boundary estimates used here are the ordinary elliptic estimates for Dirichlet and normal Neumann conditions: positive definiteness of \(g\) verifies their principal complementing condition, and the disjoint boundary components can be covered separately (Agmon et al. 1959). The maximum principles and local elliptic estimates below are used in their usual uniformly elliptic forms (Gilbarg and Trudinger 2001).

To close continuation at \(t=1\), first work on a fixed truncation. The elliptic \(W^{2,p}\) estimate and gradient interpolation give \[\|\phi\|_{W^{2,p}} \le C_p\bigl(1+\|\phi\|_{L^p}+\|d\phi\|_{L^p}\bigr) \le \tfrac12\|\phi\|_{W^{2,p}} +C'_p(1+\|\phi\|_{L^p}).\] The constants include the fixed boundary data and are uniform in \(t\). If the supremum norms of solutions on this truncation were unbounded, divide them by their supremum norms. The preceding estimate with \(p>3\), compactness, and the equation give a nonzero limit \(v\) with homogeneous mixed data satisfying \[-\Delta_gv=t_*b|dv|_g.\] This can be written \(\Delta_gv+A\cdot dv=0\) with a bounded measurable drift: take \(A=t_*b\nabla v/|dv|\) off the critical set and zero on it. The strong and Hopf maximum principles exclude a nonzero solution with the homogeneous mixed conditions. Thus the supremum norms are bounded. The \(W^{2,p}\) estimate, Schauder estimates, and the smooth equation now give the bounds needed for closedness of continuation. Existence on each truncation follows.

Every such solution is nonnegative. A negative minimum cannot be interior because the right side of Equation (19) is positive. At an inner-boundary minimum, the outward-domain derivative would be negative by the Hopf principle, whereas the prescribed derivative in that direction is \(1\). The outer value is zero.

It remains to make these bounds independent of \(R\). For fixed sufficiently large \(A\) and then sufficiently large \(r_0\), the positive radial function \[v_0(r)=r^{-1}(1-Ar^{-\delta})\] satisfies, on \(r\ge r_0\), \[\begin{align*} -\Delta_gv_0-b|dv_0|_g &=A\delta(1+\delta)r^{-3-\delta} +O(r^{-3-q})+O(r^{-3-\beta})\ge c_0w. \end{align*}\] The constants are fixed independently of the truncation. Since \(b\zeta=o(w)\), a sufficiently large multiple of \(v_0\) is a supersolution of the regularized equation. The comparison principle gives \[ 0\le\phi(x)\le C(1+M_R)v_0(r),\qquad M_R=\sup_{N\cap\{r\le r_0\}}\phi, \quad r_0\le r\le R. \tag{20}\] The notation for the compact core includes all points outside the end chart. Were \(M_R\) unbounded along expanding truncations, the normalized solutions \(\phi/M_R\) would have locally uniform \(W^{2,p}\) bounds, including at \(B\), by the preceding local estimates and Equation (20). A subsequence would converge locally in \(C^1\) to a nonnegative function with maximum one on the core, homogeneous inner Neumann data, and a bounded-drift homogeneous equation. Equation (20) makes its value tend to zero at infinity. It consequently attains a positive maximum, contradicting the strong or Hopf principle for that homogeneous equation. This proves the uniform core bound. The contradiction uses the homogeneous bounded-drift equation.

Local compactness and diagonal extraction give a smooth solution of Equation (19), with \(\phi=O(r^{-1})\). The claimed derivative count can be checked directly on annuli. Put \[\phi_{\rho}(y)=\rho\phi(\rho y),\qquad g_{\rho}(y)=g(\rho y),\qquad 1<|y|<4.\] On smaller fixed annuli the equation is \[-\Delta_{g_{\rho}}\phi_{\rho} =\rho b(\rho y) \sqrt{|d\phi_{\rho}|_{g_{\rho}}^2+ (\rho^2\zeta(\rho y))^2} +\rho^3w(\rho y).\] Its coefficients have uniform \(C^{1,1}\) bounds from the two derivatives in Equation (9); the scaled drift is \(O(\rho^{-\beta})\) and the scaled source is \(O(\rho^{-\delta})\). The bounded zeroth-order norm, interior \(W^{2,p}\) estimates, and gradient interpolation first give uniform \(C^{1,\alpha}\) bounds for some \(\alpha>0\). The right side is then uniformly \(C^\alpha\). Schauder estimates give uniform \(C^{2,\alpha}\) bounds on smaller annuli. This proves \(\phi=O_2(r^{-1})\) without using a third derivative of \(g\) or a second derivative of \(K\).

The right side of Equation (19) is \(w+O(r^{-3-\beta})\). It is integrable and proves Equation (15). The divergence theorem, with outward-domain normal \(-\nu\) on \(B\), gives \[ \lim_{r\to\infty}\int_{S_r}\partial_{n_g}\phi\,\,\mathrm dA_g =-|B|_g-\int_N \bigl(b\sqrt{|d\phi|_g^2+\zeta^2}+w\bigr)\,\,\mathrm dV_g. \tag{21}\] The Euclidean flux differs by \(O(r^{-q})\) and has the same finite limit.

For every fixed finite \(s\ge0\), the factor \(u_s\) in Equation (16) is bounded and at least one, so the transformed data are complete up to \(B\). The \(u\)-form of Lemma 3 gives \[\begin{align*} 16\pi u_s^4(\mu_s-|J_s|_{g_s}) &\ge16\pi(\mu-|J|_g) +\frac{8s}{u_s}(-\Delta_g\phi-|K|_g|d\phi|_g) \ge \frac{8s}{u_s}w,\\ \theta_{+,s} &=u_s^{-2}\left(\theta_+-\frac{4s}{u_s}\right). \end{align*}\] This proves the asserted signs for all finite \(s\), without a smallness restriction.

Since \(u_s-1=O_2(r^{-1})\), the transformed decay exponent is \(\min(q,1)>1/2\). The exact energy transformation is \[16\pi u_s^4\mu_s=16\pi\mu-8s u_s^{-1}\Delta_g\phi.\] Here \(\Delta_g\phi=O(r^{-3-\delta})\), and the additional current term is bounded by a fixed-\(s\) constant times \(|K|_g|d\phi|_g=O(r^{-3-q})\). Both are integrable in dimension three. The bounded positive conformal factor also preserves integrability in the transformed volume measure.

The leading metric change is \(4s\phi\delta\), whose ADM energy flux is \(-8s\int_{S_r}\partial_r\phi\,\,\mathrm dA_\delta\) before division by \(16\pi\). All remaining metric flux terms are \(O_s(r^{-q})+O_s(r^{-1})\). Writing \(\pi=K-(\mathop{\mathrm{tr}}_gK)g\), one has \(\pi_s=u_s^2\pi\), so the change in its Euclidean momentum flux is \(O_s(r^2r^{-1}r^{-1-q})=O_s(r^{-q})\). This proves Equation (17); Equation (21) gives \(A_\phi>0\). No \(1/r\) coefficient expansion for \(\phi\) is needed. Finally, the cut class is unchanged and every cut has area \(\int_\Gamma u_s^4\,\,\mathrm dA_g\). Bounding \(u_s\) between \(1\) and \(1+s\|\phi\|_\infty\) and taking infima proves Equation (18), including coincident and disconnected cuts.

For the final assertion, take \(b\) nonnegative, compactly supported, and at least \(|K|\); the proof is unchanged. The end equation is then simply \(-\Delta_g\phi=w\). Once the two-derivative estimate has been established, successive annular elliptic estimates use the assumed symbol bounds for \(g\) and \(w\) to give \(\phi=O_k(r^{-1})\) for every \(k\). ◻

Proof. For purposes of perimeter minimization only, extend the metrics through a collar of \(B\) and attach a fixed compact filling. The filling need not satisfy any constraint or curvature condition. Extend \(g_j\) so that the extensions converge uniformly on this fixed collar and filling. A filled enclosure is then a bounded set containing the prescribed inner obstacle.

For each fixed metric, sufficiently large spheres in its asymptotically flat chart have strictly positive outward mean curvature. Clipping a bounded enclosure at any such sphere cannot increase perimeter. To see this, integrate the divergence of the outward unit normal field of the sphere foliation over the part of the enclosure outside the sphere. Its divergence is positive. Its flux through the exterior boundary is at most the area of that boundary, whereas its flux on the cutting sphere is the negative of the new cutting area. Thus the discarded boundary area is at least the inserted area. Approximation gives the same statement for finite-perimeter sets. The direct method on the resulting bounded domain now gives a minimizer. The clipping observation makes this a minimizer against every bounded competitor, not merely those inside the chosen truncation.

Off the obstacle the boundary is locally perimeter minimizing. If it contains a point of radius \(r\) sufficiently large, the coordinate ball of radius \(c_1r\) about that point avoids the obstacle. After scaling this ball to unit size, Equation (22) gives uniform ellipticity and uniform first-derivative bounds for its area integrand. The local perimeter density estimate gives \[\operatorname{Area}_{g_j} (\partial F_j\cap B_{c_1r})\ge c_2r^2.\] Here \(c_1,c_2>0\) are independent of \(j\) and \(r\). One may obtain this estimate from the relative isoperimetric inequality and the comparison which removes, or fills, the set in concentric balls; integration of the resulting differential inequality gives the area lower bound. The fixed competitor bounds the left side above independently of \(j\). Consequently the minimizing boundaries lie in a fixed compact set.

Perimeter compactness now gives a limiting filled enclosure. Uniform metric convergence on that compact set and lower semicontinuity imply \[a_g(B)\le\liminf_j a_{g_j}(B).\] Conversely, using a fixed enclosure with \(g\)-area arbitrarily close to \(a_g(B)\) gives the reverse upper-limit inequality.

The local obstacle regularity theorem for pure area minimization with a \(C^2\) obstacle gives \(C^{1,1}\) regularity, and smoothness off contact, in dimension three; its local estimates require only the appropriate bounded geometry of the smooth metric (Huisken and Ilmanen 2001, Regularity Theorem 1.3). This applies to the fixed smooth obstacle and our smooth one-sided metrics after the collar extension. Finally, push a minimizing boundary slightly away from its contact set by a smooth vector field agreeing there with the outward obstacle normal. In a small obstacle collar its flow increases the signed distance to the obstacle. The resulting \(C^1\) embedded boundary is a positive distance from the obstacle. Smooth local graph approximation, within that distance, preserves enclosure and changes area by an arbitrarily small amount. This proves equality of the smooth and perimeter infima and also justifies using smooth fixed competitors in the upper-limit argument. ◻

Proof. In static isotropic coordinates \((T,y)\) the Schwarzschild metric is \[ -\left(\frac{1-m/(2r)}{1+m/(2r)}\right)^2\,\mathrm dT^2 +\left(1+\frac{m}{2r}\right)^4\delta_{ij}\,\mathrm dy^i\,\mathrm dy^j, \qquad r=|y|. \tag{23}\] Its perturbation \(h\) from Minkowski space has symbol decay \(r^{-1}\) on every fixed spacelike cone. Constant Lorentz transformations preserve this decay. We verify the transformation of the ADM charges directly.

In inertial coordinates for the background Minkowski metric \(\eta\), put \[\begin{align*} \mathcal F_{\ell ab} ={}&\partial_a h_{\ell b}-\partial_\ell h_{ab}\\ &+\eta_{\ell b} (\partial^d h_{ad}-\partial_a\mathop{\mathrm{tr}}_\eta h) -\eta_{ab} (\partial^d h_{\ell d}-\partial_\ell\mathop{\mathrm{tr}}_\eta h). \tag{24}\end{align*}\] This tensor is antisymmetric in its first two slots, and differentiation of the displayed formula gives \(\partial^\ell\mathcal F_{\ell ab}=2G^{\mathrm{lin}}_{ab}\). The exact Einstein tensor vanishes; its terms beyond the linearization are \(O(r^{-4})\). Thus this divergence is \(O(r^{-4})\).

Use an inertial frame in which the plane under consideration is \(x^0=0\). With \(\eta=\operatorname{diag}(-1,1,1,1)\), direct substitution in Equation (24) gives \[\begin{align*} \mathcal F_{i00}&=\partial_jh_{ij}-\partial_ih_{jj},\\ \mathcal F_{i0k} &=2\bigl(K^{\mathrm{lin}}_{ik} -\delta_{ik}\mathop{\mathrm{tr}}K^{\mathrm{lin}}\bigr) +\partial_kh_{0i}-\delta_{ik}\partial_jh_{0j}, \end{align*}\] where \[K^{\mathrm{lin}}_{ik} =\tfrac12(\partial_0h_{ik}-\partial_ih_{0k}-\partial_kh_{0i}).\] The convention is the normal rate of the spatial metric divided by two. For fixed \(k\), the final vector has zero flux through every closed sphere. Indeed it equals \[\partial_j(\delta_{jk}h_{0i}-\delta_{ik}h_{0j}),\] the divergence of a tensor antisymmetric in \(i,j\). This zero-flux identity is local to a neighborhood of the sphere: extend the potential smoothly inside before applying the divergence theorem, if necessary. The nonlinear errors in the induced metric and second fundamental form give vanishing flux errors. Therefore the limiting fluxes of \(\mathcal F_{i0b}\) are \(16\pi(E,P)\).

These fluxes are the integrals of the dual two-form of the first two slots of \(\mathcal F\), with its translation slot held fixed. They therefore transform tensorially under a constant Lorentz change of frame. Independence of the limiting spacelike cut follows directly from Stokes’ theorem: two large cuts on planes of slopes bounded strictly below one can be joined within a fixed spacelike cone by a three-dimensional cylinder of volume \(O(R^3)\), all of whose points have radius comparable to \(R\). Its divergence error is \(O(R^{-4})\), so the flux difference is \(O(R^{-1})\). Replacing the resulting ellipsoidal cuts by coordinate spheres on either plane gives the same estimate. The limiting translation charges consequently transform by the Lorentz representation. The static plane has charges \((m,0)\), as follows by inserting Equation (23) into the ADM integral. Its Lorentz orbit is precisely the set of future timelike vectors of norm \(m\), proving the assertion.

More explicitly the plane can be written \(T=v\cdot y\), \(|v|<1\). Writing \(y=Ax\), where \(A=(I-v\otimes v)^{-1/2}\), makes the induced Minkowski metric in the \(x\) coordinates equal to \(\delta\). The displayed decay then follows from the symbol estimates in Equation (23) and the formula for the second fundamental form. These coordinates will be used when identifying the reference end with the given end. ◻

Proof. We first record the ordinary divergence inverse being used. On a ball, the regularized Bogovskii operator solves \(\mathop{\mathrm{div}}u=f\) for a mean-zero compactly supported source, preserves compact support, and gains one derivative in \(W^{k,p}\); see (Costabel and McIntosh 2010, sec. 3.1, Equation (3.13), Theorem 3.2, Corollary 3.4, and Remark 3.5). To apply it on the annulus, cover \(Q\) by finitely many balls compactly contained in \(\mathcal A\), adding finitely many overlapping balls so that the overlap graph is connected. Partition the source among the balls. On a spanning tree of the overlap graph, pass each piece’s integral to its parent by subtracting and adding that integral times a fixed unit-integral bump in the overlap. Starting at the leaves leaves every piece with zero integral; the root does also because the total integral vanishes. The decomposition has bounded Sobolev norms because all cutoffs and bumps are fixed. The sum of the ball inverses gives a compact inverse on the annulus. For sources in a fixed compact set its output support lies in another fixed compact set. This construction may be repeated twice using slightly enlarged compact sets still contained in \(\mathcal A\).

For the scalar assertion, solve \(\partial_i u_i=f\) compactly. Integration by parts gives \[\int u_i=-\int x^if=0.\] Apply the divergence inverse to each component of \(u\), obtaining \(\partial_j A_{ij}=u_i\). Symmetrizing \(A\) does not change its double divergence. Denote this symmetric matrix by \(S\) and put \[h=S-\tfrac12(\mathop{\mathrm{tr}}S)\delta.\] Since \(h-(\mathop{\mathrm{tr}}h)\delta=S\), its linearized scalar curvature is \(f\). The two applications of the divergence inverse give the two-derivative gain.

For the vector assertion, translation orthogonality gives \(\int V_i=0\). Solve \(\partial_jB_{ij}=V_i\) componentwise. Orthogonality to rotations and integration by parts imply \[0=\int(x^jV_i-x^iV_j) =-\int(B_{ij}-B_{ji}).\] Thus the skew part \(S_{ij}=(B_{ij}-B_{ji})/2\) has zero integral. Apply the divergence inverse to its independent components to obtain a tensor \(D\), skew in its first two indices, such that \(\partial_kD_{ijk}=-S_{ij}\). Define \[ C_{ijk}=D_{ijk}-D_{ikj}-D_{jki}. \tag{26}\] Index interchange shows both \(C_{ijk}=-C_{ikj}\) and \((C_{ijk}-C_{jik})/2=D_{ijk}\). Consequently \[T_{ij}=B_{ij}+\partial_kC_{ijk}\] is symmetric and satisfies \(\partial_jT_{ij}=V_i\). The last statement uses the skew symmetry in \(j,k\) to cancel the double divergence of \(C\). Set \(k=T-(\mathop{\mathrm{tr}}T)\delta/2\) to invert trace reversal. The first inverse gains one derivative, the second gains another, and the last derivative of \(C\) leaves a net gain of one. All constructions preserve smoothness and have the stated support and norm bounds. ◻

Proof. The scalar-curvature expansion has remainder bounded by \[C\bigl(|h||\partial^2h|+|\partial h|^2+|K|^2\bigr),\] and the replacement of the momentum constraint by Euclidean divergence has remainder bounded by \(C(|h||\partial K|+|\partial h||K|)\). These are \(O(r^{-2-2q})\), using exactly the derivatives in Equation (9). They are integrable because \(2+2q>3\).

For any integrable function \(F\) on the end and fixed \(c>1\), \[ \frac1R\int_{r_0<|x|<cR}|x|\,|F(x)|\,\,\mathrm dx\longrightarrow0. \tag{28}\] Indeed the contribution inside any fixed radius \(M\) tends to zero after division by \(R\); the remaining contribution is at most \(c\int_{|x|>M}|F|\). Let \(R\to\infty\) and then \(M\to\infty\).

For an affine function \(f\), the vector field \[\mathcal Q_f(h)^i =f(\partial_jh_{ij}-\partial_i\mathop{\mathrm{tr}}h) -(\partial_jf)h_{ij}+(\partial_if)\mathop{\mathrm{tr}}h\] satisfies \(\partial_i\mathcal Q_f(h)^i=f\mathcal Lh\). For a Euclidean Killing field \(Y\), \[\partial_j(\pi_{ij}Y^i)=Y^i\partial_j\pi_{ij},\] because \(\pi\) is symmetric. Integrating from a fixed inner sphere and using Equation (28) proves the asserted \(o(R)\) estimates. This argument does not assert the existence of a center-of-mass or angular-momentum limit and requires no parity condition. ◻

Proof. Fix \(\tfrac12<\beta<\min(q,1)\) and work on the annulus \(\mathcal A=\{1<|y|<4\}\) with \(x=Ry\). The scaled fields are \[g_R(y)=g(Ry),\qquad k_R(y)=RK(Ry),\] and likewise for the reference data. Their constraint densities are the original densities multiplied by \(R^2\). If \(h_R=g_R-\delta\), then \[ \|h_R\|_{C^2(\mathcal A)}+\|k_R\|_{C^1(\mathcal A)} +\|h_{*,R}\|_{C^2(\mathcal A)}+\|k_{*,R}\|_{C^1(\mathcal A)} \le CR^{-\beta}. \tag{30}\] Choose \(\chi\in C^\infty(\mathcal A)\) with \(0\le\chi\le1\), equal to one near \(|y|=1\) and zero near \(|y|=4\), with all its derivatives supported in a fixed compact subannulus. Blend both fields by this cutoff.

Write the constraints in the unnormalized form \[\mathcal C(g,k) =\bigl(\mathop{\mathrm{Scal}}_g+(\mathop{\mathrm{tr}}_gk)^2-|k|_g^2, \mathop{\mathrm{div}}_g(k-(\mathop{\mathrm{tr}}_gk)g)\bigr).\] Their Euclidean linear part is \((\mathcal Lh,\mathcal Dk)\). On fields satisfying Equation (30), the nonlinear remainder is bounded in supremum norm by \(CR^{-2\beta}\). Thus the blended constraints equal the blended original constraint sources, plus a quadratic error and the compact commutators \[\begin{align*} f_R&=\mathcal L(\chi(h_R-h_{*,R})) -\chi\mathcal L(h_R-h_{*,R}),\\ V_R&=\mathcal D(\chi(k_R-k_{*,R})) -\chi\mathcal D(k_R-k_{*,R}). \end{align*}\] They have \(C^{0,1}\) norms at most \(CR^{-\beta}\). In detail the scalar commutator applied to a symmetric tensor \(h\) is \[\begin{align*} &2\partial_i\chi(\partial_jh_{ij}-\partial_i\mathop{\mathrm{tr}}h) +(\partial_i\partial_j\chi)h_{ij}-(\Delta\chi)\mathop{\mathrm{tr}}h; \end{align*}\] one derivative of this expression uses at most two derivatives of \(h\). The vector commutator is \((\partial_j\chi)(k_{ij}-(\mathop{\mathrm{tr}}k)\delta_{ij})\); one derivative uses at most one derivative of \(k\). No derivative of the complete matter sources is needed.

Every affine moment of \(f_R\) and every Killing-field moment of \(V_R\) is \(o(R^{-1})\). For the constant moments, integration by parts on \(\mathcal A\) gives \(R^{-1}\) times the difference between the physical translation fluxes on its two boundary spheres, less the integrals of the blended linear sources. The flux difference tends to zero because the reference and original ADM vectors agree. Replacing the physical momentum integrand by its Euclidean trace reversal costs \(O(R^{1-2q})\), which tends to zero. By Lemma 8, the physical linear-source integrals on this receding annulus tend to zero as well. For a linear scalar test or a rotational vector test the corresponding factor is \(R^{-2}\). Its physical boundary fluxes are \(o(R)\) by the same lemma, while the weighted source integrals are \(o(R)\) by Equation (28). This proves the claim.

Remove these small moments by fixed smooth bumps. Explicitly, for a basis \(p_1,\ldots,p_4\) of affine functions and a fixed nonnegative bump \(\omega\) positive on a ball inside the annulus, the Gram matrix \[G_{ab}=\int\omega p_ap_b\] is positive definite. Subtract \(\omega\sum_{a,b}p_a(G^{-1})_{ab}\int p_bf_R\) from \(f_R\). For the vector moments use in exactly the same way the Gram matrix \(\int\omega Y_a\cdot Y_b\) of a basis of the six Euclidean Killing fields. It too is positive definite, since a nonzero Killing field cannot vanish on an open ball. Both bump corrections have every fixed smooth norm \(o(R^{-1})\).

Apply Lemma 7 with the negative of the moment-free commutators. Since their \(W^{1,p}\) norms are \(O(R^{-\beta})\), choosing \(p>3\) gives metric and tensor corrections of size \(O(R^{-\beta})\) in \(W^{3,p}\) and \(W^{2,p}\), respectively. Sobolev embedding gives the needed \(C^{2,\alpha}\) and \(C^{1,\alpha}\) bounds. Their supports stay in a fixed compact subset of \(\mathcal A\). Extend them by zero and undo the scaling. The metric remains positive definite for large \(R\).

To check DEC, denote the original scaled sources by \((A_R,B_R)\), so \(A_R\ge2|B_R|_{g_R}\). The reference sources vanish. The corrected sources therefore have the form \[\bigl(\chi A_R,\chi B_R\bigr) +o(R^{-1})+O(R^{-2\beta})\] in supremum norm. The norm of \(B_R\) changes by at most \(CR^{-\beta}|B_R|\le CR^{-2\beta}\) when the metric is changed, since the pointwise constraint formula gives \(|B_R|\le CR^{-\beta}\). All other nonlinear correction terms have the same quadratic bound. Because \(2\beta>1\), the scaled DEC deficit is \(o(R^{-1})\). Multiplication by \(R^{-2}\) gives Equation (29). Outside the correction annulus the data are either the original DEC data or the exact vacuum reference data. Smoothness follows from smoothness of the original fields and of the compact inverse constructions, irrespective of the sizes of their unneeded higher derivatives. ◻

Proof. Use the description \(T=v\cdot y\), \(y=Ax\), from Lemma 6. Choose \(R_b=c_bR\) so large, with \(c_b\) fixed, that the region \(|y|\ge R_b\) is entirely inside the exact reference part. Let \(\psi\) be a smooth function equal to one on \((-\infty,0]\), zero on \([1,\infty)\), and taking values in \([0,1]\). Replace the plane by the graph \[ T_R(y)=(v\cdot y) \psi\left(\frac{\log(|y|/R_b)}{L}\right). \tag{33}\] Its Euclidean gradient satisfies \[|dT_R|_\delta\le |v|\bigl(1+\|\psi'\|_\infty/L\bigr).\] Choose a fixed \(L\) making the right side strictly smaller than one. The lapse and spatial coefficients in Equation (23) converge to their Minkowski values, so for all sufficiently large \(R\) this graph is uniformly spacelike. In fact for positive Schwarzschild mass the lapse is smaller than one and the spatial conformal factor exceeds one, which only improves this estimate on the static exterior.

The graph agrees with the original plane before \(R_b\) and with a constant-time slice after \(e^LR_b\). Its first derivative is bounded, and its second and third derivatives are \(O(r^{-1})\) and \(O(r^{-2})\). The induced metric is uniformly comparable with \(\delta\); differentiating its pullback formula gives the first bound in Equation (31). The second fundamental form uses the graph’s second derivatives and ambient first derivatives, divided by a uniformly positive spacelikeness factor, and gives the other bound. These estimates remain true in the \(x\) coordinates because \(A\) is a fixed invertible linear map. The constraints vanish on the entire bent part by the Gauss–Codazzi equations in exact vacuum Schwarzschild spacetime.

It remains to make the radius precise. On the bent region keep \(\rho_R=|x|=|A^{-1}y|\). Beyond the bend, where the slice is already static, write \(y=r\omega\) and put \(a(\omega)=|A^{-1}\omega|\). Choose \(R_c\) a sufficiently large fixed multiple of \(R_b e^L\), and a cutoff \(\chi_0\) equal to one before zero and zero after one. In that static region set \[\rho_R(r,\omega) =r\exp\left[ \chi_0\left(\frac{\log(r/R_c)}{L_0}\right)\log a(\omega)\right].\] For fixed sufficiently large \(L_0\) its radial derivative is bounded below by a positive constant: differentiating in \(r\) gives the positive factor \(\rho_R/r\) times \(1+L_0^{-1}\chi_0'\log a\), which is at least \(1/2\). The function and its first two Euclidean derivatives have the bounds \(\rho_R\asymp r\), \(|d\rho_R|\le C\), and \(|\partial^2\rho_R|\le C/r\). Before this interpolation, the same bounds hold for \(|A^{-1}y|\), and uniform metric comparability gives the gradient lower bound. Combining these facts with Equation (31) proves Equation (32). After the interpolation, \(\rho_R=r\) is precisely the static isotropic radius. All radius ratios used in the construction are fixed, independently of \(R\). ◻

Proof. All geometry estimates below are uniform in \(R\). Enlarge the fixed transition region slightly so that, in terms of \(t=\rho_R/R\), it lies inside an interval \((a,b)\) with \(0<a<b<\infty\). Arrange that \(t\ge b\) is in the static spherical region, and that the support of the possible deficit lies in a fixed smaller interval \(I\Subset(a,b)\). Increasing \(b\) if needed makes these assertions follow from Lemma 10. We take \(a=1\): the compact supports in the annular correction leave a fixed neighborhood of \(t=1\) free of deficit. The factor constructed below will consequently be constant throughout the original region \(r\le R\), and is extended by that constant over the compact core of \(N\).

Let \(\eta_R\to0\) be as in Equation (29), replacing it by a positive majorant if necessary. We will choose \(\varepsilon_R=C_0\eta_R\) and construct \[u_R=1+\frac{\varepsilon_R}{R}F_R(\rho_R/R),\] where \(F_R\) and the relevant derivatives are bounded independently of \(R\). Write \(z_R=-F_R'\). On \([a,b+1]\), initially define \(z_R\) by \[ z_R'=C_1z_R+\omega_1, \qquad z_R(a)=0, \tag{35}\] where \(\omega_1\) is smooth, nonnegative, zero near \(a\), and at least one on \(I\). Take \(C_1\) sufficiently large, depending only on the constants in Equations (31) and (32). The solution is nonnegative and has uniform bounds on this fixed interval. For a function with \(F_R'=-z_R\) the chain rule gives \[\begin{align*} -\Delta_{g_R}u_R-|K_R|_{g_R}|du_R|_{g_R} &=\frac{\varepsilon_R}{R^3} \left[z_R'|d\rho_R|_{g_R}^2 +Rz_R\Delta_{g_R}\rho_R -Rz_R|K_R|_{g_R}|d\rho_R|_{g_R}\right]\\ &\ge\frac{\varepsilon_R}{R^3} (c_1z_R'-C_2z_R). \end{align*}\] Choose \(C_1\) so that this is nonnegative and is at least \(c_1\varepsilon_RR^{-3}\) on \(I\).

The preceding differential inequality need not be imposed on the spherical tail. There \(K_R=0\) and the Schwarzschild radial Laplace flux coefficient is \((r+m/2)^2\). Put \(c_R=m/(2R)\) and, on the still spherical interval near \(b\), define \[A_R(t)=(t+c_R)^2z_R(t).\] It has positive derivative wherever \(z_R>0\). Continue it to a positive constant on \([b+1,\infty)\) while keeping \(A_R'\ge0\). For an explicit smooth continuation, continue the solution of Equation (35) to \(b+1\), multiply the derivative of \((t+c_R)^2z_R(t)\) by a nonnegative cutoff equal to one near \(b\) and zero near \(b+1\), and integrate with the initial value \(A_R(b)\). Define \(z_R=A_R(t)/(t+c_R)^2\) on this tail, and set it to zero for \(t\le a\). All joins are smooth, the functions are nonnegative, and their bounds on finite intervals are uniform in \(R\). The constant tail value, denoted \(A_R^\infty\), is bounded independently of \(R\).

Now define \[F_R(t)=\int_t^\infty z_R(s)\,\,\mathrm ds.\] It is nonnegative, uniformly bounded, and constant before \(a\). The preceding construction gives a superharmonic \(u_R\) on the spherical tail, because its signed radial flux is \(-\varepsilon_RA_R(t)\) and is nonincreasing. It also gives Equation (34). For large \(R\), \(1\le u_R\le2\). Increasing \(C_0\) ensures \[-\Delta_{g_R}u_R-|K_R||du_R| \ge \frac{u_R}{8}\, \bigl[16\pi(|J_R|_{g_R}-\mu_R)\bigr]_+.\] On the original inner region the conformal factor is constant and the original data satisfy DEC. Elsewhere this inequality and Lemma 3 prove DEC for the repaired data. Using \(u_R\), rather than estimating \(\log u_R\) separately, incorporates the gradient-square term exactly.

On the final end, \[u_R=1+\frac{\varepsilon_RA_R^\infty}{r+m/2}.\] Writing \(a_R=\varepsilon_RA_R^\infty=o(1)\), one obtains the exact identity \[ u_R^4\left(1+\frac{m}{2r}\right)^4\delta =\left(1+\frac{m/2+a_R}{r}\right)^4\delta. \tag{36}\] Thus the final mass is \(m+2a_R=m+o(1)\) and the momentum is zero. Multiplication of \(K_R\) by \(u_R^2\) preserves its compact support. On the inner boundary, where \(u_R\) is constant, the expansion is simply multiplied by \(u_R^{-2}\), so its weak sign is preserved. ◻

Proof. Apply Proposition 9, Lemma 10, and Lemma 11 along any sequence \(R_j\to\infty\). The resulting DEC exteriors have compactly supported second fundamental form and an exact rest Schwarzschild end with energy tending to \(m\). They agree with the original data on expanding compact subsets except for a constant conformal factor tending to one. Hence they converge smoothly on each fixed compact subset. Their boundary expansion is weakly negative.

Apply the last part of Lemma 4 separately to each of these fixed exteriors, using the same chosen \(0<\delta<1\) and a compactly supported drift coefficient. Choose its positive parameter \(s\) in Equation (16) small enough that the energy change, \(\|\log(1+s\phi)\|_\infty\), and the supremum of its first derivative multiplied by the fixed original coordinate radius are at most \(1/j\). Also require the change in the first \(j\) smooth norms on the first \(j\) members of a compact exhaustion to be at most \(1/j\). All these quantities are finite for the fixed \(j\)-th exterior, so these choices are possible. No estimate uniform in the receding bending radius is required for this final strictification.

The resulting data satisfy (i). The strict conformal factor has symbol decay of all orders and does not enlarge the support of \(K\). On the far Schwarzschild end the drift vanishes, so \(-\Delta\phi=w=r^{-3-\delta}\). Since the rest Schwarzschild metric has zero scalar curvature and \(K=0\) there, the exact \(u\)-formula gives \[R_{g_j}=8s(1+s\phi)^{-5}r^{-3-\delta}.\] This proves the scalar-curvature bound in (ii). The Ricci bound follows from the two-derivative metric decay. The energy change tends to zero and the momentum change is zero. The prescribed smallness gives smooth local convergence, proving (ii) and the local convergence and charge assertions in (iii).

The metrics before strictification satisfy the uniform comparisons and scaled first-derivative bounds of Lemma 5, in the fixed original end chart: the reference plane, the bend, and the radius interpolation involve only fixed linear maps and fixed radius ratios. The radial repair preserves these bounds. The final strictification parameters were chosen to preserve them as well. That lemma therefore gives the convergence of enclosing infima.

If the reduced energy inequality is available, applying it to these data gives \[E_j\ge\sqrt{a_{g_j}(B)/(16\pi)}.\] Passing to the limit proves \(m\ge\sqrt{a_g(B)/(16\pi)}\). Only the energies and enclosing infima enter this passage to the original data. ◻

Proof. Choose a countable dense set \(\{x_i\}\) and a number \(L\) larger than the diameter of the compact space. Set \(d_c(x)=\min\{L,\operatorname{dist}(x,E_c)\}\), with the value \(L\) when \(E_c\) is empty. For each \(i\), \(c\mapsto d_c(x_i)\) is a bounded nonincreasing real function, so its right discontinuities form a countable set. Avoid their countable union. Then \(d_{c_j}(x_i)\to d_c(x_i)\) for all \(i\) whenever \(c_j\downarrow c\). Every \(d_c\) is \(1\)-Lipschitz. A finite net chosen from the dense set therefore upgrades this convergence to uniform convergence on the compact space.

For nonempty \(E_c\), if points \(y_j\in E_{c_j}\) stayed a fixed distance from \(E_c\), uniform convergence evaluated at \(y_j\) would contradict \(d_{c_j}(y_j)=0\). The opposite Hausdorff inclusion follows from \(E_c\subset E_{c_j}\). For empty \(E_c\), uniform convergence to \(L\) is incompatible with any nonempty \(E_{c_j}\), since its distance function has a zero. This proves the stated convention as well. ◻

Proof. Use normal graphs \(\Sigma(v)\) over \(\Sigma\) and pull their expansion back to \(\Sigma\). The map \[\Phi:C^{2,\alpha}(\Sigma)\longrightarrow C^{0,\alpha}(\Sigma), \qquad \Phi(v)=\Theta_Q(\Sigma(v))-c\] is smooth near zero and has linearization \(L=D\Phi(0)\), the MOTS stability operator for \(Q-(c/2)g\). Its principal part is \(-\Delta_\Sigma\). Stability gives a principal eigenvalue \(\lambda\geq0\) and a positive smooth eigenfunction \(\varphi\). If \(\lambda>0\), take \(v=s\varphi\). Smooth dependence gives \(\Phi(s\varphi)=s\lambda\varphi+O(s^2)\) in \(C^0\), which is positive for all sufficiently small \(s>0\). The graph velocity is positive as well.

Suppose \(\lambda=0\). The kernel of \(L\) is spanned by \(\varphi\), and the adjoint kernel is spanned by a positive smooth function \(\varphi^*\). These are the principal-eigenfunction facts for a scalar elliptic operator on a connected closed surface. For example, simplicity of the kernel also follows by writing a kernel element as \(v=\varphi w\) and applying the strong maximum principle to the resulting equation for \(w\), whose zeroth-order term vanishes. Consider the augmented map \[\mathcal G(v,a) =\left(\Phi(v)-a,\ \int_\Sigma v\,\,\mathrm dA\right).\] Its derivative at \((0,0)\) is \[ (v,a)\longmapsto \left(Lv-a,\ \int_\Sigma v\,\,\mathrm dA\right). \tag{47}\] This map is an isomorphism. Indeed, to solve \(Lv-a=f\) with \(\int v=b\), the Fredholm compatibility condition uniquely fixes \[a=-\frac{\int_\Sigma\varphi^*f\,\,\mathrm dA} {\int_\Sigma\varphi^*\,\,\mathrm dA}.\] The equation for \(v\) then has a solution; adding a multiple of \(\varphi\) imposes its prescribed integral uniquely. Elliptic estimates and this one-dimensional normalization give a bounded inverse between the indicated Hölder spaces.

The implicit-function theorem consequently supplies smooth \(v(s),a(s)\) with \(\mathcal G(v(s),a(s))=(0,s)\). Differentiating at zero and pairing with \(\varphi^*\) gives \(a'(0)=0\) and \[v'(0)=\frac{\varphi}{\int_\Sigma\varphi\,\,\mathrm dA}>0.\] After shortening the interval, all these graphs form an outward foliation and have constant expansion \(c+a(s)\). If \(a(s)\leq0\) at any positive \(s\) in this interval, replace only this frontier component by \(\Sigma(v(s))\). The collar is disjoint from all other components and designated barriers, so the new smooth bounding domain strictly contains the maximal region and its every free boundary component has expansion at most \(c\). This contradicts maximality. Therefore \(a(s)>0\) for every such \(s>0\).

In either case positive graph velocity and compactness give the asserted bounds on \(|\nabla s|\). Smooth elliptic bootstrapping of the graph equation gives smooth leaves in the zero-eigenvalue case. ◻

Proof. Take a support \(\phi\) to \(D_z\) at \(x'\). There is a nearest point \(x\in D\) and a length-\(z\) minimizing unit-speed geodesic \(\gamma:[0,z]\to N\) from \(x\) to \(x'\). For the radii under consideration the entire segment lies in a fixed smooth compact neighborhood and is shorter than its injectivity radius. The ball \(\overline B_z(x)\) is contained in \(D_z\) and touches its exterior support at \(x'\). Its tangent plane there therefore agrees with that of the support and \[ \frac{\nabla\phi(x')}{|\nabla\phi(x')|}=\dot\gamma(z). \tag{50}\] Let \(P_t:T_xN\to T_{\gamma(t)}N\) denote parallel transport and put \(P=P_z\).

For every \(X\in T_xN\) solve the Jacobi boundary problem \[ J_X''+\operatorname{Rm}(J_X,\dot\gamma)\dot\gamma=0, \qquad J_X(0)=X,\qquad J_X(z)=PX. \tag{51}\] Here primes denote covariant differentiation along \(\gamma\). The short interval and absence of conjugate points make this problem uniquely solvable. Its estimates can be seen without division by an uncontrolled quantity. In a parallel frame write \(j_X(t)=P_t^{-1}J_X(t)\) and \(\mathcal R(t)\) for the curvature coefficient. The integral equation is \[j_X(t)=X+tA_X- \int_0^t(t-s)\mathcal R(s)j_X(s)\,\,\mathrm ds, \qquad A_X=\frac1z\int_0^z(z-s)\mathcal R(s)j_X(s)\,\,\mathrm ds.\] It implies, by absorption for small \(z\), \[ \sup_{0\leq t\leq z}|j_X(t)|\leq2|X|, \qquad |A_X|\leq Cz|X|. \tag{52}\] Also \(\langle J_X,\dot\gamma\rangle\) is affine in \(t\) and has equal endpoint values. Thus \(\langle A_X,\dot\gamma(0)\rangle=0\).

We may therefore prescribe the first jet of a smooth unit vector field \(V\) near \(x\) by \[V(x)=\dot\gamma(0),\qquad \nabla_XV(x)=A_X.\] To realize it with uniform second-jet bounds, choose a smooth local orthonormal frame obtained by radial parallel transport from \(x\). In that frame take the affine coefficient vector with the prescribed value and first derivatives, and normalize its length. The orthogonality just proved means normalization preserves its prescribed first derivatives. The frame has uniformly bounded derivatives on a fixed small coordinate ball, and Equation (52) therefore gives uniformly bounded second derivatives of \(V\). This construction uses the radial direction and the ambient geometry, with no derivatives of the support \(\phi\).

Define the local endpoint map \[F_z(y)=\exp_y\bigl(zV(y)\bigr).\] Its differential at \(x\) is the terminal value of the Jacobi field with initial values \(X,\nabla_XV(x)\). By Equation (51), it has the exact property \[ F_z(x)=x',\qquad (\,\mathrm dF_z)_x=P, \qquad |(\nabla\,\mathrm dF_z)_x|\leq Cz. \tag{53}\] The last bound follows by differentiating the smooth map \((y,v,z)\mapsto\exp_y(zv)\) twice in \(y\), including the first and second derivatives of \(V\). At \(z=0\) the map is the identity and its covariant second derivative vanishes. Its \(z\)-derivative is uniformly bounded for the bounded jets just constructed, giving the displayed \(Cz\) bound. The exact middle identity, rather than a near-isometry estimate, is essential.

The map \(F_z\) takes \(D\) locally into \(D_z\), since its defining geodesic from any \(y\in D\) has length \(z\). Consequently \(\psi=\phi\circ F_z\) is an exterior support to \(D\) at \(x\). Its derivative is \(P^*\,\mathrm d\phi\), so its gradient length is exactly \(|\nabla\phi(x')|\) and, by Equation (50), its normal is \(\dot\gamma(0)\). The covariant chain rule gives \[\mathop{\mathrm{Hess}}\psi(X,Y) =\mathop{\mathrm{Hess}}\phi(PX,PY) +\,\mathrm d\phi\bigl((\nabla\,\mathrm dF_z)(X,Y)\bigr) \quad\hbox{at }x.\] Trace over an orthonormal basis of \(\dot\gamma(0)^\perp\) and divide by the common gradient length. The first term is exactly the corresponding normalized tangential trace at \(x'\), and the remaining term has absolute value at most \(Cz\). Finally, the tensor trace changes by at most \(Cz\), by the \(C^1\) bound on \(Q\) and parallel transport. It follows that \[\bigl|\mathcal E_Q[\psi](x) -\mathcal E_Q[\phi](x')\bigr|\leq Cz.\] The assumed support inequality at \(x\) proves Equation (49). There is no term containing \(z|\mathop{\mathrm{Hess}}\phi|\). ◻

Proof. All distance neighborhoods in this proof are relative to \(N\). Since \(D\) contains a collar of \(\Gamma_-\) and avoids \(\Gamma_+\), their new frontiers, at sufficiently small radii, lie in a compact subset of the interior. A minimizing segment ending on a new frontier has its nearest point on the interior frontier of \(D\); it cannot start on \(\Gamma_-\) because its covered collar has fixed positive width. Thus the proof of Lemma 17 applies unchanged. Choose \(\beta>0\) so small that its conclusion holds for \(0<z\leq\beta\), all these frontiers avoid the prescribed boundary, and \[ c+C\beta<c'. \tag{55}\]

We first construct a smooth inner enclosure without imposing any expansion bound on it. Smoothly approximate the continuous distance function \(d_D=\operatorname{dist}(\cdot,D)\) on \(N\) uniformly, with error less than \(\beta/32\), using a finite coordinate cover, mollification, and a partition of unity. Choose a regular value in \((\beta/3,\beta/2)\) of the resulting smooth function. Its sublevel set \(U\) has smooth interior boundary and, for example, satisfies \[ D_{\beta/4}\subset U\subset D_{3\beta/4}. \tag{56}\] The boundary \(\Gamma_-\) is contained in its relative interior and \(\Gamma_+\) is excluded. If a component of \(N\setminus\overline U\) meets no component of \(\Gamma_+\), fill it. This only deletes components of the smooth free boundary. After this filling the second inclusion in Equation (56) need not hold for its interior, but every remaining free boundary component still lies in the indicated thin neighborhood, and the first inclusion still holds.

Choose \(\eta\in C^\infty(N)\), \(0\leq\eta\leq1\), equal to one on that free boundary and supported in \(\{d_D<\beta\}\). Its existence uses the positive separation between this compact smooth boundary and the closed set \(\{d_D\geq\beta\}\); it requires no smoothness of \(d_D\). For a sufficiently large finite constant \(A\), the tensor \[ Q_A=Q-\tfrac12(c'+A\eta)g \tag{57}\] makes the free boundary of \(U\) a strictly negative inner barrier: \[\Theta_{Q_A}(\partial U) =\Theta_Q(\partial U)-c'-A<0.\] All prescribed outer faces remain strictly positive for \(Q_A\), because \(\eta=0\) there and their original expansion exceeds \(c'\). Apply the barrier part of Theorem 13 on each component of \(N\setminus\overline U\). Each meets \(\Gamma_+\) by the filling step and has nonempty inner boundary by connectedness of \(N\) and \(U\ne\varnothing\). We obtain smooth MOTSs \(\Sigma\) bounding a domain \(E\) which contains \(\overline U\) and avoids \(\Gamma_+\). They satisfy, with their normal out of \(E\), \[ \Theta_Q(\Sigma)=c'+A\eta\geq c'. \tag{58}\]

We claim \(\Sigma\) misses \(\{d_D<\beta\}\). Otherwise set \(z=\min_{\Sigma}d_D<\beta\). The first inclusion in Equation (56) gives \(z\geq\beta/4>0\). Moreover \(D_z\) lies on the inner side of \(\Sigma\). To see this directly, if \(d_D(y)<z\), a shortest path from \(y\) to \(D\) of length less than \(z\) cannot cross \(\Sigma\): such a crossing would have distance from \(D\) less than \(z\). Since \(D\subset E\), the whole path and \(y\) lie in \(E\). Closure gives \(D_z\subset\overline E\).

At a point attaining the minimum, a defining function for the smooth boundary \(\Sigma\) is therefore an exterior support to \(D_z\), with the same outward normal as in Equation (58). The parallel support bound and Equation (55) imply \[\Theta_Q(\Sigma)\leq c+Cz<c',\] contradicting Equation (58). This excludes the entire modification range, so \(\eta=0\) on \(\Sigma\) and proves Equation (54). The contained \(D_{\beta/4}\) gives strict enclosure of \(D\). If desired, fill any remaining complementary components which meet no designated outer face; this deletes smooth boundary components and preserves all stated properties. ◻

Proof. The large-sphere comparison above confines all negative-threshold regions to a fixed compact set, independently of how close the thresholds are to zero. Choose a slightly larger compact set \(\mathcal K\) containing this range and \(\mathop{\mathrm{supp}}K\) in its interior. Let \[m_0=\min_{\mathcal K}\mathfrak m>0, \qquad T_0=\sup_{\mathcal K}|\mathop{\mathrm{tr}}K|.\] First restrict both negative thresholds to an interval so close to zero that \[\max\{|c_b|,|c_w|\}\,T_0<m_0/8,\] while retaining Equation (40). Lemma 14 allows the required successive choices in that interval. Construct the two regions, select \(\Omega\), and fix their disjoint collars inside \(\mathcal K\) as above. All ensuing constants may depend on these fixed choices.

Here is an explicit offset construction. On each collar choose a smooth nonincreasing cutoff \(\chi(s)\), equal to one near \(s=0\) and zero near the collar’s far end. Choose \(\kappa_B>0\) so that \(1-\kappa_Bs>0\) throughout that collar. For a common amplitude \(a>0\), define the contribution of a black collar to be \(a\chi(s)(1-\kappa_Bs)\), and that of a white collar to be \(-a\chi(s)(1-\kappa_Bs)\). Extend by zero outside the disjoint collars and sum. This produces a smooth function on the manifold with boundary \(\Omega\), supported in \(\mathcal K\), with \(C_b=a\), \(C_w=-a\), and \(k_B=a\kappa_B\) on the respective collars. These remain valid bounds when a type is absent. Its \(C^1\) norm tends to zero with \(a\), even though the collar widths have already been fixed.

For \(h\in[b_-,b_+]\) and \(C\in[C_w,C_b]\), \[|h+C|\leq\max\{|c_b|,|c_w|\}+C_b-C_w.\] Choose \(a\) so small that \((C_b-C_w)T_0+\|\nabla C\|_\infty<m_0/8\). The first two terms of Equation (61) are then at most \(m_0/4\) on \(\mathcal K\) and vanish outside \(\mathcal K\). Finally take a fixed smooth positive weight equal to \(r^{-4}\) far out and multiply it by a sufficiently small positive constant. The end lower bound for \(\mathfrak m\) and \(\delta<1\), together with compact positivity, allow the choice \(\rho\leq\mathfrak m/4\) everywhere. This proves Equation (61). All remaining assertions were proved in constructing the regions and collars. ◻

Proof. Introduce an auxiliary coordinate \(s_0\) and the stationary Lorentz metric \[{\bf g}=-l^2(\,\mathrm ds_0-\,\mathrm df)^2+\bar g =-U^2\,\mathrm ds_0^2+g(\,\mathrm dx+\beta\,\mathrm ds_0,\,\mathrm dx+\beta\,\mathrm ds_0), \qquad \beta=l^2\nabla f=Uw.\] Its \(s_0\)-slices have future normal \(N_0=(\partial_{s_0}-\beta)/U\) and second fundamental form \[ k=-\frac{\mathop{\mathrm{sym}}\nabla\beta}{U} =-H^f-p(\,\mathrm dt\otimes w^\flat+w^\flat\otimes\,\mathrm dt). \tag{73}\] Write \(\theta=\mathop{\mathrm{tr}}k\) and, for symmetric tensors, put \[\mathscr B(A,B)=\langle A,B\rangle-(\mathop{\mathrm{tr}}A)(\mathop{\mathrm{tr}}B),\qquad P_A=A-(\mathop{\mathrm{tr}}A)g,\qquad Q=K-k.\] The normal expansion equation and contracted Gauss equation are \[\begin{split} \mathop{\mathrm{Ric}}_{\bf g}(N_0,N_0)&=-N_0(\theta)-|k|^2+U^{-1}\Delta U,\\ \mathop{\mathrm{Scal}}_g&=\mathop{\mathrm{Scal}}_{\bf g}+2\mathop{\mathrm{Ric}}_{\bf g}(N_0,N_0)+|k|^2-\theta^2. \end{split}\] Consequently \(\mathop{\mathrm{Scal}}_{\bf g}=\mathop{\mathrm{Scal}}_g+|k|^2+\theta^2+ 2N_0(\theta)-2U^{-1}\Delta U\). Stationarity gives \(UN_0(\theta)=-\beta(\theta)\), and \(\mathop{\mathrm{div}}\beta=-U\theta\) follows from (73). Thus \[ U\mathop{\mathrm{Scal}}_{\bf g} =U\bigl(\mathop{\mathrm{Scal}}_g+\mathscr B(k,k)\bigr) -2\mathop{\mathrm{div}}(\nabla U+\theta\beta). \tag{74}\] In the coordinates \(s_0-f\) the same metric is a static warped product. Its mixed Christoffel symbols are \(\mathbf{\Gamma}^{s}_{is}=\partial_i\log l\) and \(\mathbf{\Gamma}^{i}_{ss}=l\bar\nabla^il\); contraction gives \(\mathop{\mathrm{Scal}}_{\bf g}=\mathop{\mathrm{Scal}}_{\bar g}-2l^{-1}\bar\Delta l\). Since \(\bar g^{-1}=A_d\) and \(\,\mathrm dV_{\bar g}=(U/l)\,\mathrm dV_g\), \[ \bar\Delta\phi=\frac lU\mathop{\mathrm{div}}\left(\frac Ul A_d\nabla\phi\right). \tag{75}\] Direct differentiation of \(U\) and (73) gives \[\nabla U-\frac Ul A_d\nabla l=-k(\beta,\cdot).\] In fact both sides equal \[\frac{l^4}{U}\mathop{\mathrm{Hess}}f(\nabla f,\cdot) +\frac{l^3}{U}\bigl[|\nabla f|^2\,\mathrm dl +(\nabla f)(l)\,\mathrm df\bigr],\] as follows by differentiating \(U^2=l^2+l^4|\nabla f|^2\). Substitution in (74) yields \[U\mathop{\mathrm{Scal}}_{\bar g} =U\bigl(\mathop{\mathrm{Scal}}_g+\mathscr B(k,k)\bigr)+2\mathop{\mathrm{div}}(P_k\beta).\] The constraints read \(16\pi\mu=\mathop{\mathrm{Scal}}_g-\mathscr B(K,K)\) and \(8\pi J=\mathop{\mathrm{div}}P_K\). Using \(P_K:\nabla\beta=-U\mathscr B(K,k)\), expansion therefore gives \[ U\mathop{\mathrm{Scal}}_{\bar g} =U[16\pi(\mu+J(w))+\mathscr B(Q,Q)]-2\mathop{\mathrm{div}}(P_Q\beta). \tag{76}\] Indeed the derivative of \(P_K\) cancels its term in \(\mathop{\mathrm{div}}(P_K\beta)\), and \(\mathscr B(Q,Q)+2\mathscr B(K,k)=\mathscr B(K,K)+\mathscr B(k,k)\).

The three-dimensional conformal formula is \[\tfrac12e^{4t}\mathop{\mathrm{Scal}}_{\hat g} =\tfrac12\mathop{\mathrm{Scal}}_{\bar g}-4\bar\Delta t-4|\,\mathrm dt|_{A_d}^2.\] For \(Y=P_Qw\), replacing \(U\) by \(u=e^{2t}U\) gives \(-U^{-1}\mathop{\mathrm{div}}(UY)=-u^{-1}\mathop{\mathrm{div}}(uY)+2Y(t)\). Since \(u/(U/l)=L\), Equation (75) similarly gives \[-4\bar\Delta t-4|\,\mathrm dt|_{A_d}^2 =-u^{-1}\mathop{\mathrm{div}}(4uA_d\nabla t)+4(p+1)|\,\mathrm dt|_{A_d}^2.\] Before the flux modification, the scalar-curvature expression is \[ \begin{split} 8\pi(\mu+J(w))+\tfrac12\mathscr B(Q,Q) +2P_Q(w,\nabla t)+4(p+1)|\,\mathrm dt|_{A_d}^2\\ -u^{-1}\mathop{\mathrm{div}}\bigl(u(P_Qw+4A_d\nabla t)\bigr). \end{split} \tag{77}\]

At \(\sigma>0\), \[ 4\,\mathrm dZ=(4+pv)\,\mathrm dt+v\,\mathrm d\log\sigma. \tag{78}\] Write \(x=\partial_e t\) and \(y=\nabla_\perp t\). The blocks of \(Q\) are \[\begin{gathered} Q_{\perp\perp}=T,\qquad Q_{e\perp}=M+pay,\qquad Q_{ee}=q+2pax,\\ \mathop{\mathrm{tr}}Q=F+(1-\chi)q+2pax. \end{gathered}\] Hence \((P_Qw)_\perp=a(M+pay)\) and \((P_Qw)_e=a(\chi q-F)\). Together with \(aH^f(e,\cdot)=v\,\mathrm d\log\sigma\), Equation (78) proves component by component \[ V/u=P_Qw+4A_d\nabla t+wF. \tag{79}\] Also \(uw=e^{2t}l^2\nabla f\), so \[ \mathop{\mathrm{div}}(uw)/u =F+(1-\chi)q-\mathop{\mathrm{tr}}K+2(p+1)ax. \tag{80}\] Replacing the flux in (77) adds \(w(F)+F\mathop{\mathrm{div}}(uw)/u\). The remaining quadratic expression is \[ \begin{split} \mathcal T={}&\tfrac12\mathscr B(Q,Q)+2P_Q(w,\nabla t) +4(p+1)|\,\mathrm dt|_{A_d}^2\\ &+F\left[F+\frac{p+4}{4+pv}S(w,w)+2(p+1)w(t)\right]. \end{split} \tag{81}\] Here \((1-\chi)q=(p+4)S(w,w)/(4+pv)\), which gives a smooth formula at \(w=0\).

For the explicit coefficients, substitute \(|T|^2=|T^{\operatorname{tf}}|^2+(F-\chi q)^2/2\) in \(\mathscr B(Q,Q)/2\). The transverse terms become \((M+pay)\cdot(M+(p+2)ay)\). The coefficients of \(q^2,Fq,F^2\) are respectively \(\chi-\chi^2/4,-\chi/2,3/4\). The \(qx\) coefficient is \(2(p+1)\chi a\), the \(Fx\) terms cancel, and the remaining \(x^2\) term is \(4(p+1)dx^2\). These are the terms in (72). ◻

Proof. With \(x=\partial_e t\) and \(y=\nabla_\perp t\), exact completion gives \[ \begin{split} \mathcal T={}&\tfrac12|T^{\operatorname{tf}}|^2 +|M+(p+1)ay|^2+[4(p+1)-v]|y|^2\\ &+4(p+1)d\left(x+\frac{\chi aq}{4d}\right)^2\\ &+\tfrac34\left(F-\frac{\chi q}{3}\right)^2 +\frac{4d(9-d)}{3(4+pv)^2}q^2. \end{split} \tag{84}\] Completing first in \(x\) leaves \(3\chi q^2/(4+pv)\); completing next in \(F\) leaves \(3\chi/(4+pv)-\chi^2/12=4d(9-d)/[3(4+pv)^2]\). The latter coefficient is at least \(32d/[3(4+n)^2]\). It controls \(dq^2\) polynomially. The \(F\)-square then controls \(F^2\), since \(\chi^2q^2\le dq^2\). The transverse gradient coefficient is at least three, and the shifted \(M\)-square controls \(M\) with an additional factor \((1+n)^2\). The squared \(d\)-norm of the shift in the \(x\)-square is \(\chi^2vq^2/(16d)\le dq^2/16\), so it also controls \(dx^2\). Finally \(|T|^2=|T^{\operatorname{tf}}|^2+(F-\chi q)^2/2\). This proves (82). If \(\,\mathrm dt=0\), the sole cross term is \(-\chi Fq/2\). Using \(|\chi Fq|/2\le\chi q^2/4+\chi F^2/4\) and \(0<\chi\le1\) proves (83). ◻

Proof. The tangential Hessian is \(\mathop{\mathrm{Hess}}f|_{TB}=(\partial_\nu f)\operatorname{II}_g\). Thus \(\mathop{\mathrm{tr}}_\perp H^f=w_\nu H\) and \(\chi q=F-P_B-w_\nu H\). The normal component of (70), using (78), is \[V_\nu/u=4d\partial_\nu t+\chi w_\nu q =4d\partial_\nu t+w_\nu(F-P_B)-vH.\] The \(\bar g\)-normal is \(\sqrt d\,\nu\). Its mean curvature is \(\sqrt d\,H\), because the tangential metric is unchanged and its normal first variation is multiplied by \(\sqrt d\). The conformal mean-curvature formula gives the asserted mean curvature for \(\hat g\). ◻

Proof. Fix a smooth \(j:\mathbb R\to[0,1]\) which vanishes for \(t\le0\) and equals one for \(t\ge1\). For \(0\le\lambda\le1\) put \[V_\lambda=uA_\chi(4\nabla Z+\lambda K(w,\cdot)),\qquad \mathcal B=-H+w_\nu(F-P_B)-N_Bd.\] Before the last stage use the boundary condition \[ \begin{split} (V_\lambda)_\nu/u ={}&[1-(1-\lambda)j(t)]\\ &\cdot[\mathcal B-(1-\lambda)(A_\chi K(w,\cdot))_\nu]. \end{split} \tag{89}\] Outer values remain \(f=Z=0\) throughout.

We specify two smooth collar modifications. Let \(\nu_s=\nabla s/|\nabla s|\) be the outward unit normal to a collar leaf; the leaf coordinate \(s\) here is local and is separate from the homotopy parameter. Choose disjoint collar cutoffs equal to one at the faces. Take \(\eta>0\) so small that \(b_-+3\eta<b_+\). A black contribution to \(D_0(x,w,h)\) is a negative constant times a collar cutoff, times a smooth directional cutoff supported where \(w\cdot\nu_s<-1/2\) and equal to one at \(-1\), times a nonincreasing height cutoff equal to one below \(b_-+\eta\) and zero above \(b_-+2\eta\). A white contribution has positive sign, its directional cutoff supported where \(w\cdot\nu_s>1/2\) and equal to one at \(1\), and a nondecreasing height cutoff equal to zero below \(b_+-2\eta\) and one above \(b_+-\eta\). Decrease \(\eta\) to separate these ranges. Their sum \(D_0\) is smooth, bounded, supported in the collars, and nondecreasing in \(h\). It is nonpositive for \(h\le b_-+\eta\), nonnegative for \(h\ge b_+-\eta\), and vanishes on black outward and white inward directions, for every length \(0\le|w|\le1\).

Choose the magnitudes larger than \(2\sup_B|H|+1\). Equation (63) then gives, for \(F=h+C+D_0\) at the assigned boundary height \(b=b_\pm\), \[ F(x,+\nu,b)\ge P_B+H,\qquad F(x,-\nu,b)\le P_B-H. \tag{90}\] These are limiting directional evaluations at \(a=1,\chi=0\). The compatible direction has equality; \(D_0\) corrects the other one. Its derivatives on bounded height ranges are independent of \(n\). Choose a smooth compactly supported extension \(b(x)\), constant equal to \(b_\pm\) on each smaller collar. On these collars let \(D_1(x,w)=A_1w\cdot\nu_s\), extended by a collar cutoff, with \(A_1>\sup_B|H|+1\). Then \(D_1(x,0)=0\) and \(D_1(x,\pm\nu)=\pm A_1\) on \(B\).

The four stages are the following.

1. For \(0\le s\le1\), set \(\lambda=1-s\), keep \(F=h+C\) and \(h|_B=b\), and use \(\mathop{\mathrm{div}}V_\lambda=u\Xi\) and (89).

2. For \(1\le s\le2\), set \(\lambda=0\), \(\alpha=s-1\), retain \(h|_B=b\), and replace the trace right side by \(F=h+C+\alpha D_0(x,w,h)\). Keep the same scalar equation and boundary prescription.

3. For \(2\le s\le3\), set \(\lambda=0\), \(\zeta=s-2\), prescribe \(h|_B=(1-\zeta)b\), and use \[ \begin{split} F={}&h+(1-\zeta)[C+D_0(x,w,h+\zeta b(x))]\\ &+\zeta[\mathop{\mathrm{tr}}_{A_\chi}K+D_1(x,w)]. \end{split} \tag{91}\] Keep the same scalar equation and boundary prescription. At a face, \(h+\zeta b=b\). The value at \(w=\pm\nu\) is the convex combination of the preceding directional value and \(P_B+D_1(x,\pm\nu)\), so (90) persists.

4. For \(3\le s\le4\), retain the trace problem at \(\zeta=1\) and multiply both \(\Xi\) and the right side of (89) by \(\eta_s=4-s\).

The prescriptions agree at the common endpoints. Their \(h\)-derivatives are \(1\), \(1+\alpha\partial_hD_0\), or \(1+(1-\zeta)\partial_hD_0\), all at least one.

At \(\zeta=1\) the trace equation is \(\mathop{\mathrm{tr}}_{A_\chi}H^f=h+D_1(x,w)\) with zero Dirichlet values. At a positive interior maximum of \(f\), one has \(w=0\), \(A_\chi=I\), \(D_1=0\), and \(l\Delta f=\tau f>0\), a contradiction. A negative minimum is excluded in the same way. Thus \(f=0\) and \(t=Z\) in the last stage. At \(s=4\) the remaining equation is \(\mathop{\mathrm{div}}(4L(Z)\nabla Z)=0\), with zero inner conormal flux and \(Z=0\) on \(S_R\). Multiplication by \(Z\) and integration give \(Z=0\). For the fixed-point construction in Section 6, the last map therefore has output identically zero. ◻

Proof. The collar modifications vanish at \(w=0\). At an interior extremum the trace equation reads either \(\mathop{\mathrm{tr}}K+l\Delta f=h+C\) or \(\mathop{\mathrm{tr}}K+l\Delta f=h+(1-\zeta)C+\zeta\mathop{\mathrm{tr}}K\). The maximum and minimum inequalities bound \(h\) by fixed bounds for \(|\mathop{\mathrm{tr}}K|+|C|\). Boundary values range between \(b_-,0,b_+\). This proves the height bound without using the floor. Boundedness of \(D_0,D_1\) and \(|\mathop{\mathrm{tr}}_{A_\chi}K|\le3|K|\) proves the assertion for \(F\). ◻

Proof. Take \(r_0\) outside the supports of \(K,C,D_0,D_1\). There \(F=h\) in the first three stages, and \(h=0\) in the last one. For \(\psi=r^{-\gamma}\), with axis \(\nabla\psi/|\nabla\psi|\), the controlled end gives, uniformly for \(0<\chi\le1\), \[\mathop{\mathrm{tr}}_{A_\chi}\mathop{\mathrm{Hess}}\psi =\gamma((\gamma+1)\chi-2)r^{-\gamma-2} +O(r^{-\gamma-3})<0\] after enlarging \(r_0\). At a comparison contact, coefficients use the given \(Z\) and the common gradient of test and solution. The positive factor \(l/(\tau\sqrt D)\) preserves this sign. Height monotonicity therefore makes \(C(\psi-R^{-\gamma})\) an upper barrier and its negative a lower barrier. Choose \(C\) to dominate \(C_0\) on \(r=r_0\) for every \(R\ge2r_0\). Differentiating at the common outer value gives (93); tangential derivatives there vanish.

At \(S_R\), \(Z=0\). The monotonicity of its defining expression implies \(t>-\epsilon\) if \(\ell^2|\nabla f|^2<e^{8\epsilon}-1\). It suffices to require \(C^2\ell^2\tau^{-2}R^{-2-2\gamma}<e^{8\epsilon}-1\), which is a permissible lower bound depending on fixed \(n,\epsilon\). ◻

Proof. Consider the graph of \(f\) in the Riemannian product with warped fiber \((\Omega\times\mathbb R,g+l^2\,\mathrm db_0^2)\). Its induced metric is \(\bar g\). At \(\,\mathrm dt=0\) one also has \(\,\mathrm dl=0\), and its second fundamental form, up to orientation, is \(H^f\). Write \(E=l^{-1}\partial_{b_0}\). The graph unit normal is \(\sqrt d\,E-ae\). The ambient curvature formulas at this point are \[\begin{gathered} \mathop{\mathrm{Scal}}_{\mathrm{amb}}=\mathop{\mathrm{Scal}}_g-2\Delta l/l,\\ \mathop{\mathrm{Ric}}_{\mathrm{amb}}|_{T\Omega}=\mathop{\mathrm{Ric}}_g-\mathop{\mathrm{Hess}}l/l,\qquad \mathop{\mathrm{Ric}}_{\mathrm{amb}}(E,E)=-\Delta l/l, \end{gathered}\] with zero mixed Ricci terms. Hence the ambient contribution to half the graph scalar curvature is \[\tfrac12\mathop{\mathrm{Scal}}_g-v\mathop{\mathrm{Ric}}_g(e,e) -v\,\mathop{\mathrm{tr}}_\perp\mathop{\mathrm{Hess}}l/l.\] The last term is nonpositive: at \(\,\mathrm dt=0\), \(\mathop{\mathrm{Hess}}l=lp\,\mathop{\mathrm{Hess}}t\) and \(p\ge0\). The second-form contribution is \([(\mathop{\mathrm{tr}}_{\bar g}H^f)^2-|H^f|_{\bar g}^2]/2\). Since the inverse graph metric is \(A_d\), its expansion is exactly (95). Finally the conformal contribution at this point is \(-4\bar\Delta t\le0\). This proves (94).

For the algebraic estimates we must retain the degenerate normal coefficient. First evaluate \(\mathcal S\) on \(S=K+H^f\), whose trace in the transverse plane is \(F-\chi q\). Subtracting (95) from (72) at \(\,\mathrm dt=0\) gives \[ \begin{split} \mathcal T-\mathcal S(S) ={}&|T^{\operatorname{tf}}|^2+(1+d)|M|^2\\ &+\chi(1+d-\chi/2)q^2-dFq+F^2/2. \end{split} \tag{98}\] The normal coefficient is at least \(\chi\), while \(|dFq|\le\chi q^2/2+d^2F^2/(2\chi)\). Because \[d^2/\chi=d(4+pv)/4\le1+n/4,\] Equation (83) implies \(\mathcal T-\mathcal S(S)\ge c\mathcal T-C(1+n)F^2\) with universal \(c>0\).

To replace \(S\) by \(H^f=S-K\), polarize (95). The difference is bounded by \[C|K|(|T^{\operatorname{tf}}|+|M|+|F|+d|q|)+C|K|^2.\] Indeed the only extra normal product is \(dq\,\mathop{\mathrm{tr}}_\perp K\); all remaining products involve transverse entries, \(dM\), or \(\mathop{\mathrm{tr}}_\perp S=F-\chi q\). By (83) and \(d/\sqrt\chi\le C\sqrt{1+n}\), this bound is at most \(C\sqrt{1+n}|K|\sqrt{\mathcal T}+C|K|^2\). Young’s inequality absorbs a fixed small fraction of \(\mathcal T\) and costs only \(C(1+n)|K|^2\). This proves (96).

If \((1+p)v\le\eta_*\), then \(d\) and \(\chi\) are within \(C\eta_*\) of one. At \(d=\chi=1\) the normal block in (98) is \(3q^2/2-Fq+F^2/2\); its symmetric matrix has smallest eigenvalue \((2-\sqrt2)/2>0\). Choose \(\eta_*>0\) small enough that its coefficients remain within half this spectral gap. The transverse coefficients are already bounded below. For \(K=0\), Equation (83) therefore gives \(\mathcal T-\mathcal S(H^f)\ge c_*\mathcal T\) in this regime. In the remaining regime \(v>\eta_* /(1+p)\), use (96) with \(K=0\) and absorb \((1+n)F^2\) into \(C(1+n)^2vF^2\). This proves (97). All formulas extend to \(\sigma=0\) by their invariant expressions; there \(d=\chi=1\) and the full quadratic expressions are independent of the temporary choice of axis. ◻

Proof. At this contact \(\,\mathrm dt=0\) and \(p=n\) is locally constant as a function of \(t\) near its value \(-\epsilon\). Differentiating \(w\) in an orthonormal frame gives \[ \nabla_iw_j=H^f_{ij}-w_jH^f_{ik}w^k. \tag{100}\] In the \(e\)-frame the normal vector slot is multiplied by \(d\). Equations (83) and \(d/\sqrt\chi\le C\sqrt{1+n}\) therefore show \[ |\nabla w|\le C(1+n)^N(|K|+\sqrt{\mathcal T}). \tag{101}\] No bound for the uncontracted normal Hessian is asserted.

Each first-three-stage right side can be written as a smooth function \(F(x,w,h,p)\). Its \(h\)-derivative is at least one. On the bounded height range supplied by Lemma 24, its explicit \(x\)- and \(w\)-derivatives are polynomially bounded and compactly supported, apart from the derivative of the term \(h\). This follows directly from the fixed collar cutoffs and \(A_\chi=I-(p+4)w\otimes w/(4+pv)\), whose denominator is at least four. For example its first \(w\)-derivatives on \(|w|\le1\) are bounded by \(C(1+n)^2\), and the other explicit modifications have bounds independent of \(n\). Combining these bounds with (101) and Young’s inequality therefore produces a fixed finite power of \(1+n\). Since \(\nabla p=0\) at the contact and \(w(h)=\tau a\sigma\), the chain rule and (101) give, for every fixed \(\eta>0\), \[ w(F)\ge\tau a\sigma-\eta\mathcal T -C_\eta(1+n)^N\mathbf1_{\mathcal K}. \tag{102}\] The dependence \(D_0(x,w,h+\zeta b(x))\) only adds the fixed compact derivative of \(b\) and does not change this conclusion.

The omitted flux is \((1-\lambda)uA_\chi K(w,\cdot)\). Let \(B_K=A_\chi K(w,\cdot)\). Its derivative is bounded by a polynomial constant times \(1+\sqrt{\mathcal T}\) on \(\mathcal K\), by (101); derivatives of \(p\) vanish. The weight term requires the contraction rather than a bound for \(|\nabla\log u|\). At the contact, \[\nabla\log u=H^f(w,\cdot),\qquad \langle\nabla\log u,B_K\rangle =a^2\bigl(\chi H^f_{ee}K_{ee} +H^f_{e\perp}\cdot K_{e\perp}\bigr).\] This involves only \(\chi H^f_{ee}\) and \(H^f_{e\perp}\) and is controlled by (83). It follows that \[ |\mathop{\mathrm{div}}(uB_K)|/u \le\eta\mathcal T+C_\eta(1+n)^N\mathbf1_{\mathcal K}. \tag{103}\]

Combine the scalar identity with (94). The difference \(8\pi\mu-\mathop{\mathrm{Scal}}_g/2\) equals \(((\mathop{\mathrm{tr}}K)^2-|K|^2)/2\) and is compactly supported; so is \(J\). The term \(F\mathop{\mathrm{tr}}K\) is compactly bounded by Lemma 24. On \(\mathcal K\), Equation (96) therefore contributes \(c_*\mathcal T-C(1+n)^N\). Outside \(\mathcal K\), \(K=0\), \(F=h\), and (97) together with Lemma 25 gives \(c_*\mathcal T-C(1+n)^Nv r^{-2\gamma}\). The remaining Ricci term is bounded below by \(-C\mathbf1_{\mathcal K}-Cv r^{-3}\); enlarge \(\mathcal K\) if necessary. Both end weights are bounded by a multiple of \(\rho_0+\rho_1\). Finally subtract the omitted flux and use (102)–(103), choosing their two \(\eta\)’s with sum at most \(c_*/2\). Fix \(0<\delta_0<\min(c_*/4,1/4)\). Decreasing it slightly if necessary gives (99). ◻

Proof. Choose \(R_{\mathrm{floor}}\) to include the requirement of Lemma 25. A floor contact cannot occur on \(S_R\). In the first three stages, an inner contact has \(j(t)=0\). Equation (89) then cancels the omitted flux on both sides and gives \(V_\nu/u=\mathcal B\). Equation (85) forces \(H+4\partial_\nu t=-N_B\), so \(\partial_\nu t<0\). This contradicts the nonnegative derivative into \(\Omega_R\) at a boundary minimum.

Consider an interior contact. Let \(E_n=C(1+n)^N[\mathbf1_{\mathcal K}+v(\rho_0+\rho_1)]\) denote the error in (99). Because \(\rho_0\) has a positive minimum on \(\mathcal K\), \(\mathbf1_{\mathcal K}\le C_{\mathcal K}\rho_0\). Also \(v\rho_0\le\rho_0\), and \(\rho/\rho_0\) is globally bounded. We can therefore choose a polynomial \(C_n\) so large that, pointwise for all \(0\le v\le1\), \[ \mathcal P\ge E_n+2\rho+\rho_0. \tag{104}\] At the floor \(m_0=1\), so the scalar equation and the lower estimate give \[\begin{split} 0 &\ge \delta_0\mathcal T+(1-\delta_0)\tau a\sigma +\mathcal P-E_n-\rho\\ &\ge \rho+\rho_0>0, \end{split}\] a contradiction. Enlarging \(C_n\) below preserves this inequality and preserves its polynomial dependence.

In the last stage \(f=0\), \(t=Z\), \(u=L\), and \(v=0\). At an interior minimum, \(\mathcal T=(|K|^2+(\mathop{\mathrm{tr}}K)^2)/2\). Increase \(C_n\) polynomially if necessary so that \(\delta_0\mathcal T+\rho-C_n\rho_0<0\) at all such points. For \(\eta_s>0\), the scalar equation at a floor minimum has left side \(4L\Delta t\ge0\) and strictly negative right side. At an inner floor minimum its flux is \(4\partial_\nu t=\eta_s(-H-N_B)<0\), also impossible. If \(\eta_s=0\), multiplication of \(\mathop{\mathrm{div}}(4L(t)\nabla t)=0\) by \(t\), with zero outer value and zero inner flux, gives \(t=0\). Thus the last stage has no floor contact either. ◻

Proof. We prove the lower assertion. It is enough to consider an arbitrary sequence \(n_i\to\infty\), \(R_i\to\infty\) and solutions in either of the first two stages. The elementary height and end estimates in Lemmas 24 and 25 give \[ |h_i|\le C_0,\qquad |h_i|\le C(r^{-\gamma}-R_i^{-\gamma})\quad\hbox{on the end}, \qquad 3/4<\gamma<1. \tag{107}\] Near each white face extend \(h_i\) to the other side by the constant \(b_+\). Extend the fixed geometric coefficients smoothly there. The extended functions are continuous. Define the lower relaxed limit \[\underline h(x)=\lim_{j\to\infty} \inf\{h_i(y):i\ge j,\ \operatorname{dist}(x,y)<1/j\}.\] It is lower semicontinuous, bounded, and satisfies the limiting end bound. No equicontinuity of \(h_i\) is asserted.

Choose \(b_-<k'<\min\{0,b_-+\eta\}\), where \(\eta\) is the low-height sign range of the added term \(D_0\). Suppose that a smooth function \(\phi\) touches \(\underline h\) from below at \(x\), with \(\underline h(x)\le k'\) and \(d\phi(x)\ne0\). Exclude for the moment the black faces. Subtracting a small fourth-order localization term from \(\phi\) makes the contact strict without changing its two-jet. Minimization on a small closed ball then produces points \(x_i\to x\) and constants \(c_i\to0\) such that \(\phi+c_i\) touches \(h_i\) from below at \(x_i\), and \(h_i(x_i)\to\underline h(x)\). The boundary of this ball cannot contain \(x_i\) by strictness. Nor can \(x_i\) lie on a white face or on the extended side: their values there are \(b_+>k'\). Thus every sufficiently large \(i\) uses the interior trace equation.

At these contacts, \[\nabla f_i=\tau_i^{-1}\nabla\phi, \qquad d_i\le\frac{\tau_i^2}{\ell_i^2|d\phi(x_i)|^2} \longrightarrow0, \qquad a_i\longrightarrow1, \qquad \chi_i\longrightarrow0.\] Here \(a_i=l_i|\nabla f_i|/\sqrt{D_i}\) is the deformation variable. Positivity of \(A_{\chi_i}\) and the Hessian inequality at a lower contact imply \[\mathop{\mathrm{tr}}_{A_{\chi_i}}K+ \frac{l_i}{\tau_i\sqrt{D_i}} \mathop{\mathrm{tr}}_{A_{\chi_i}}\mathop{\mathrm{Hess}}\phi \le F_i(x_i).\] In the second stage \(D_0\le0\) at these values; in the first stage it is absent. Consequently \(\limsup F_i(x_i)\le k'+C_b\). Passing to the limit gives \[ \frac{\mathop{\mathrm{tr}}_{(d\phi)^\perp}\mathop{\mathrm{Hess}}\phi}{|d\phi|} +\mathop{\mathrm{tr}}_{(d\phi)^\perp}K\le k'+C_b. \tag{108}\] This is the expansion of the level surface of \(\phi\), oriented toward increasing \(\phi\). The only Hessian used in this passage is the fixed test Hessian. Removing the localization proves the assertion for non-strict contacts as well.

We now transfer the test inequality to a closed sublevel set. Fix \(b_-<k<k'\) and put \(E_k=\{\underline h\le k\}\). Set \[ \psi_j(s)=\frac{1}{1+\exp[-j^2(s-k-j^{-1})]}, \qquad v_j=\psi_j(\underline h). \tag{109}\] The lower relaxed limit of \(v_j\) is the function \(v\) which equals zero on \(E_k\) and one on its complement. Indeed, at a point of \(E_k\) the fixed-point values \(\psi_j(\underline h(x))\) tend to zero. If \(\underline h(x)>k\), lower semicontinuity supplies a neighborhood where \(\underline h>k+\delta\) for some \(\delta>0\), and \(v_j\) tends uniformly to one there.

Let \(\varphi\) be a nonzero-gradient lower test of \(v\) at a zero value. The same strict-contact localization gives lower tests \(\varphi+c_j\) of \(v_j\) at \(y_j\to x\), with \(v_j(y_j)\to0\). Since \(\psi_j(k')\to1\), one has \(\underline h(y_j)<k'\) eventually. Locally about \(y_j\), the test value is in \((0,1)\), so \(\psi_j^{-1}(\varphi+c_j)\) is a smooth lower test of \(\underline h\). An increasing change of defining function does not change its oriented level expansion: if \(\theta'>0\), then \[\mathop{\mathrm{Hess}}(\theta\circ\varphi) =\theta'\mathop{\mathrm{Hess}}\varphi+\theta''d\varphi\otimes d\varphi, \qquad \frac{\mathop{\mathrm{tr}}_{(d\varphi)^\perp}\mathop{\mathrm{Hess}}(\theta\circ\varphi)} {|d(\theta\circ\varphi)|} =\frac{\mathop{\mathrm{tr}}_{(d\varphi)^\perp}\mathop{\mathrm{Hess}}\varphi}{|d\varphi|}.\] Equation (108) therefore holds for the test \(\varphi+c_j\) and passes to \(\varphi\). A smooth exterior support of \(E_k\), with defining function negative on the containing inner side, is a lower test of \(v\): shrink its neighborhood so that its positive values are less than one. Thus every such support has expansion at most \(k'+C_b\).

This argument also rules out a hidden layer at a white face. The reversed white face is an exterior support of \(E_k\cap\bar\Omega\) at any contact there, with expansion \[-H_B+P_B=-c_w>0.\] For \(k'\) sufficiently close to \(b_-\), this is strictly greater than \(k'+C_b\). The high extension ensured that all preceding low-value tests came from the interior equation, so the support contradiction applies at the face itself. Hence \(E_k\) avoids every white face. The end estimate makes \(E_k\) compact.

Adjoin the entire closed region \(\mathcal B_{c_b}\) and fill bounded complementary pockets. At a point on its smooth black frontier, every exterior support of the union also contains the black region locally. The tangency comparison therefore bounds its expansion by \(c_b<k'+C_b\). At all other frontier points the preceding sublevel argument applies. Filling pockets only removes frontier pieces and preserves this property. The resulting compact closed filled set contains the required original inner barrier with a collar, because the black region already does. It avoids the outer barriers. Lemma 18 places it in a smooth black trapped region at any threshold \(c''>k'+C_b\) sufficiently close to \(c_b\). It is consequently contained in \(\mathcal B_{c''}\).

Finally let \(Q\) be as in the lemma. Hausdorff right continuity at \(c_b\) gives a \(\delta>0\) such that \(\mathcal B_{c_b+\delta}\cap Q=\varnothing\); decrease \(\delta\) so that the enlarged threshold is still negative and all barriers remain strict. Choose \(k,k',c''\) with \[b_-<k<k',\qquad k'+C_b<c''<c_b+\delta.\] The construction implies \(E_k\cap Q=\varnothing\) for every sequence under consideration. If the uniform estimate \(h>k\) on \(Q\) failed, choose violating solutions with \(n_i\to\infty\), \(R_i\to\infty\) and \(x_i\in Q\), \(h_i(x_i)\le k\). A subsequence has \(x_i\to x\in Q\) and \(\underline h(x)\le k\), a contradiction. This gives (106), after decreasing its positive margin.

For the upper assertion apply the entire argument to \((-h,-K,-C)\), interchanging colors. The reversed black faces have strictly positive white expansion \(-c_b\). Their exclusion and compactness give positive distance from the black region before applying the white version of Lemma 18. Adjoining the full white region is therefore legitimate in the fixed black complement. Empty white regions and missing face types cause no change: only an actually nonempty sublevel would supply a trapped seed and hence a contradiction. ◻

Proof. Write \(s\) for the smooth leaf parameter in an outward collar and \(N=\nabla s/|ds|\). All geometric coefficients of this fixed collar are bounded, with \(|ds|\) bounded above and below. At a comparison contact with \(h_0=b_-+\psi(s)\), \(\psi'>0\), the shared gradient gives \[ d\le C\frac{\tau^2}{\ell^2(\psi')^2},\qquad 1-a\le d,\qquad n\chi\ge c_1d\quad(n\ge4). \tag{111}\] If the slopes are in a fixed positive interval, the first bound is \(d\le C\ell\). None of these comparisons uses an upper bound for \(Z\).

The trace operator on this test is \[\begin{align*} &\mathop{\mathrm{tr}}_{A_\chi}\left(K+ \frac{l}{\tau\sqrt D}\mathop{\mathrm{Hess}}h_0\right)\\ &\quad=P_s+aH_s+\chi K(N,N) +a\chi\left( \frac{\mathop{\mathrm{Hess}}s(N,N)}{|ds|} +|ds|\frac{\psi''}{\psi'}\right), \tag{112}\end{align*}\] where \(P_s\) is the tangential trace of \(K\) on the leaf. Thus it is the leaf expansion \(H_s+P_s\), with an error of absolute value at most \(C d\), plus the displayed logarithmic-slope term. The same formula for a negative slope has \(a\) replaced by \(-a\) in the Hessian terms. This is an evaluation at the contact variables; it does not differentiate the unknown \(t\).

On a black collar \(H_s+P_s\ge c_b\) and \(C=C_b-k_Bs\). Choose small positive slopes so that \(\psi(0)=0\) and \(\psi(s)\le k_Bs/2\). At the outer end of a sufficiently short fixed collar, Lemma 29 lets this test lie below the solution. In the compatible direction the add-on \(D_0\) vanishes. The residual of the trace equation on the test is therefore at least \[ k_Bs/2-Cd+a\chi|ds|\frac{\psi''}{\psi'}. \tag{113}\] Choose a smooth nonnegative \(\beta_n(s)\) equal to \(An\) for \(0\le s\le1/n\), supported in \(0\le s\le2/n\), and satisfying \(\int\beta_n\le2A\). Define \[\psi'(s)=m\exp\left(\int_0^s\beta_n(q)\,\mathrm dq\right).\] Choose the fixed \(A\) large enough that the final term of (113) dominates \(Cd\) on \(s\le1/n\), using (111) and \(a\ge1/2\) there. On \(s\ge1/n\), the first term dominates \(Cd\le C\ell\) for large \(n\); the taper contributes with the good sign. Since the slope multiplier is at most \(e^{2A}\), a fixed sufficiently small \(m>0\) enforces all the previous small-slope requirements. This constructs a strict lower barrier with slopes bounded above and below independently of \(n\).

An upper barrier from \(b_-\) is constructed in the same collar with \[\psi'(s)=M\exp\left(-\int_0^s\beta_n(q)\,\mathrm dq\right).\] The smooth leaf expansion differs from its boundary value by \(O(s)\). Choose the fixed \(M\) so large that \(\psi(s)\ge C_2s\) dominates this variation, the offset variation, and the height bound at the outer collar end. The logarithmic-slope term is now negative. On \(s\le1/n\) it dominates \(Cd\); on the rest of the collar the negative margin proportional to \(s\) does so. The upper barrier is strict at every possible positive contact. Its slopes lie in \([Me^{-2A},M]\), an interval independent of \(n\).

To check the comparison signs, at a negative interior minimum of \(h-h_0\) the Hessian of \(h\) dominates that of \(h_0\), the gradients and \(Z\) agree, and hence so do \(t,l,A_\chi\). The right side is strictly increasing in the height. Thus a strict positive residual for a lower barrier contradicts the equation. At a positive maximum the inequalities reverse, proving the upper-barrier assertion. Taking the one-sided derivative at \(s=0\) proves the black part of (110). Replacing \((h,K,C)\) by \((-h,-K,-C)\) proves the white part.

In the third stage the boundary height is the assigned interpolated constant, say \(b_\zeta\). Construct tests \(b_\zeta+\psi(s)\) and \(b_\zeta-\psi(s)\) with large positive \(\psi'\) and \((\log\psi')'=-\beta_n\). At \(s=0\), the two directional inequalities of Proposition 23 say precisely that the positive-slope test has nonpositive limiting residual and the negative-slope test nonnegative limiting residual. At every comparison contact, the fixed positive barrier slope and Equation (111) give \(d\le C\ell\), so \(a\ge1/2\) for large \(n\). On \(1/2\le|w|\le1\), the first \(w\)-derivatives of \(A_\chi\) in Equation (67) are bounded independently of \(0\le p\le n\). The other collar terms have fixed smooth coefficients on the bounded height range. Thus, at fixed boundary height the coefficient and geometric errors are \(O(s+d+\chi)\), since \(|w-(\pm N)|=1-a\le d\) and the radial comparison segment stays in this range. Changing the test height supplies, by height monotonicity, a residual of magnitude at least \(\psi(s)\) with the required sign. Large fixed slopes dominate the \(O(s)\) error and the outer-end height bound. The logarithmic-slope term has the required sign for both tests, as follows from (112); it dominates the \(O(d)\) error on the initial \(1/n\) interval. The remaining margin dominates it beyond that interval. This proves the two-sided upper derivative bound, without using sublevel exclusion in the third stage.

Finally, on each own-color collar the lower barrier just constructed gives the required one-sided height bound. On the rest of a fixed compact region use Lemma 29; the opposite height bound on an own-color collar follows from that lemma as well, since the closed collar is disjoint from the other region. A finite cover proves \(b_-\le h\le b_+\) on the compact regions in the statement. ◻

Proof. For \(\sigma>0\), write \(e_f=\nabla f/\sigma\), the axis denoted by \(e\) in Section 4. Set \[W=l\bar\nabla f=\frac{l\nabla f}{D}=\sqrt d\,a e_f.\] Its norm in \(\bar g\) is \(a\le1\). For any covector \(\xi\), \[ |\xi(W)|\le |\xi|_{A_d} \le\sqrt{1+n/4}\,|\xi|_{A_\chi}. \tag{116}\] Direct differentiation, using \(uW=Lw\), gives \[\begin{align*} \frac{\mathop{\mathrm{div}}(uW)}u &=\sqrt d\left[ \mathop{\mathrm{tr}}_{A_d}H^f+(dp+p_L)a\partial_{e_f}t\right], \tag{117}\\ \frac{\mathop{\mathrm{div}}(\sqrt D\,W)}{\sqrt D} &=\sqrt d\left[ \mathop{\mathrm{tr}}_{A_d}H^f+dp\,a\partial_{e_f}t\right]. \tag{118}\end{align*}\] For example, \(\mathop{\mathrm{div}}w=\mathop{\mathrm{tr}}_{A_d}H^f+dp\,a\partial_{e_f}t\), while differentiating \(L\) adds \(p_La\partial_{e_f}t\) to the first formula. Coercivity in Lemma 21 bounds both right sides by \(\Pi_n(1+\sqrt{\mathcal T})\). More explicitly, writing \(q=(K+H^f)_{e_fe_f}\), the normal Hessian term has coefficient \(d^{3/2}\), so it is bounded by \(\sqrt d|q|+C\). The other Hessian terms are transverse traces, and the gradient term is at most \((2n+2)\sqrt d|\partial_{e_f}t|\). All are controlled by the stated coercivity estimate. This also establishes the formulas and bounds at zero gradient by their smooth direction-free extensions.

Let \(\varepsilon_B=1\) on a black collar and \(-1\) on a white collar. The signed slopes give \(w_\nu=\varepsilon_B a\) on \(B\) and \(a\ge1/2\) for large \(n\). Thus \[uW_\nu=\varepsilon_B La, \qquad \sqrt D\,W_\nu=\varepsilon_B a.\] The outward integration normal is \(-\nu\). Apply the divergence theorem to \(-\varepsilon_B uW\varphi\), with a fixed collar cutoff. Its boundary flux is \(La\varphi\). Equations (116) and (117), and the fixed cutoff derivative, give (114). Replace \(u\) by \(\sqrt D\) and use (118) for (115). No factor \(\ell^{-1}\) was used. ◻

Proof. Extend the fixed base patch smoothly and embed a compact enlargement isometrically in Euclidean space by Nash’s embedding theorem (Nash 1956, Theorem 2). Its product with the vertical line gives an isometric Euclidean immersion of the graph. The second fundamental form of the base embedding is bounded, so its contribution to the graph mean curvature vector is bounded. The scalar mean curvature in the product is \[H_\Gamma= \frac{\sqrt D}{l\sqrt{1+\sigma^2}} \left(\mathop{\mathrm{tr}}_{e_f^\perp}H^f+d_0H^f(e_f,e_f)\right).\] The prefactor is bounded by \(C(n)\), and \(d_0/\sqrt d=\sqrt{1+l^2\sigma^2}/(1+\sigma^2)\le e\). Coercivity therefore gives \(|\boldsymbol H_\Gamma|\le C(n)(1+\sqrt{\mathcal T})\). This estimate involves controlled quadratic energy, not a pointwise bound for the Hessian.

The ordinary Michael–Simon inequality (Michael and Simon 1973), in the compact-support form stated in (Simon 2014, chap. 4, Section 5, Theorem 5.7), gives after the boundary extension described below, for nonnegative \(\varphi\), \[\left(\int_\Gamma\varphi^{3/2}\,\mathrm d\Gamma\right)^{2/3} \le C\left[ \int_\Gamma(|\nabla_\Gamma\varphi| +|\boldsymbol H_\Gamma|\varphi)\,\mathrm d\Gamma +\int_{\partial\Gamma}\varphi\,\mathrm dA\right].\] One can obtain the boundary form directly from the compact-support version by smoothly extending each smooth graph across its boundary and cutting off on a collar tending to zero: the cutoff derivative integral converges to the boundary integral, and the added curvature integral tends to zero. This argument is applied to each smooth graph before the uniform estimate is taken. The only boundary in the support is an inner face, where \(f\) is constant and \(\,\mathrm dA=\,\mathrm dA_g\). Equation (115) and the fixed-\(n\) graph comparisons bound its contribution by \[C(n)\int_\Gamma [(1+\sqrt{\mathcal T})\varphi +|\nabla_\Gamma\varphi|]\,\mathrm d\Gamma.\] Apply the resulting inequality to \(\varphi=|\omega|^4\). Cauchy–Schwarz bounds its right side by \[C(n)\|\omega\|_6^3 \left(\int_\Gamma [|\nabla_\Gamma\omega|^2+(1+\mathcal T)\omega^2] \,\mathrm d\Gamma\right)^{1/2}.\] The left side is \(\|\omega\|_6^4\); division, with the zero case separate, and squaring prove the assertion. ◻

Proof. It remains to extract the bounds for the conformal variable and the gradient. Since \(Z=t+\log D/8\) and \(D\ge1\), the floor gives \(-\epsilon\le t\le Z\). The upper bound for \(Z\) gives \[\sigma^2=\frac{e^{8(Z-t)}-1}{l^2} \le\ell^{-2}\bigl(e^{8(C(n)+\epsilon)}-1\bigr).\] This proves the asserted fixed-\(n\) gradient bound. The height bounds are Lemma 24. The finite collection of sublevel and collar estimates and the polynomial absorption choose \(n_0\) once. Enlarge \(R_{\min}(n)\) to include the outer estimates and all fixed compact patches used above. The next section derives Hessian and continuity estimates from these bounds and uses them to prove existence. ◻

Proof. The proof combines a Cordes estimate, a divergence identity for the gradient, and improvement of affine approximations. We first establish these estimates, then handle boundary reflection and the iteration.

Proof. Write \(\mathcal L=-\mathop{\mathrm{div}}(aD)\) and, for \(s\in\{-1,1\}\), set \(V_s=e^{skZ}\). The weak chain rule gives \[ \mathcal L V_s =\mathop{\mathrm{div}}(skV_sb)-skV_se -k^2V_s\bigl(aDZ\cdot DZ+b\cdot DZ\bigr). \tag{162}\] Choose \(k\ge\max\{1,2C_0/\lambda\}\) and use \(B|DZ|\le(\lambda/2)|DZ|^2+B^2/(2\lambda)\). Since \(e^{-kM}\le V_s\le e^{kM}\), we obtain \[ \mathcal L V_s\le\mathop{\mathrm{div}}F_s+\nu, \qquad |F_s|\le ke^{kM}B, \qquad \nu\le C\,\mathrm dx+C\mu. \tag{163}\]

On a ball \(B_R\) compactly contained in the patch, solve \(\mathcal L w_s=\mathop{\mathrm{div}}F_s+\nu\) with zero Dirichlet data. The bounded vector source has a solution of supremum norm at most \(CR\): apply the \(L^p\) source estimate, \(p>3\), whose scaled factor is \(R^{1-3/p}\), to \(\|F_s\|_p\le CR^{3/p}\). For the positive measure source use the Green bound \(0\le\mathcal G(x,y)\le C|x-y|^{-1}\) for uniformly elliptic measurable divergence coefficients (Littman et al. 1963, sec. 7, Theorem (7.1) and Remark 2). To use that interior comparison uniformly for all \(x,y\in B_R\), extend the coefficients elliptically to \(B_{2R}\). Domain monotonicity bounds the zero-Dirichlet Green function on \(B_R\) by the one on \(B_{2R}\), where \(B_R\) is a fixed compact interior region after rescaling. This gives the stated uniform bound. Integration over concentric annuli gives \[ \sup_{x\in B_R}\int_{B_R}\mathcal G(x,y)\,\mathrm d\nu(y) \le C\int_0^{2R}\frac{\nu(B_t(x))}{t^2}\,\mathrm dt +CR^{-1}\nu(B_{2R}(x)) \le C(R^2+R^\beta). \tag{164}\] One may first approximate the measure by smooth positive densities. The uniform potential bound and the energy identity \(\int aDw\cdot Dw=\int w\,\mathrm d\nu\) give a bounded sequence in \(W^{1,2}_0\) and hence the required weak solution. Thus, with \(\gamma=\min\{1,\beta\}>0\), \[ \|w_s\|_\infty\le\varepsilon_R, \qquad \varepsilon_R=CR^\gamma. \tag{165}\]

Let \(M_R=\sup_{B_R}Z\) and \(m_R=\inf_{B_R}Z\). The functions \[ H_+=e^{kM_R}-e^{kZ}+w_++\varepsilon_R, \qquad H_-=e^{-km_R}-e^{-kZ}+w_-+\varepsilon_R \tag{166}\] are nonnegative and satisfy \(\mathcal L H_\pm\ge0\). Weak Harnack controls their \(L^q\) means on \(B_{R/2}\) by their infima on \(B_{R/4}\). Each exponential deficit is comparable, with constants depending on \(kM\), to \(M_R-Z\) or \(Z-m_R\), respectively. At least one of the sets \[\{Z\le(M_R+m_R)/2\}\cap B_{R/2},\qquad \{Z\ge(M_R+m_R)/2\}\cap B_{R/2}\] has at least half the measure of \(B_{R/2}\). Apply weak Harnack to the corresponding deficit. It follows that either the upper supremum decreases, or the lower infimum increases, by at least \(c(M_R-m_R)-C\varepsilon_R\) on \(B_{R/4}\). Consequently \[ \operatorname{osc}_{B_{R/4}}Z \le(1-c)\operatorname{osc}_{B_R}Z+CR^\gamma. \tag{167}\] Iteration proves a Hölder estimate with any sufficiently small positive exponent below both \(\gamma\) and \(-\log(1-c)/\log4\).

For the boundary estimate, flatten a face and put \(S=\operatorname{diag}(1,1,-1)\). An even extension of \(Z\) uses \(a_-=Sa_+S\) and \(b_-=Sb_+\). Its flux has a plane jump whose density is twice the prescribed normal flux, hence bounded. An odd extension after subtracting a constant Dirichlet value uses \(a_-=Sa_+S\), \(b_-=-Sb_+\), and the sign-reflected bulk source. Its normal flux is continuous. The transformed metrics and volume densities are absorbed into the displayed coefficients. A bounded plane density has ball mass \(O(r^2)\), so it is admissible with exponent one. This proves the asserted boundary version. ◻

Proof. Start with the bounds for \(f,Z,t,\sigma\) from Proposition 33. At fixed parameters they bound \(u,l,D\) above and away from zero, and give a uniform \(\chi_0>0\). After dividing the scalar trace equation by \(l/\sqrt D\), its right side is \[ G_f=\frac{\sqrt D}{l} \bigl(F_s-\mathop{\mathrm{tr}}_{A_\chi}K\bigr). \tag{168}\] It is bounded throughout the homotopy, including its added collar terms. Lemma 34 therefore gives a uniform \(C^{1,\alpha}\) estimate for \(f\) and \[ \int_{B_r}|\mathop{\mathrm{Hess}}f|^2\le Cr^{1+2\alpha} \tag{169}\] on the uniform patches, including boundary patches where \(f\) is constant.

After multiplying by the Riemannian volume density, the second equation has coordinate principal coefficients \(a^{ij}=4\sqrt{\det g}\,uA_\chi^{ij}\), a bounded flux vector \(b\), and a source bounded in absolute value by \[ C\bigl(1+|DZ|^2+|D^2f|^2\bigr). \tag{170}\] Indeed \(t\) is a smooth function of \(x,Z,Df\) on the bounded variable range. Thus \(Dt\) is a bounded linear combination of \(DZ,D^2f\) and fixed smooth terms; the expression \(\mathcal T\) is quadratic in these quantities. All other source terms have lower order. The same volume density is included in \(b,e\); it is smooth and bounded above and away from zero on the chosen patches, so it preserves these bounds and uniform ellipticity. The prescribed conormal flux at the inner boundary is bounded. At the outer face \(Z=0\). Apply Lemma 35, with \(\mu=C|D^2f|^2\,\mathrm dx\) and the bounded reflected plane density. Equation (169) supplies its measure growth hypothesis. We obtain a uniform positive Hölder exponent for \(Z\).

Now \(Z\) and \(Df\) are Hölder, so all coefficients and the right side of the scalar \(f\)-equation are Hölder on their bounded variable range. Interior and constant-Dirichlet boundary Schauder estimates give \(f\in C^{2,\theta}\) for a uniform \(\theta>0\). Expand the \(Z\)-equation in nondivergence form. Its principal coefficients are uniformly elliptic and \(C^{0,\theta}\), while its right side is bounded by \(C(1+|DZ|^2)\). On an inner face \(Df\) is normal. Therefore \(A_\chi\) has zero normal–tangential components there and normal eigenvalue \(\chi\). Its boundary equation solves for \[ \partial_\nu Z=\mathcal H_s(x,Z,f,Df), \tag{171}\] where \(\mathcal H_s\) is smooth on a fixed bounded variable range. Its derivatives with respect to \(x\) and its displayed scalar arguments are bounded at fixed parameters. The outer condition remains constant Dirichlet. These are different, disjoint smooth boundary components; there is no change of boundary condition along an edge or a corner.

It remains to absorb the quadratic gradient growth. Fix \(p>3\). The linear local \(W^{2,p}\) estimate for the leading nondivergence operator with Equation (171) or outer Dirichlet data has the form, on fixed nested patches \(U_x\Subset V_x\), \[ \|Z\|_{W^{2,p}(U_x)} \le C\left(1+\|DZ\|_{L^{2p}(V_x)}^2 +\|Z\|_{W^{1,p}(V_x)}\right). \tag{172}\] Here the boundary norm is controlled by \[ \|\mathcal H_s(x,Z,f,Df)\|_{W^{1-1/p,p}(\partial V_x)} \le C\bigl(1+\|Z\|_{W^{1,p}(V_x^+)}\bigr), \tag{173}\] using trace and composition after extending its smooth expression through the collar. The derivatives of \(f,Df\) entering this extension are already bounded. The leading matrix has a fixed Hölder modulus, so its small oscillation on sufficiently small patches permits the usual freezing and absorption in the linear estimate. The normal boundary operator is uniformly oblique and satisfies the complementing condition. These are the classical local elliptic boundary estimates (Agmon et al. 1959, 1964).

Let \([Z]_{C^{0,\theta}}\le H_0\). On balls or half-balls of radius \(h\), the scaled Gagliardo–Nirenberg inequality, applied after subtracting a constant, gives \[ \begin{split} \|DZ\|_{L^{2p}(B_h)}^2 &\le C\operatorname{osc}_{B_h}Z\, \|D^2Z\|_{L^p(B_h)}\\ &\quad+Ch^{3/p-2}(\operatorname{osc}_{B_h}Z)^2. \end{split} \tag{174}\] One can use a bounded extension operator on a half-ball; its constants are uniform in the boundary charts. This is the interpolation inequality of (Nirenberg 1959, Lecture II, Equation (2.2)) with its lower-order term on a bounded patch. Cover \(V_x\) by radius-\(h\) patches of bounded overlap and take the \(\ell^p\) sum of Equation (174). It follows that \[ \|DZ\|_{L^{2p}(V_x)}^2 \le CH_0h^\theta\|D^2Z\|_{L^p(V_x^+)} +CH_0^2h^{2\theta-2}. \tag{175}\] Also, for any \(\eta>0\), \(\|Z\|_{W^{1,p}(V_x)} \le\eta\|D^2Z\|_{L^p(V_x^+)}+C_\eta\). Each enlarged patch \(V_x^+\) is covered by a bounded number of the patches \(U_y\), independently of \(R\). Set \(S=\sup_y\|Z\|_{W^{2,p}(U_y)}\), finite for each smooth finite-domain solution. Equations (172) and (175) give \[ S\le C(H_0h^\theta+\eta)S+C(h,\eta). \tag{176}\] Choose \(h\) and then \(\eta\) so the first coefficient is less than \(1/2\). We obtain a uniform \(W^{2,p}\) bound and hence a \(C^{1,1-3/p}\) bound for \(Z\).

The coefficients in both equations and in the normal boundary condition now have the regularity required for Schauder bootstrap. First the nondivergence \(Z\)-equation has Hölder source, giving \(Z\in C^{2,\theta'}\); the scalar \(f\)-equation then gains another derivative. Repetition gives every finite classical derivative supported by the smooth data. The same procedure holds on the uniform unit patches of the asymptotic end and on the large outer spheres, whose normal-coordinate constants are uniform. This proves the asserted independence of \(R\). The reduced background has end symbol bounds of every order by Proposition 12; the bootstrap here does not assume additional derivatives of the original asymptotic data. ◻

Proof. The implicit definition of \(t\) has derivative \(\partial Z/\partial t=1+pv/4>0\). Thus \(t\) and all the coefficients are smooth functions of \(x,Z,Df\), also at \(Df=0\). There is no dependence of the principal coefficient on the height \(f\) itself. For bounded height the added collar terms in \(F_s\) are bounded, and its derivative with respect to \(h=\tau f\) is at least one. At \(Df=0\) the equation and this height sign give upper and lower constant barriers. Hence every scalar solution satisfies \[ |h|\le C_0, \tag{178}\] where the constant can be chosen independently of \(Z\) and the homotopy parameter. We include the fixed boundary values in its choice.

We first derive the coefficient bounds at an unbounded slope. If \(|Z|\le M\) and \(\sigma\to\infty\), the implicit equation forces \(t\to-\infty\) uniformly in \(Z\). There \(p=n\) and \(l=e^{nt}\), so its exact form is \[ e^{8Z}=e^{8t}+e^{(2n+8)t}\sigma^2. \tag{179}\] Consequently \[ \begin{split} t&=-\frac{\log\sigma}{n+4}+O(1),\qquad l\asymp\sigma^{-n/(n+4)},\\ d&\asymp\sigma^{-8/(n+4)},\qquad \chi\asymp\sigma^{-8/(n+4)},\qquad \frac{\sqrt D}{l}\asymp\sigma. \end{split} \tag{180}\] The constants are uniform for \(|Z|\le M\) at fixed \(n\). Since \(n\ge4\), the exponent \(8/(n+4)\) is at most one. Combining the large-slope estimates with compactness of the remaining variable range yields \[ \chi\ge\frac{c}{1+\sigma},\qquad |G_f|\le C(1+\sigma) \tag{181}\] whenever Equation (178) holds.

We prove a global gradient estimate using this radial ellipticity. Let \(\delta\) be smaller than a fixed collar radius, half the distance between distinct boundary components, and a normal injectivity radius of a smooth extension of the finite domain. On a boundary distance collar use \[ \psi(r)=k^{-1}\log(1+Br),\qquad \psi''=-k(\psi')^2. \tag{182}\] The upper and lower barriers are the assigned face value plus or minus \(\psi\). On these tests the axis is normal to the distance leaves, so their principal trace is \(\chi\psi''+\psi'\Delta r\). At slopes at least one, Equation (181) gives \[ -k\chi(\psi')^2\le-\tfrac12ck\psi'. \tag{183}\] Choose \(k\) large enough to dominate all the bounded geometry terms and the linear growth in Equation (181). Next shrink \(\delta\) so that \(1/(2k\delta)\) exceeds any fixed large-slope threshold used here. Finally choose \(B\) large enough that \(B\delta\ge1\) and \(\psi(\delta)\) exceeds twice the height oscillation. Then \(\psi'\ge1/(2k\delta)\) on this collar and the two barriers compare by the height monotonicity. This gives a uniform boundary gradient bound.

For the interior estimate, increase \(B\) if necessary and compare two points using \[ f(x)-f(y)-\psi(\operatorname{dist}(x,y)), \qquad 0<\operatorname{dist}(x,y)<\delta. \tag{184}\] The distance is taken in the fixed smooth extension. The boundary barriers exclude a positive maximum with one endpoint on the boundary. The inequality \(\psi(\delta)>2\operatorname{osc}f\) excludes the other boundary of the two-point domain. At a positive interior maximum let \(r=\operatorname{dist}(x,y)\) and \(q=\psi'(r)\). The two gradients have length \(q\) and are parallel transports along the connecting geodesic. Vary both endpoints simultaneously in parallel transverse directions. The second variation formula for the short geodesic yields \[ \mathop{\mathrm{tr}}_{e_x^\perp}\mathop{\mathrm{Hess}}f(x) -\mathop{\mathrm{tr}}_{e_y^\perp}\mathop{\mathrm{Hess}}f(y)\le Cqr. \tag{185}\] Varying each endpoint separately along that geodesic gives \(f_{;e_xe_x}(x)\le\psi''(r)\) and \(f_{;e_ye_y}(y)\ge-\psi''(r)\). Subtracting the two scalar equations therefore gives \[ -2C(1+q) \le G_f(x)-G_f(y) \le Cqr+(\chi_x+\chi_y)\psi''(r). \tag{186}\] By Equations (181) and (182), the last term is at most \(-ckq\) at the large comparison slopes. Our choice of \(k\) gives a contradiction. Thus Equation (184) is nonpositive. Reversing \(x,y\) proves the same bound for the absolute difference and, on letting \(y\to x\), a global Lipschitz bound for \(f\). This argument used no continuity modulus for \(\chi_x-\chi_y\); only the common transverse eigenvalues and the radial lower bound were needed.

For existence interpolate the normalized equation with \(\Delta f=\tau f\), retaining the given boundary values. Explicitly, for \(a\in[0,1]\) take principal matrix \((1-a)I+aA_\chi\) and right side \((1-a)\tau f+aG_f\). The height sign, the linear gradient growth, and Equation (181) persist uniformly in \(a\). The height and logarithmic gradient bounds just proved are therefore uniform. After these bounds the interpolating equations are uniformly elliptic. Their matrices still have the axial form, with axial eigenvalue \((1-a)+a\chi\), so Lemma 34 applies. Schauder estimates then give a uniform \(C^{2,\theta}\) bound at fixed input \(Z\). The coefficients and right side are now Lipschitz in position on bounded input sets, which permits their \(C^{0,b}\) estimates and an upgrade to \(C^{2,b}\). Differentiating once, or using the next Schauder estimate, yields \(f\in C^{3,b}\) because \(Z\in C^{1,b}\).

The scalar linearization has the form \(a^{ij}D_{ij}\varphi+b^iD_i\varphi-c\varphi\) with \(c\ge c_1\tau>0\) on the bounded variable range. Indeed the principal matrix is independent of the height, while \(\partial_fG_f\ge\tau\sqrt D/l>0\). The Dirichlet maximum principle gives uniqueness and a zero kernel; the linear elliptic Dirichlet theory gives an inverse. The implicit-function theorem gives openness of the interpolation parameter set. The uniform classical bounds and compactness give closedness, starting from the invertible Laplace endpoint. This proves existence. The same maximum principle applied to the difference of two solutions proves uniqueness of the nonlinear solution. The implicit-function theorem, or compactness and uniqueness, gives continuous dependence on \(Z\) and on the homotopy parameter. At the finitely many concatenation points the scalar equations agree, so the same continuity holds there. ◻

Proof. We first make the radius convention precise. Fix the prepared data, all thresholds and collars, and \(\epsilon\). Increase \(n_0\) so that \[ n_0\ge\max\{4,\lceil2/\epsilon\rceil\} \tag{187}\] and so that the floor test and all estimates of Proposition 33 apply for \(n\ge n_0\). Let \(R_{\mathrm{ap}}(n)\) be a sufficient lower radius for those estimates, and let \(R_{\mathrm{geo}}\) contain all compact boundary faces and supports used in the construction. The end barrier of that proposition gives a constant \(C_{\mathrm{out}}\), independent of \(n,R,s\), such that every admissible or floor-contact homotopy solution has \[ |\nabla f|\le C_{\mathrm{out}}\tau^{-1}R^{-1-\gamma} \quad\text{on the outer sphere}. \tag{188}\] The scalar last stage has \(f=0\) and satisfies the same bound. Choose, enlarging to a radius of a smooth truncation if necessary, \[ \begin{split} R_{\min}(n)=\max\biggl\{&n,R_{\mathrm{geo}},R_{\mathrm{ap}}(n),\\ &\left(\frac{2C_{\mathrm{out}}\ell} {\tau\sqrt{e^{8\epsilon}-1}} \right)^{1/(1+\gamma)}\biggr\}. \end{split} \tag{189}\] At \(Z=0\), the value of its implicit expression at \(t=-\epsilon\) is \(-\epsilon+\frac18\log(1+\ell^2\sigma^2)\). Equations (188) and (189) make this strictly negative. Strict monotonicity of that expression in \(t\) therefore excludes an outer floor contact. The term \(n\) in Equation (189) ensures \(R_{\min}(n)\to\infty\).

The sublevel and collar estimates apply to arbitrary smooth admissible or floor-contact solution sequences with \(n\to\infty\) and expanding outer cutoffs, uniformly over the compact ranges of the homotopy parameters. If the needed conclusion failed for arbitrarily large \(n\) with \(R\ge R_{\min}(n)\), one could select such a violating sequence. The chosen radius convention would make it precisely an expanding sequence excluded by those proofs. Thus a single \(n_0\), followed by the radius choices above, suffices for every homotopy parameter. No upper bound for \(Z\) is inserted in the earlier contact estimates: there \(\sigma\asymp\tau^{-1}\) and \(l\ge\ell\) alone give \(d\lesssim(\tau/\ell)^2\).

Fix now \(n\ge n_0\) and \(R\ge R_{\min}(n)\) for the rest of the degree argument. Let \[ X_b=\{Z\in C^{1,b}(\overline{\Omega_R}): Z=0\text{ on the outer face}\}, \qquad 0<b<1. \tag{190}\] For an input \(Z\in X_b\), first solve \(f=\mathcal S_s(Z)\) by Lemma 37. Compute \(t,u,A_\chi,\mathcal T\) and the full homotopy source from this input pair. Write \(B_s\) for the frozen drift summand in the flux, \(E_s\) for the full right side of its divergence equation, and \(q_s\) for the prescribed inner flux. Thus before the final scaling \(B_s=\lambda_suA_\chi K(w,\cdot)\) and \(E_s=u\Xi\); on the final interval \(E_s\) and \(q_s\) have the simultaneous scalar factor prescribed in Proposition 23. Define \(\mathcal K_s(Z)=\widetilde Z\) by the linear mixed problem \[ \begin{cases} \mathop{\mathrm{div}}(4uA_\chi\nabla\widetilde Z+B_s)=E_s &\text{in }\Omega_R,\\ (4uA_\chi\nabla\widetilde Z+B_s)_\nu=q_s &\text{on }B,\\ \widetilde Z=0 &\text{on the outer face}. \end{cases} \tag{191}\] The normal \(\nu\) in the second line points into \(\Omega_R\); the variational formulation uses its negative as the integration normal. This changes the sign of the boundary functional, not coercivity. On a bounded input set, the scalar gradient estimate bounds the coefficients above and away from zero. The nonempty outer Dirichlet face gives a Poincaré inequality, so the linear problem is uniquely solvable by its coercive bilinear form.

On bounded input sets in \(C^{1,b}\), Lemma 37 bounds \(f\) in \(C^{3,b}\). Hence the leading coefficient and drift in Equation (191) are \(C^{1,b}\), the bulk source is \(C^{0,b}\), and the prescribed boundary data are \(C^{1,b}\). The derivatives of \(t(x,Z,Df)\) involve only \(DZ\) and \(D^2f\), so this assertion includes the quadratic expression \(\mathcal T\). Linear regularity gives a bounded output set in \(C^{2,b}\). The inclusion \(C^{2,b}\hookrightarrow C^{1,b}\) is compact on the finite domain. Continuous dependence of the scalar solve and the linear problem makes \((s,Z)\mapsto\mathcal K_s(Z)\) a continuous compact homotopy on bounded sets. These assertions hold also at the concatenation parameters because the defining equations coincide there. A fixed point solves the coupled system; repeated Schauder estimates make it smooth.

Admissibility is evaluated after the scalar solve. Define the open subset of the parameter–input product \[ \mathcal O=\{(s,Z)\in[0,4]\times X_b: \min_{\overline{\Omega_R}}t_s[Z]>-\epsilon, \ \|Z\|_{C^{1,b}}<M_1\}. \tag{192}\] It is relatively open because the scalar solve and the implicit function defining \(t\) depend continuously on \((s,Z)\). Propositions 33 and 36 give a uniform bound in \(C^{1,b}\) for all fixed points satisfying \(t\ge-\epsilon\). If initially needed, use their higher classical bounds to reach the chosen exponent \(b\). Choose \(M_1\) larger than this bound. No fixed point lies on the norm boundary of \(\mathcal O\). No fixed point lies on its floor boundary either: outer contacts were excluded above and all remaining contacts are excluded by Proposition 28.

To apply degree on the moving admissible set, we need compactness of its fixed points. The set of fixed points in \(\overline{\mathcal O}\) is compact: their \(X_b\) norms are bounded, the parameter interval is compact, and \(Z=\mathcal K_s(Z)\) with a compact homotopy. Any limit has \(t\ge-\epsilon\) by continuity; it is a smooth fixed point and cannot lie on the floor. Thus these fixed points are compactly separated from the excluded boundary. The excision form of Leray–Schauder homotopy invariance applies to the sections \(\mathcal O_s\); equivalently, cover the compact fixed-point set by finitely many product neighborhoods contained in \(\mathcal O\) and use ordinary homotopy invariance and excision on the resulting finite parameter subdivision. This is the compact perturbation-of-identity degree (Leray and Schauder 1934).

At \(s=4\), the scalar equation has the unique solution \(f=0\). The drift vanishes, and both \(E_4\) and \(q_4\) vanish for every input. Equation (191) then has output identically zero. Moreover, \(t_4[0]=0>-\epsilon\), so \(0\in\mathcal O_4\) and \[ \deg(I-\mathcal K_4,\mathcal O_4,0)=1. \tag{193}\] Homotopy invariance gives the same degree at \(s=0\). There is therefore a fixed point in \(\mathcal O_0\), which is the desired smooth solution with \(t>-\epsilon\).

The bounds of Proposition 36 are independent of \(R\) at fixed \(n\). A diagonal subsequence on a compact exhaustion, using boundary charts at the fixed inner faces, converges smoothly to a solution on \(\overline\Omega\). The floor inequality passes to \(t\ge-\epsilon\). The next section establishes the end normalization, decay, and flux for these limits. ◻

Proof. The compact convergence, including at the fixed inner boundary, follows from Theorem 38 and its uniform fixed-\(n\) classical estimates. All equations and inner boundary conditions pass to this limit. To preserve the end normalization, we prove the decay estimates on the truncations before passing to infinity.

Outside a fixed large sphere, \(K=C=0\), and the first equation is \[\mathop{\mathrm{tr}}_{A_\chi}\mathop{\mathrm{Hess}}_g f=\frac{\tau\sqrt D}{l}\,f.\] At fixed \(n\) the preceding estimates make the left side uniformly elliptic, and its positive height coefficient is bounded below by a positive constant depending on \(n\). It is bounded above as well. For sufficiently small \(c_n>0\), the functions \(\exp[-c_n(r-r_0)]\) are positive supersolutions of the corresponding linear equation outside a sufficiently large \(r_0\): their radial second derivative has size \(c_n^2\), while their tangential second derivatives have size \(c_n/r\), and the fixed positive height term dominates both. Multiplying by a constant bounds the solution on \(S_{r_0}\); the zero outer Dirichlet value is also bounded. The maximum principle applied to both signs of \(f\) gives the uniform exponential estimate. The fixed-\(n\) unit-patch estimates, followed by the local homogeneous linear estimates for this equation, give the same decay for derivatives. The corresponding boundary estimates apply on the large outer spheres. Iterating the smooth equations gives each required finite derivative order.

The relation \(Z=t+\tfrac18\log(1+l^2|\nabla f|^2)\) now shows that \(t-Z\) is exponentially small, with derivatives. The graph errors have the same property. On the end the second equation becomes \[ \Delta_gZ+b_n(Z)|dZ|_g^2=O(r^{-4}),\qquad b_n(s)=p_L(s)-\delta_0\bigl(p_L(s)-1\bigr). \tag{195}\] Indeed \(A_\chi=I\) and \(u=L(Z)\) up to exponentially small errors, whereas \(\mathcal T=4(p_L(Z)-1)|dZ|^2\) up to such errors. The \(\rho_0\) penalty weight is \(O(r^{-4})\); the term with \(\rho_1\) is multiplied by the exponentially small \(v\). All coefficients in this statement are bounded on the fixed-\(n\) solution range. Differentiably controlled exponential errors can absorb any fixed polynomial power of \(r\).

Define an increasing smooth function on that range by \[\Phi_n(0)=0,\qquad \Phi_n'(s)=\exp\!\left(\int_0^s b_n(z)\,\mathrm dz\right).\] Its derivative is bounded above and below by positive fixed-\(n\) constants. The chain rule in (195) gives \[\Delta_g\Phi_n(Z)=O(r^{-4}).\] The function is bounded and vanishes at the outer cutoff. To obtain a bound independent of that cutoff, choose \(0<\eta<1\). For large \(r\), a positive multiple of \(r^{-1}-r^{-1-\eta}\) satisfies \[-\Delta_g(r^{-1}-r^{-1-\eta})\ge c r^{-3-\eta}.\] Here the Euclidean leading term is \(\eta(1+\eta)r^{-3-\eta}\), and the metric error is \(O(r^{-4})\). It dominates the right side after increasing the multiple and the fixed inner radius. Comparison for both signs yields \(|\Phi_n(Z)|\le C'(n) r^{-1}\), where \(C'(n)\) is a fixed-\(n\) constant with no polynomial claim. Consequently \(Z=O(r^{-1})\).

At sufficiently large radius this estimate and the exponential graph error give \(t>-\epsilon+1/n\); hence the floor penalty vanishes there. Rescale each annulus of radii comparable to \(r\) to unit size. The equation for \(\Phi_n(Z)\) has a bounded rescaled source and uniformly controlled metric coefficients. Local \(W^{2,p}\) estimates with \(p>3\) first give the gradient decay. Its source is then a smooth weight times a smooth function of the bounded variables, plus the already controlled exponential errors. Schauder estimates, or a further Sobolev bootstrap followed by them, give \(Z=O_2(r^{-1})\). Inverting \(\Phi_n\) preserves these bounds, and the same holds for \(t\).

Since \(L(0)=1\), we have \(u=1+O(r^{-1})\) and \(A_\chi=I+O(e^{-c_n r})\). With \(K=0\) on the end, the definition of \(V\) gives (194). Moreover \(\mathop{\mathrm{div}}V=u\Xi\) is integrable there: \(\mathcal T=O(r^{-4})\), \(\rho=O(r^{-4})\), the penalty has vanished, and \(\tau a\sigma\) decays exponentially. The divergence theorem on annuli proves the existence of the outward flux limit. ◻

Proof. At the physical endpoint the curvature identity gives \[\begin{align*} \tfrac12e^{4t}R_{\hat g} ={}&8\pi(\mu+J(w))+(1-\delta_0)\mathcal T +(1-\delta_0)\tau a\sigma\\ &+w\cdot\nabla C-F\mathop{\mathrm{tr}}K-\rho+m_0(t)\mathcal P. \end{align*}\] The height bounds on the compact supports of \(K\) and \(C\) give \(h\in[b_-,b_+]\). Since \(|w|\le1\), the terms \(|F\mathop{\mathrm{tr}}K|+|\nabla C|+\rho\) are dominated by \(8\pi(\mu-|J|)\), by the choices in Proposition 19 and the choice of \(\rho\). Outside those supports the same conclusion uses \(8\pi(\mu-|J|)-\rho\ge0\). All remaining terms are nonnegative. This proves \(R_{\hat g}\ge0\).

The boundary identity and the prescribed flux give \(H+4\partial_\nu t=-N_B<0\). Since \(f\) is constant on each face, this is exactly the mean-curvature sign for \(\hat g\). Also \(\hat g\ge e^{-4\epsilon}g\), so any escaping curve has infinite length; the compact boundary is included. Smoothness and completeness therefore follow.

Lemma 39 gives the stated metric decay. The reduced construction supplies \(R_g=O(r^{-3-\delta})\) with symbol estimates on the rest end. Since graph errors are exponentially small, the conformal formula gives \[R_{\hat g} =e^{-4t}\bigl(R_g-8\Delta_gt-8|dt|_g^2\bigr) +O(e^{-c_n r}).\] Equation (195), the vanishing far-end penalty, and the gradient estimate give \(\Delta_gt=O(r^{-4})\). This proves the asserted scalar-curvature decay, including the pointwise falloff required by the AF convention of (Bray 2001, Definition 21 and Theorem 19).

Exponential graph errors contribute no ADM flux. The conformal change and \(t=O_2(r^{-1})\) give \[E_{\hat g} =E-\frac1{2\pi}\lim_{r\to\infty} \int_{S_r}\partial_\nu t\,\mathrm dA_g.\] Replacing the metric normal and area form by their Euclidean versions costs \(o(1)\), and the \(O(r^{-3})\) error in (194) has vanishing sphere integral. This proves (196). ◻

Proof. The normal \(\nu\) on \(B\) points into \(\Omega\), so the divergence theorem has the sign \[ \mathfrak F_n =\int_\Omega u\Xi\,\mathrm dV_g+\int_BV_\nu\,\mathrm dA_g. \tag{197}\] Each integral is finite for a fixed \(n\), by Lemma 39 and compact smoothness.

At a black face, \(F-P_B=H\) and \(w_\nu=a\). At a white face, \(F-P_B=-H\) and \(w_\nu=-a\). Thus at either type of face, \[V_\nu/u=-(1-a)H-N_Bd\ge-Cd.\] The signed boundary gradient bounds give \(\sigma\ge c/\tau\), so \[\sqrt d\le\frac1{l\sigma}\le C\tau/\ell,\qquad \int_BV_\nu\,\mathrm dA_g\ge -C\frac{\tau}{\ell}\int_BL\,\mathrm dA_g.\] Use the polynomial collar trace estimate from Section 5 with the constant test function: \[\int_B L\,\mathrm dA_g \le C(1+n)^C \int_{\mathcal C}u(1+\sqrt{\mathcal T})\,\mathrm dV_g,\] where \(\mathcal C\) is a fixed compact union of collars. Its polynomial constant is independent of the upper \(Z\) bound. Since \(\rho\) has a fixed positive minimum on \(\mathcal C\), \[\int_{\mathcal C}u(1+\sqrt{\mathcal T})\,\mathrm dV_g \le C\int_\Omega u(\delta_0\mathcal T+\rho)\,\mathrm dV_g.\] The factor \(C(1+n)^C\tau/\ell =C(1+n)^C\ell^{1/2}\) tends to zero. For large \(n\), the entire negative boundary term in (197) is therefore at most one quarter of \(\int_\Omega u(\delta_0\mathcal T+\rho)\,\mathrm dV_g\).

It remains to estimate the floor penalty. On its support, \(-\epsilon\le t\le-\epsilon+1/n\), so \(l\) and \(L\) are comparable to \(\ell\), with constants independent of \(n\). Directly from their definitions, \[\begin{split} u&=L\sqrt{1+l^2\sigma^2} \le C(\ell+\ell^2\sigma),\\ uv&=\frac{Ll^2\sigma^2}{\sqrt{1+l^2\sigma^2}} \le C\ell^2\sigma,\\ ua\sigma&=Ll\sigma^2\ge c\ell^2\sigma^2 . \end{split}\] Consequently \[um_0\mathcal P \le C_n\bigl[\ell\rho_0+\ell^2\sigma(\rho_0+\rho_1)\bigr].\] Young’s inequality, with the square term chosen to be a quarter of \(\delta_0\tau ua\sigma\), yields \[um_0\mathcal P \le \tfrac14\delta_0\tau ua\sigma+ C_n\left[\ell\rho_0+ \frac{\ell^2}{\tau}(\rho_0^2+\rho_1^2)\right].\] Squaring the constant only changes the polynomial \(C_n\). The integrals of \(\rho_0\), \(\rho_0^2\), and \(\rho_1^2\) are finite: on the end their orders are respectively \(r^{-4}\), \(r^{-8}\), and \(r^{-4\gamma}\), with \(4\gamma>3\). Integrating leaves at most \(C(1+n)^C(\ell+\ell^2/\tau)\), after absorption.

Substitute both bounds in (197). Positive fractions of the three bulk terms \(\delta_0\mathcal T,\rho,\delta_0\tau a\sigma\) remain. Dropping them proves the claim. Finally \(\ell^2/\tau=\ell^{1/2}\), so the error tends to zero exponentially against every polynomial loss. ◻

Proof. For each fixed sufficiently large \(n\), minimize \(\hat g\)-area among filled cuts enclosing \(B\), using a large outer sphere as an auxiliary barrier. The area-compactness and obstacle arguments of Section 2 apply to this smooth AF metric. A fixed enclosing competitor bounds the area, and local perimeter density estimates prevent minimizers from escaping to infinity. The positive mean curvature of sufficiently large outer spheres excludes contact there, as in Lemma 5. For the inner barrier, let \(U_s\) be a short outward normal collar of the filled obstacle and \(Y\) its outward unit foliation field. Strict negativity gives \(\mathop{\mathrm{div}}_{\hat g}Y<0\) throughout the collar. If a perimeter minimizer \(E\) omitted a positive-volume portion \(D=U_s\setminus E\), Gauss–Green and \(|Y|_{\hat g}\le1\) would give \[P_{\hat g}(E\cup U_s)-P_{\hat g}(E) \le\int_D\mathop{\mathrm{div}}_{\hat g}Y\,\mathrm dV_{\hat g}<0.\] Thus \(E\) contains a whole inner collar; obstacle regularity upgrades this inclusion to its regular representative. Its free boundary is disjoint from the obstacle. The resulting cut \(\Gamma_n\) is compact, smooth, and minimal in dimension three; it may be disconnected. It is outer area-minimizing by its minimizing property. Equivalently one may first use the ordinary trapped-region theorem for \((\hat g,0)\) and then take the outer minimizing enclosure. The boundary version of the Riemannian Penrose theorem applies to the exterior of \(\Gamma_n\), by Lemma 40 and (Bray 2001, Theorem 19).

For every two-plane the restriction of \(\bar g=g+l^2df^2\) dominates that of \(g\). The floor therefore gives \[|\Gamma_n|_{\hat g} \ge e^{-4\epsilon}|\Gamma_n|_g \ge e^{-4\epsilon}A_*.\] The last inequality uses that \(B\), and hence \(\Gamma_n\), cuts off the original filled obstacle. Lemmas 40 and 41 now give \[E+\frac{\varepsilon_n}{8\pi} \ge E_{\hat g} \ge\sqrt{\frac{|\Gamma_n|_{\hat g}}{16\pi}} \ge e^{-2\epsilon}\sqrt{\frac{A_*}{16\pi}}.\] Let \(n\to\infty\) with \(\epsilon\) fixed, and then \(\epsilon\downarrow0\). This proves the reduced inequality.

Finally apply Proposition 12 to any exterior in Theorem 2. The reduced energies converge to its invariant ADM mass and the enclosing infima converge to its original enclosing infimum. Applying the reduced bound to each member of that sequence and passing to the limit gives Equation (8). ◻

Proof. A sufficiently large coordinate sphere gives finiteness. For positivity, choose a small disk in one component of \(B\) and a smooth proper embedded ray from its center to the end, normal to \(B\) initially and straight in the asymptotic coordinates outside a compact set. The ray has a tubular neighborhood with disk cross-sections; its coordinate width can be fixed and positive on the straight tail. In product coordinates on this tube, pull back a smooth two-form compactly supported in the transverse disk and with integral one. Extend it by zero through the lateral boundary of the tube. The resulting two-form \(\omega\) is smooth and closed on the exterior, including its boundary, and has bounded pointwise norm. The bound follows from compactness on the initial part and asymptotic flatness on the tail.

Orient the transverse disk so that the flux of \(\omega\) through a large coordinate sphere is one. A smooth enclosing cut in the interior and a sufficiently large sphere bound a compact region disjoint from \(B\). Stokes’ theorem therefore gives \(\int_\Gamma\omega=1\). The same identity holds for coincident cuts by smooth outward approximation. Consequently \[1=\left|\int_\Gamma\omega\right| \le \|\omega\|_{L^\infty(g)}\operatorname{Area}_g(\Gamma).\] Taking the infimum proves positivity. If the filled perimeter closure of the cut class is used, its infimum agrees with the smooth infimum by Definition 1 and the approximation established in the exterior reduction. ◻

Proof. Use the conformal family of Lemma 4, which requires no sign assumption on the ADM vector. For every finite \(s\ge0\), its data \((g_s,K_s)\) satisfy all exterior hypotheses except possibly timelikeness, with \[E_s=E+sA_\phi,\qquad P_s=P,\qquad A_\phi>0.\] Equation (18) and Lemma 45 give \(a_{g_s}(B)\ge a_g(B)>0\). Suppose that \(E\le|P|_\delta\) and put \(s_0=(|P|_\delta-E)/A_\phi\ge0\). For every \(s>s_0\), Equation (17) places \((N,g_s,K_s)\) in the timelike class of the already proved Theorem 2. That theorem and the preceding area bound imply \[\sqrt{(E+sA_\phi)^2-|P|_\delta^2} \ge\sqrt{\frac{a_g(B)}{16\pi}}>0.\] Letting \(s\downarrow s_0\) gives a contradiction. Therefore \(E>|P|_\delta\), and applying Theorem 2 to the original data proves the claimed inequality. ◻

Proof of Theorem 44. Let \(D_0\) be the connected component of \(D^+\) containing the distant chosen end. Its boundary is a union of components of \(S\) and is nonempty: otherwise \(D_0\) would be both open and closed in the connected manifold \(M\), hence equal to \(M\), contrary to \(S\ne\varnothing\). Put \(X_0=\overline{D_0}\), including its smooth compact boundary. Because \(S\) is compact, each sufficiently distant end of \(M\) lies entirely in \(D_0\) or entirely outside it. Outside a compact set, \(X_0\) is precisely the union of the original distant ends contained in \(D_0\). In particular, \(X_0\) has finitely many ends, all asymptotically flat.

For every end of \(X_0\) other than the chosen one, choose a sufficiently distant open coordinate tail, and let \(N\) be \(X_0\) with all these open tails removed. These tails can be chosen disjoint and outside \(S\). Removing them preserves connectedness: an excursion of a path into a removed tail can be replaced by a path on its connected coordinate sphere. The region \(N\) is a smooth manifold with nonempty compact boundary \(B\), consisting of \(\partial D_0\) and the added coordinate spheres, and its only end is the chosen one.

The region is closed in the complete manifold \((M,g)\) and has smooth boundary. It is complete for its intrinsic distance with the boundary included: an intrinsic Cauchy sequence is ambient Cauchy, its ambient limit lies in \(N\), and smooth boundary charts compare the intrinsic distance locally with Euclidean distance on a half-ball. The restricted constraints, integrability, and decay satisfy the exterior hypotheses. The chosen ADM vector is still \((E,P)\).

On the inherited part of \(B\), the normal into \(N\) agrees with the normal into \(D^+\), so the assumed expansion is nonpositive. On an added sphere in another end, the normal into \(N\) points toward decreasing radius. Consequently \[H_B=-\frac2r+O(r^{-1-q}),\qquad \operatorname{tr}_B K=O(r^{-1-q}),\] and its future outward expansion is strictly negative when its radius is sufficiently large. Corollary 46 therefore applies to \((N,g,K)\) and gives \[ E>|P|_\delta,\qquad \sqrt{E^2-|P|_\delta^2}\ge\sqrt{\frac{a_g(B)}{16\pi}}. \tag{200}\]

Finally, \[ a_g(B)\ge A_{\min}(S). \tag{201}\] To see this, first take an enclosing cut \(\Gamma\) lying in the interior of \(N\). The component on its distant chosen-end side is an open region \(D'^+\subset D_0\subset D^+\), with compact smooth boundary consisting of components of \(\Gamma\). It is a competitor in Equation (198); its area is at most that of \(\Gamma\). If a cut coincides with any part of \(B\), approximate it by smooth cuts pushed into the interior of \(N\), with areas converging to its area. This also handles cuts coinciding with a newly introduced sphere. Taking the infimum proves Equation (201). Other components of \(D^+\) cause no difficulty, since the original competitor class permits the chosen-end component alone. Combining Equations (200) and (201) proves the theorem. ◻

Proof. The original exterior is complete as a metric space with boundary, as follows. An intrinsic Cauchy sequence is Cauchy in the complete ambient manifold, and its ambient limit remains in the closed exterior. A smooth interior or boundary half-ball chart then gives convergence in the intrinsic distance. Its compact boundary and unique end place it in Definition 1. Since \(S\) is an enclosing cut and every enclosing cut has area at least \(A\), its enclosing infimum is \(A\). Theorem 2 gives the non-strict inequality. Equality would imply \(\ell=1\) by Theorem 48, contradicting \(\ell\geq2\). ◻

Proof. A fixed enclosing competitor bounds all minimizing perimeters from above. A minimizing boundary that reaches arbitrarily large end radius has a point with a ball of radius comparable to that radius which is disjoint from the obstacle. The scaled metric bounds and the interior perimeter density estimate give a lower area bound equal to a positive constant times the square of that radius. This contradicts the fixed upper bound. Thus all minimizing boundaries for the paths under consideration stay in a fixed compact set; this is also the compactness argument in Lemma 5. One may minimize on a large compact truncation, whose sphere is a strict mean-curvature barrier, and then remove the truncation. Perimeter compactness gives a limiting filled set. The fixed-obstacle condition is closed under this convergence, and lower semicontinuity gives a minimizing boundary.

The regularity input is the smooth-obstacle perimeter theorem: in ambient dimension three, a perimeter minimizer outside a \(C^2\) obstacle is \(C^{1,1}\) and is smooth minimal away from contact (Huisken and Ilmanen 2001, Regularity Theorem 1.3(iii)). These minimizers realize the smooth-cut infimum as well. Every smooth filled enclosing cut is an admissible finite-perimeter competitor. Conversely, flow a regular minimizing boundary for a small positive time by a smooth vector field pointing strictly outward along \(S\) and supported in its collars. The moved filled set contains the obstacle in its interior. A sufficiently close smooth \(C^1\) approximation of that compact embedded boundary still encloses the obstacle, and its area tends to the original perimeter as the flow time and approximation error tend to zero. Filling any bounded complementary pockets only decreases area. The resulting smooth cuts prove equality of the two infima. This construction applies to each nearby metric as well.

The local obstacle estimates and the interior density estimates apply uniformly on the fixed compact region. For a convergent sequence of minimizing sets, lower semicontinuity and the upper bound from a fixed minimizing competitor give convergence of the perimeters. Express the normal vector measures in a fixed smooth local \(g\)-orthonormal frame. These are smooth matrix multiples of the distributional derivatives of the set indicators, so they converge weakly, and their Euclidean total variations are precisely the convergent \(g\)-perimeters. Strict convergence of these vector measures gives convergence of the integrals of every continuous function of position and unit normal; in particular it gives convergence for every continuous function of the unoriented tangent plane. This is the strict-measure continuity theorem, applied locally and then with a partition of unity (Ambrosio et al. 2000). The same argument works when the metrics vary: their uniform convergence converts convergence of the varying perimeters into strict convergence for the fixed metric.

For any rectifiable cut of bounded area the pointwise area-element expansion gives, uniformly over those cuts, \[ |T|_{g_s}=|T|_g+s a'_T(h)+O(s^2)|T|_g. \tag{214}\] Testing a fixed member of \(\mathcal T\) gives the upper bound in Equation (213). For the opposite bound, take minimizers \(T_s\) for \(g_s\). Any sequence \(s\downarrow0\) has a subsequence converging as above to some \(T\in\mathcal T\). Since \(|T_s|_g\ge A\), Equation (214) gives \[\frac{a(s)-A}{s}\ge a'_{T_s}(h)+O(s) \longrightarrow a'_T(h) \ge\min_{T'\in\mathcal T}a'_{T'}(h).\] This proves the derivative formula and the asserted continuity.

Finally, if \(D_1,D_2\) are two minimizing filled sets, then their union and intersection are admissible and perimeter submodularity gives \[P(D_1\cup D_2)+P(D_1\cap D_2) \le P(D_1)+P(D_2)=2A.\] Each term on the left is at least \(A\), so both are minimizers. Different tangent planes at an intersection off the obstacle would give a corner in one of these two minimizing boundaries, contradicting their interior regularity. Thus the tangent planes agree there. ◻

Proof. Adjoin an end point to \(\mathcal A\), collapsing every ray that escapes spatial compact sets to that point; if \(\mathcal A\) is compact, take an additional isolated end point. The bounded fibers make the result a compact space. Equations (219) and (220) show that \(C'_H/w\) extends continuously to it, with value \(8a(H)\). Take the disjoint compact union of this space, \(S\), and \(\mathcal T\), and associate to \(H\) the continuous function whose values on the three pieces are \[ \frac{C'_H}{w},\qquad -\theta'_H,\qquad c a'_T(h)-16\pi\mathcal E'(H), \tag{222}\] respectively. Smooth dependence of the boundary expansion and Lemma 51 give the remaining continuity.

We first prove that no image of this linear map is strictly positive at every point of the compact test space. Suppose otherwise, and write \(H=aQ+H_0\). Positivity at the end point gives \(a>0\). Realize its tangent by the actual path \[ g_s=e^{4as\phi}(g+s h_0),\qquad K_s=e^{2as\phi}(K+s p_0). \tag{223}\] The metric remains positive and uniformly comparable to \(g\) for small \(|s|\). The varied data retain the required asymptotic decay with exponent \(\min\{q,1\}>1/2\), and their charge variations have finite limits. The conformal factor has its stated flux. Thus the path has differentiable ADM fluxes.

Here are the estimates which ensure nonlinear feasibility. On the end the tensors in \(H_0\) are \(O_2(r^{-1})\) and \(O_1(r^{-2})\), and their far-end Euclidean linear constraints vanish. Terms in their linear constraint variation containing a background error cost \(O(s r^{-3-q})\), while new quadratic terms cost \(O(s^2r^{-4})\). Changing the orthonormal identification has the same bound: the background momentum is \(O(r^{-2-q})\), and the metric variation is \(O(r^{-1})\). Applying the exact conformal law to the intermediate data \((g+s h_0,K+s p_0)\), and keeping the nonnegative background term with its positive scaling factor, therefore gives, uniformly over the full unit ball of test vectors, \[ e^{4as\phi}C_s =C+s\bigl(8a w+o(w)\bigr)+O(s^2w) \quad\hbox{on the end}. \tag{224}\] The \(o(w)\) is uniform as \(r\to\infty\) at fixed \(H\), and the quadratic bound is uniform for small \(s\). The gradient square in the conformal formula is \(O(r^{-4})=O(w)\) because \(\delta<1\). One may use the composition of the two natural isometries of the unit balls in deriving this estimate; it tests the same DEC. At active rays its derivative agrees with Equation (215), since the difference of two isometric identifications is tangent to the unit sphere and is annihilated by \(J\) at an active ray.

It follows from Equation (224) that the DEC holds on a fixed far end for all sufficiently small positive \(s\). On the remaining compact ball bundle, the derivative is strictly positive on the closed active set. A neighborhood of that set has the same positive derivative with a fixed smaller margin. On its compact complement the original \(C\) has a positive minimum. Taylor’s formula now gives the DEC throughout the compact region as well. The strict inequality \(-\theta'_H>0\) on compact \(S\) gives \(\theta_{+,s}<0\) for small positive \(s\). The new sources are integrable: outside a compact set their changes, after the harmless positive rescaling of the original sources, have the integrable bounds just displayed. The ADM vector stays future timelike. Hence these are data to which Theorem 2 applies.

Finally, strict positivity on the compact cut space gives \[c\min_{T\in\mathcal T}a'_T(h)-16\pi\mathcal E'(H)>0.\] By Lemma 51 and Equation (204), this says that the right derivative of \(16\pi(\mathcal E(s)-\sqrt{a(s)/(16\pi)})\) is strictly negative. The expression is zero at \(s=0\), whereas the supporting-energy inequality and Theorem 2 make it nonnegative for small positive \(s\). This is a contradiction.

The image of the linear test map is therefore disjoint from the open cone of strictly positive continuous functions. The separating form of the Hahn–Banach theorem gives a nonzero continuous linear functional, nonnegative on nonnegative functions, which annihilates the image. The Riesz representation theorem represents it by nonnegative finite measures on the three compact pieces. Its mass on the objective piece \(\mathcal T\) cannot be zero: otherwise its value on the image of \(Q\), which is strictly positive on both remaining pieces including their end point, would be positive. Normalize this objective mass to one. On the finite active rays divide the corresponding measure by \(w\) and call the result \(\Lambda\). It is locally finite; the original finite measure makes its pairing with \(C'_H\) finite. Put the end-point mass, multiplied by eight, into \(z\). Rearranging the annihilation identity gives Equation (221) with the stated signs. ◻

Proof. Choose a compact neighborhood of the support of \(s\). In a Gaussian time collar put \[\mathbf g_\epsilon=-\,\mathrm dt^2+g_\epsilon(t),\qquad g_\epsilon(0)=g,\qquad \partial_tg_\epsilon(0)=2K.\] Choose the second time derivative so that its spatial Einstein components at time zero are \[ (\mathbf G_\epsilon)_{ij}=8\pi(D_\epsilon)_{ij}, \qquad D_\epsilon=\frac{J\otimes J}{\mu+\epsilon}. \tag{228}\] This is a free choice of jets, not a choice of evolution satisfying an energy condition. To see its solvability, the terms in the spatial Einstein tensor containing \(\partial_tK\) are its trace reversal, with a fixed nonzero overall sign. On symmetric tensors in dimension three this map is invertible: its eigenvalue is one on the tracefree part and minus two on the pure-trace part. All remaining terms are determined by \(g,K\) and their spatial derivatives. Thus smooth second jets realizing Equation (228) exist; a quadratic extension with a time cutoff realizes them by a smooth Lorentzian metric for sufficiently small time. The constraints give at time zero \[ \mathbf G_\epsilon(n,n)=8\pi\mu, \qquad \mathbf G_\epsilon(n,e_i)=8\pi J_i, \tag{229}\] where \(n=\partial_t\) and \(e_i\) is a spatial orthonormal frame.

The regularization has the precise compact convergence needed below. Define \(D=J\otimes J/\mu\) where \(\mu>0\), and \(D=0\) at \(\mu=0\). The DEC implies \[ |D_\epsilon|\le\mu,\qquad |\nabla D_\epsilon|\le2|\nabla J|+|\nabla\mu|. \tag{230}\] The second inequality follows by differentiating the quotient and using \(|J|\le\mu\) in each term. Smooth nonnegative \(\mu\) has \(\nabla\mu=0\) at its zero set. There also \(\nabla J=0\): in a smooth frame \(|J_i|\le\mu\), so each directional derivative of \(J_i\) vanishes. It follows first that \(D\) is differentiable with zero derivative at that set, and then from Equation (230) that this derivative is continuous there. On a compact set, split into \(\mu\ge\delta\) and \(\mu<\delta\). On the first part the quotient and its derivatives converge uniformly as \(\epsilon\to0\). On the second part the bound in Equation (230) is uniformly small as \(\delta\downarrow0\), by compactness and continuity at the zero set. Therefore \[ D_\epsilon\longrightarrow D\quad\hbox{in }C^1 \hbox{ on every compact set}. \tag{231}\]

Vary the slice by the graphs \(t=a s(x)\) at \(a=0\), using its induced metric and second fundamental form. Their variation is compactly supported where \(s\) and its derivatives are supported, and its metric component is \(2sK\). We prove that its first DEC variation tends to zero at an active ray \((x,v)\).

If \(\mu(x)>0\), activity implies \(\mu=|J|\), \(|v|=1\), and \(v=-J^\sharp/\mu\) at that point. Write \(\zeta=n+v\). On the portion of the initial slice where \(\mu>0\) the limiting tensor prescribed above is \[ \mathbf G_0=\frac{8\pi}{\mu}\alpha\otimes\alpha, \qquad \alpha(n)=\mu,\qquad \alpha(e_i)=J_i. \tag{232}\] The covector \(\alpha\) is causal and is nonnegative on future causal vectors. At the active point, \(\zeta\) is null and \(\alpha(\zeta)=0\). Parallel transport \(\zeta\) along any spatial curve in the initial slice, using the spacetime connection. It remains future null, so its contraction with \(\alpha\) is a nonnegative function with a minimum zero at \(x\). Thus \[(\nabla_i\alpha)(\zeta)=0,\qquad (\nabla_i\mathbf G_0)(e_i,\zeta)=0 \quad\hbox{at }x.\] The spatial spacetime connection at time zero depends only on \(g,K\), not on \(\epsilon\). Equation (231) therefore also gives convergence of these spatial covariant derivatives. Contracted Bianchi, applied before taking the limit, states in an orthonormal frame that \[ (\nabla_n\mathbf G_\epsilon)(n,\zeta) =\sum_i(\nabla_{e_i}\mathbf G_\epsilon)(e_i,\zeta). \tag{233}\] Only the normal derivative of the constraint components is used here; Bianchi expresses it through the spatial first derivatives controlled above. Its limiting value is zero.

The derivative of the contraction defining the DEC on the graph also differentiates its two vector arguments. The term varying the first argument vanishes because \(\mathbf G_0(\cdot,\zeta)=0\). For the second argument, its isometrically transported spatial part still has length one, so the varied \(\zeta\) stays null. At the active point \(\alpha\) is proportional to the covector obtained by lowering \(\zeta\); hence \(\alpha(\dot\zeta)=0\). This kills the second argument term. Together with Equation (233) it proves the desired limit at such a ray.

If \(\mu(x)=0\), then \(J=0\) and their spatial first derivatives vanish. The limiting tensor and its spatial first derivatives are zero. Equation (233) now kills the required normal derivative, and both argument terms vanish. This applies to all active \(v\), including \(|v|<1\). Finally, all the expressions just used involve only fixed compactly bounded jets, \(D_\epsilon\), and its spatial first derivatives. The compact \(C^1\) convergence makes the convergence uniform, including near \(\mu=0\). This proves Equation (227). ◻

Proof. Use the variations of Lemma 53 in Equation (221). Their ADM and boundary variations vanish. Their supports lie in a fixed compact set, on which \(\Lambda\) is finite, so uniform convergence permits passage to the limit in its pairing. The metric variation is \(2sK\), independent of the regularization. It follows that \[ \int_{\mathcal T}\int_T s\mathop{\mathrm{tr}}_TK\,\mathrm dA_g\,\mathrm d\omega(T)=0 \quad\hbox{for every }s\in C_c^\infty(\operatorname{Int}M_{\mathrm{ext}}). \tag{234}\]

Consider the positive aggregate area measure \(\beta=\int_{\mathcal T}\,\mathrm dA_T\,\mathrm d\omega(T)\) off \(S\). By Lemma 51 and the observation following its proof, the function \(F(x)=\mathop{\mathrm{tr}}_{\Pi(x)}K(x)\), where \(\Pi(x)\) is the common tangent plane, is continuous on the union of the cuts, locally away from \(S\). In particular it is measurable there. Equation (234) says the signed Radon measure \(F\beta\) is zero. Thus \(F=0\) \(\beta\)-almost everywhere. Fubini’s theorem on a countable compact exhaustion shows that for \(\omega\)-almost every cut its tangential trace of \(K\) is zero at area-almost every point off \(S\). Each such cut is smooth minimal there, so continuity improves this to \[H_T=0,\qquad \mathop{\mathrm{tr}}_TK=0\quad\hbox{on }T\setminus S.\] There can be no cancellation in this argument between cuts with different tangent planes: the common-plane property is exactly what made \(F\) a single function of position.

Fix one of these cuts. Near a point of contact with \(S\), the \(C^{1,1}\) obstacle graph and the smooth graph of \(S\) have the same first derivatives on their coincidence set. Their second derivatives agree almost everywhere on that set. One way to see the latter assertion is to apply the fact that a Lipschitz function vanishing on a measurable set has derivative zero almost everywhere on that set, first to the graph difference and then to each of its first derivatives. The outward orientations agree at contact. Since \(S\) is a MOTS, the cut consequently satisfies \(H_T+\mathop{\mathrm{tr}}_TK=0\) almost everywhere across contact, as well as on its complement. In a graph chart this is a uniformly elliptic quasilinear equation for a \(C^{1,1}\) function. Its coefficients are Hölder in position and first derivatives. Interior elliptic regularity followed by differentiation therefore makes the graph smooth. Thus the whole cut is a smooth MOTS.

Suppose the cut touches a component \(S_i\). At each contact point, the two smooth graph equations have the same orientation and one graph lies on one side of the other. Their difference satisfies a linear uniformly elliptic equation with bounded coefficients. The strong comparison principle implies local coincidence. The contact set \(T\cap S_i\) is therefore open in \(S_i\); it is also closed by compactness. Connectedness of \(S_i\) gives \(S_i\subset T\). Local coincidence and compactness then make \(S_i\) a connected component of \(T\). Consequently, every component of \(T\) either equals one of the original \(S_i\) or is disjoint from \(S\) and lies entirely in \(\operatorname{Int}M_{\mathrm{ext}}\). This argument does not use the genus of \(S_i\).

The cut is smooth and encloses the entire obstacle. Its filled set has no bounded complementary pocket, since filling such a pocket would remove positive perimeter and contradict minimality. If \(T\) had any interior component, it would thus be an enclosing smooth MOTS with at least one interior component and with every remaining component either interior or equal to an original \(S_i\). This is exactly a cut excluded by the partial-coincidence outermostness hypothesis, even if it retains some original boundary components. Hence \(T\) has no interior component. There is a subset \(I\subset\{1,\ldots,\ell\}\) such that \[T=\bigsqcup_{i\in I}S_i,\qquad \sum_{i\in I}|S_i|_g=|T|_g=A =\sum_{i=1}^{\ell}|S_i|_g.\] Each \(|S_i|_g\) is strictly positive. The equality of these finite sums therefore forces \(I=\{1,\ldots,\ell\}\), so \(T=S\). It follows that \(\omega\)-almost every cut is \(S\), proving the lemma. ◻

Proof. For a compact variation \(p=\dot K\) with \(h=0\), direct differentiation and integration of the momentum divergence give the coefficient \[2\bigl[u(\tau g-K)-\mathop{\mathrm{sym}}\nabla X+(\mathop{\mathrm{div}}X)g\bigr]\] against \(p\). It is zero distributionally. Taking its trace gives \(\mathop{\mathrm{div}}X=-u\tau\), and substituting back proves Equation (206).

Next use the compact simultaneous conformal variation \((h,p)=(4\psi g,2\psi K)\). Its background scaling term is \(-64\pi\psi(u\mu+J(X))\), which vanishes by Equation (226). The remaining pairing is \[-8\langle u,\Delta\psi\rangle +8\langle X,K(\nabla\psi,\cdot)\rangle=0.\] This is Equation (207) in distributions. Taking a divergence of the symmetric-gradient equation and using \(\mathop{\mathrm{div}}X=-u\tau\) also gives \[ \begin{split} \Delta X_j+\mathop{\mathrm{Ric}}_{jk}X^k &=(\tau\delta_j{}^i-2K_j{}^i)\nabla_i u +u(\nabla_j\tau-2\nabla^iK_{ij}). \end{split} \tag{235}\] Together with the lapse equation this is a second-order elliptic system with scalar Laplacian principal part and smooth lower-order couplings. Locally the measure moments belong to some negative Sobolev space. The local Laplace estimate applied to this system improves their Sobolev order by one, since the right sides have at most one derivative of the unknowns. Iterating with nested compact cutoffs puts them in every local Sobolev space. Sobolev embedding then gives smoothness. The distributional inequalities and complementarity now hold pointwise in the interior.

We give the full metric calculation, including the dependence of the momentum test vectors on the metric. Put \(P=K-\tau g\). For compactly supported variations the constraint pairing is the first variation of \[ I(g,K)=\int \left[u(R+\tau^2-|K|^2)+2X^i\nabla^jP_{ij}\right]\,\mathrm dV_g, \tag{236}\] together with the frame correction \[ -8\pi\int h(X,J^\sharp)\,\mathrm dV_g. \tag{237}\] The integration may be localized to a fixed region containing the variation. Changing its volume form instead of retaining the original fixed measure adds one half of its background integrand times \(\mathop{\mathrm{tr}}h\). That integrand is \(16\pi(u\mu+J(X))=0\), so this replacement does not change the pairing. Equation (237) follows directly from Equation (215) and has the displayed negative sign.

Integrating the shift term once by parts makes it \(-\int P^{ij}(\mathcal L_Xg)_{ij}\,\mathrm dV_g\) up to a boundary term unaffected by these variations. In taking a pure metric variation, \(u\), contravariant \(X\), and covariant \(K\) are fixed. The elementary variation formulas used are \[\begin{align*} \dot R&=-\mathop{\mathrm{Ric}}^{ij}h_{ij} +\nabla^i\nabla^jh_{ij}-\Delta(\mathop{\mathrm{tr}}h),\\ \dot\tau&=-K^{ij}h_{ij},\qquad \bigl(|K|^2\bigr)^{\boldsymbol\cdot} =-2(K^2)^{ij}h_{ij},\qquad (\,\mathrm dV)^{\boldsymbol\cdot}=\tfrac12\mathop{\mathrm{tr}}h\,\mathrm dV,\\ (P^{ij})^{\boldsymbol\cdot} &=-h^i{}_aK^{aj}-h^j{}_aK^{ia} +(K^{ab}h_{ab})g^{ij}+\tau h^{ij},\\ (\mathcal L_Xg)^{\boldsymbol\cdot}&=\mathcal L_Xh. \end{align*}\] For example, integrating the last term by parts and combining it with the variation of \(P^{ij}\) gives the following coefficient from the entire shift integral: \[(\mathcal L_XK)_{ij}-(\mathop{\mathrm{div}}X)K_{ij} -\bigl[X(\tau)+K^{ab}\nabla_aX_b\bigr]g_{ij}.\] The scalar-curvature terms contribute \(\mathop{\mathrm{Hess}}u-(\Delta u)g-u\mathop{\mathrm{Ric}}\). Consequently the vanishing metric coefficient is the explicit equation \[ \begin{split} 0={}&\nabla_i\nabla_j u-(\Delta u)g_{ij}-u\mathop{\mathrm{Ric}}_{ij} +2u(K^2-\tau K)_{ij}\\ &+\frac u2(R+\tau^2-|K|^2)g_{ij} +(\mathcal L_XK)_{ij}-(\mathop{\mathrm{div}}X)K_{ij}\\ &-\bigl[X(\tau)+K^{ab}\nabla_aX_b\bigr]g_{ij} -8\pi X_{(i}J_{j)}. \end{split} \tag{238}\] This calculation can equally be read distributionally, since each coefficient is linear in the moments and their derivatives.

For completeness, its simplification uses no vacuum assumption. Equation (206) gives \[\mathop{\mathrm{div}}X=-u\tau,\quad K^{ab}\nabla_aX_b=-u|K|^2,\quad \mathop{\mathrm{tr}}(\mathcal L_XK)=X(\tau)-2u|K|^2.\] Taking the trace of Equation (238) and using \(16\pi\mu=R+\tau^2-|K|^2\) and \(J(X)=-u\mu\) yields \[ \Delta u+X(\tau)=\frac u2(R+\tau^2+|K|^2). \tag{239}\] Substitution of this identity into Equation (238) gives exactly Equation (208).

These equations have the asserted finite-type property. The right side of Equation (208) is linear in \(u,X,\nabla X\), with smooth background coefficients, since \[(\mathcal L_XK)_{ij} =X^k\nabla_kK_{ij} +K_{kj}\nabla_iX^k+K_{ik}\nabla_jX^k.\] The second derivatives of \(X\) are also fixed by first jets. To make this explicit, put \(B_{ij}=\nabla_{(i}X_{j)}=-uK_{ij}\). Commuting covariant derivatives gives \[ \begin{split} \nabla_i\nabla_jX_k ={}&\nabla_i B_{jk}+\nabla_j B_{ik}-\nabla_k B_{ij}\\ &+\tfrac12\left( [\nabla_i,\nabla_j]X_k -[\nabla_i,\nabla_k]X_j -[\nabla_j,\nabla_k]X_i\right). \end{split} \tag{240}\] The commutators are curvature times \(X\), and differentiating \(B=-uK\) uses only \(u,\nabla u\) and the background first derivative of \(K\). Thus every derivative of \((u,X,\nabla u,\nabla X)\) is a homogeneous linear expression in that same first jet. This proves the final assertion. ◻

Proof. In a smooth normal collar \((s,y)\) of \(S\), all background coefficients in the first-jet system of Lemma 55 are smooth up to \(s=0\). Along a normal line the system is an ordinary linear differential equation for the full first jet, with bounded coefficients. Solve it down to \(s=0\) from a fixed interior collar surface. Smooth dependence on the initial data and on \(y\) supplies a smooth extension, which agrees with the original solution by uniqueness along each normal line. In particular all its one-sided derivatives have the values supplied by the system. The same system also proves that an interior first jet which vanishes at one point vanishes everywhere on the connected interior, by continuation along piecewise smooth paths.

We next establish the asymptotic form without having prescribed growth to the measures. Let \(Y\) denote all coordinate components of \((u,X)\). The original \(O_2(r^{-q})\), \(O_1(r^{-1-q})\) bounds, the constraints, and Equations (208) and (240) give \[ |\partial^2Y| \le C r^{-1-q'}|\partial Y| +C r^{-2-q'}|Y|, \qquad \tfrac12<q'<\min\{q,1\}. \tag{241}\] Only the original derivative bounds occur here: the curvature and \(\partial K,J\) are \(O(r^{-2-q'})\), and the coefficients of first jets are \(O(r^{-1-q'})\).

Fix a large sphere. Radial integration of Equation (241), with the supremum of \(|Y|/r+|\partial Y|\) up to radius \(r\), bounds this supremum by its initial value plus its integral against \(C r^{-1-q'}\,\mathrm dr\). Gronwall’s inequality therefore gives, uniformly in angle, \[|Y|=O(r),\qquad |\partial Y|=O(1),\qquad |\partial^2Y|=O(r^{-1-q'}).\] Each coordinate gradient has a limit along a radial ray, with error \(O(r^{-q'})\). Two points on a sphere can be joined on that sphere by a path of length \(O(r)\); the last Hessian estimate shows that their gradient limits agree. Thus there is one constant matrix \(L\) such that \[\partial Y=L+O(r^{-q'}),\qquad Y=Lx+O(r^{1-q'}).\] The scalar field \(u\) is nonnegative on every large sphere, so its linear coefficient is zero. Then \(u=O(r^{1-q'})\), and \(|X|\le u\) eliminates every linear coefficient of \(X\) as well. Now Equation (241) improves to \(|\partial^2Y|=O(r^{-1-2q'})\). Since the gradients have limit zero, radial integration gives \(|\partial Y|=O(r^{-2q'})\). As \(2q'>1\), the values have limits along radial rays. Their differences on a large sphere are \(O(r^{1-2q'})\), so there is a single constant limit \(Y_\infty\). With \(Y\) bounded, one more use of Equation (241) gives \[ Y=Y_\infty+O(r^{-q'}),\qquad \partial Y=O(r^{-1-q'}),\qquad \partial^2Y=O(r^{-2-q'}). \tag{242}\]

At this point the original measure moments could still have a component supported on \(S\). Subtract the integration by parts of the smooth interior fields from the compact identity (221). Test with \(p=0\) and a metric variation whose value and full first jet vanish on \(S\). The area and expansion variations and all the smooth integrated boundary terms then vanish. The remaining scalar-curvature variation on \(S\) contains \[-\partial_\nu^2(\mathop{\mathrm{tr}}_S h).\] This is an arbitrary smooth function on \(S\): in collar coordinates one may take the tangential trace to be a prescribed multiple of \(s^2\) times a cutoff. All momentum and frame terms involve only the value or first jet of \(h\) and hence vanish on \(S\). It follows that the scalar moment has no boundary measure. Domination \(|X|\le u\) as measures eliminates a vector boundary measure also.

We can therefore integrate the smooth adjoint by parts all the way to \(S\) and to a large coordinate sphere. Use a prototype from Equation (217), with its cutoff chosen to vanish near \(S\). Its area and expansion variations vanish, and \(a(H)=0\). Its constraint pairing is integrable by Equation (219) and Equation (242). Interior terms vanish by the adjoint equations. If the constants in the latter equation are \((u_\infty,X_\infty)\), its outer boundary term is \[ \begin{split} \int_{S_R}\bigl[&u_\infty(\partial_jh_{ij}-\partial_i h_{jj}) +2X_\infty^j(p_{ij}-(\mathop{\mathrm{tr}}_\delta p)\delta_{ij})\bigr] n^i\,\mathrm dA_\delta+o(1). \end{split} \tag{243}\] For clarity, every omitted term vanishes under the stated decay: the derivatives of \(u,X\) are \(O(r^{-1-q'})\) and multiply \(h=O(r^{-1})\); their nonconstant values multiply \(\partial h,p=O(r^{-2})\); and the metric-variation momentum terms contain \(Kh=O(r^{-2-q'})\). Multiplication by sphere area still leaves \(O(r^{-q'})\). Thus the limit of Equation (243) is \(16\pi(u_\infty E'+X_\infty^iP_i')\).

The scalar prototype has \(E'=1/2\) and \(P'=0\); for the momentum prototype in direction \(p\), \(E'=0\) and \(P'=2p/3\). The latter follows by integrating \(B(p)_{ij}n^j=r^{-2}(p_i+(p\cdot n)n_i)\). Equation (221) for these four independent fluxes consequently gives \[u_\infty=b^0,\qquad X_\infty^i=b^i.\] This proves Equation (205) and in particular shows that the adjoint is nontrivial.

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

Proof. Fix one component and use the local notation \(S\) of the statement. On \(S\), \(X=u\nu\), hence \(N=0\) and its tangential derivatives vanish. The normal-normal component of the KID equation gives \[\langle\nabla_\nu X,\nu\rangle=-uK(\nu,\nu).\] Consequently \[\partial_\nu N =2u\partial_\nu u-2u\langle\nabla_\nu X,\nu\rangle =2u\bigl(\partial_\nu u+uK(\nu,\nu)\bigr) =2u\kappa.\] This proves Equation (256), including positivity on an exterior collar when \(u|_S>0\).

If \(u=0\) on \(S\), then \(X=0\) there and all its tangential covariant derivatives vanish. The normal-normal and normal-tangential components of \(\mathop{\mathrm{sym}}\nabla X=0\) imply \(\nabla_\nu X=0\) on \(S\). Smooth division by \(s\) and then \(s^2\) yields Equation (257); the boundary equation gives \(a|_S=\kappa>0\). Compactness of each \(S_i\) makes its collar assertion uniform. Since there are finitely many disjoint compact components, shrink their collars to be pairwise disjoint and choose a common positive collar width below all of these finitely many widths. Thus every original boundary component has a timelike collar, including when the boundary lapse alternatives differ between components. Finally, Equation (251) gives \(N\to1\). ◻

Proof. The Killing equation makes \(\mathcal G\) antisymmetric. The differentiated Killing equation and \(\mathop{\mathrm{Ric}}_{\mathbf g}(\xi,\cdot)=0\) imply \(\Box_{\mathbf g}\xi=0\). Since \(\mathbf g(\xi,\xi)=-N\), differentiation twice gives \(\Box_{\mathbf g}N=-2|\mathcal G|_{\mathbf g}^2\). For a \(t\)-independent scalar, the inverse metric in Equation (252) has spatial block \(C\) and volume density \(u\sqrt{\det g}\). Its wave operator is therefore \(u^{-1}\mathop{\mathrm{div}}_g(uC\,\,\mathrm d\cdot)\), proving Equation (258).

For completeness, put \(\theta=\,\mathrm dt-A\). The Killing one-form is \(\xi^\flat=-\lambda^2\theta\), and \[\xi^\flat\wedge\,\mathrm d\xi^\flat=-\lambda^4\theta\wedge F.\] The Killing identity \[\,\mathrm d*_\mathbf g(\xi^\flat\wedge\,\mathrm d\xi^\flat) =2*_\mathbf g\bigl(\xi^\flat\wedge\mathop{\mathrm{Ric}}_{\mathbf g}(\xi,\cdot)\bigr)\] has zero right side. Taking the Hodge star in the orthonormal time form \(\lambda\theta\) shows that its left side, up to an irrelevant overall orientation sign, is \(\,\mathrm d(\lambda^3 *_hF)\). It follows that \[\mathop{\mathrm{div}}_h\bigl(\lambda^3 C^{ik}C^{j\ell}F_{k\ell}\bigr)=0.\] Using \(\lambda^3\,\mathrm dV_h=uN\,\mathrm dV_g\) gives Equation (259). For an antisymmetric two-tensor the extra connection term on the free index vanishes, so this change of volume density gives exactly the stated \(g\)-divergence. ◻

Proof. On each such compact neighborhood, \(u\) and \(B\) are bounded below. Indeed \(u>0\) in \(\Omega\) and \(B=u\) on \(\{N=0\}\). The tensor \(C\) acts as the identity on \(T\) and has eigenvalue \(N/u^2\) on \(e\).

We first justify a useful estimate for the smooth nonnegative function \(N\): \[ |\,\mathrm dN|_g^2\leq C N \tag{262}\] on a compact set after slightly enlarging its neighborhood. Choose a uniform normal-coordinate radius and a constant \(L\) bounding the Hessian and large enough that \(|\,\mathrm dN|/L\) is smaller than that radius. Taylor’s inequality on a geodesic in direction \(-\nabla N/|\nabla N|\), at length \(|\,\mathrm dN|/L\), gives \(0\leq N-|\,\mathrm dN|^2/(2L)\). This proves Equation (262).

Let \(n=(\partial_t-X)/u\) be the future unit normal to the stationary slices, and complete \(e\) by an orthonormal pair \(e_1,e_2\) in \(T\). Since \(\xi=un+Be\), \[\mathcal G_{in} =-\frac{N_i/2+B\mathcal G_{ie}}u.\] Expanding all temporal and spatial terms in the full squared norm gives \[ \begin{aligned} -2|\mathcal G|_{\mathbf g}^2 ={}&\frac{|\,\mathrm dN|_g^2}{u^2} +\frac{4B}{u^2}\sum_{a=1}^2N_a\mathcal G_{ae}\\ &-\frac{4N}{u^2}\sum_{a=1}^2\mathcal G_{ae}^2 -4\mathcal G_{12}^2. \end{aligned} \tag{263}\] The fields and their derivatives are bounded on the compact neighborhood. Equations (258), (262), and (263) therefore imply \[ \frac1u\mathop{\mathrm{div}}_g(uC\,\,\mathrm dN) \leq C_0\bigl(N+|\,\mathrm d_TN|\bigr). \tag{264}\]

Multiply by \(u\eta^2/(N+\varepsilon)\) and integrate by parts. With \(E_\varepsilon\) denoting the integral in Equation (260), the result is \[E_\varepsilon -2\int\frac{u\eta C(\,\mathrm d\eta,\,\mathrm dN)}{N+\varepsilon}\,\mathrm dV_g \leq C_0\int u\eta^2\,\mathrm dV_g +C_0\int\frac{u\eta^2|\,\mathrm d_TN|}{N+\varepsilon}\,\mathrm dV_g .\] Cauchy’s inequality bounds the cutoff term by \(E_\varepsilon/4+C\int u|\,\mathrm d\eta|_C^2\,\mathrm dV_g\). Since \(|\,\mathrm d_TN|\leq|\,\mathrm dN|_C\), the last term is bounded by \(E_\varepsilon/4+C\int u\eta^2\,\mathrm dV_g\). This proves Equation (260). Fatou’s lemma on the positive set gives Equation (261), using the local lower bound for \(u\). ◻

Proof. First note the exact contraction identity with its free index lowered using \(g\): \[ g_{ik}Q^{kj}A_j =\frac1u\left( X^j(\,\mathrm dX^\flat)_{ij} -\frac{B^2}{N}(\,\mathrm d_TN)_i \right). \tag{265}\] It follows by inserting \[F=N^{-1}\,\mathrm dX^\flat-N^{-2}\,\mathrm dN\wedge X^\flat \quad\text{and}\quad CX=\frac N{u^2}X\] into \(g_{ik}Q^{kj}A_j\). Both terms on the right of Equation (265) are orthogonal to \(X\). At \(X=0\) its second numerator is interpreted as \(B^2\,\mathrm dN-\,\mathrm dN(X)X^\flat\), so no choice of \(e\) is needed there.

The end asymptotes give a constant one-form \(A_\infty\) with \[A-A_\infty=O_1(r^{-q_0}),\qquad F=O(r^{-1-q_0}).\] Choose a smooth function equal to a linear primitive of \(A_\infty\) near infinity and zero outside a larger end neighborhood. Its differential \(\alpha\) is closed, equals \(A_\infty\) near infinity, and vanishes near \(S\) and all possible interior zeros of \(N\). These zeros are contained in a fixed compact subset of \(\Omega\), by Lemma 58. Put \(\beta=A-\alpha\), so \(\,\mathrm d\beta=F\).

For each component choose a collar cutoff \(\chi_i\) that is zero near \(S_i\) and one outside its collar. Take \(\chi=\chi_{\mathrm{end}}\chi_{\mathrm{null}} \prod_{i=1}^{\ell}\chi_i\), initially supported away from \(S\), infinity, and \(\{N=0\}\). The boundary collars are pairwise disjoint; on the support of \(d\chi_i\) all other boundary factors equal one. Testing Equation (259) by \(\chi\beta_j\) gives \[ \frac12\int_{\mathcal P}\chi uN C^{ik}C^{j\ell}F_{ij}F_{k\ell}\,\mathrm dV_g =-\int_{\mathcal P}Q^{ij}(\partial_i\chi)\beta_j\,\mathrm dV_g. \tag{266}\] The integrand on the left is nonnegative.

We remove the interior cutoff first, with the other two cutoffs fixed. Let it transition from zero to one on \(\varepsilon<N<2\varepsilon\), with derivative bounded by \(C/\varepsilon\). Near that band \(\beta=A\). Equation (265), its orthogonality to \(X\), and \(N\asymp\varepsilon\) bound its contribution by \[C\int_{\{\varepsilon<N<2\varepsilon\}} \bigl(1+|\,\mathrm d_T\log N|^2\bigr)\,\mathrm dV_g\] on a fixed compact set. This tends to zero by Lemma 60. Indeed the bands tend pointwise to the empty set within \(\mathcal P\), and their indicators are dominated by an integrable function there. This reasoning remains valid when \(\{N=0\}\) has positive measure.

Next remove the boundary cutoffs, keeping the end cutoff fixed. In the collar of \(S_i\) let \(s\) be the original normal distance and let \(\chi_i\) transition on \(s\in(\delta,2\delta)\) with derivative bounded by \(C_i/\delta\). For this local computation write \(S=S_i\). If \(u|_S>0\), then \(N\asymp s\), \(X=u\nu+O(s)\), and \(\,\mathrm d_TN=O(s)\). The flux in Equation (265) is bounded. Its orthogonality to \(X\) makes its \(g\)-normal component \(O(s)\), so multiplication by the cutoff derivative \(O(\delta^{-1})\) and integration over the collar gives a term tending to zero. If \(u|_S=0\), Equation (257) gives \[u\asymp s,\quad X=O(s^2),\quad \,\mathrm dX^\flat=O(s), \quad N\asymp s^2,\quad \,\mathrm dN=O(s).\] The entire flux is then \(O(s^2)\), with the same conclusion. Each boundary error therefore tends to zero independently of which lapse alternative holds on that component. The derivative of the finite product is the finite sum of these collar contributions; letting \(\delta\downarrow0\) makes their sum tend to zero.

Finally use a radial cutoff on \(R<r<2R\). Here \(Q=O(r^{-1-q_0})\) and \(\beta=O(r^{-q_0})\); its contribution is \(O(R^{1-2q_0})\), which tends to zero. Fatou’s lemma in Equation (266) now makes its nonnegative integrand vanish everywhere on \(\mathcal P\). Since \(uN>0\) and \(C\) is positive definite there, \(F=0\). ◻

Proof. The unique end of \(M_{\mathrm{ext}}\) lies in \(\mathcal P\) outside a compact set. Any component \(V\) of \(\mathcal P\) other than \(\mathcal E\) therefore has compact closure in the original closed exterior \(M_{\mathrm{ext}}\). Its frontier there lies in \(\{N=0\}\cup S\): an interior frontier point with \(N>0\) has a connected positive neighborhood meeting \(V\) and hence belongs to \(V\), a contradiction. The continuous function \(\lambda=\sqrt N\) is zero on that frontier. The frontier is nonempty; otherwise \(V\) would be open and closed in the connected open exterior \(\Omega\), which also contains the far end in \(\mathcal E\).

Choose any point of \(V\), where \(\lambda>0\). Compactness of the closure and the zero frontier values imply that the positive maximum of \(\lambda\) is attained at an interior point of \(V\). The strong maximum principle for the smooth locally elliptic equation \(\Delta_h\lambda=0\) makes \(\lambda\) constant on \(V\), contradicting its zero frontier values. Thus no such component exists. Every boundary collar from Lemma 58 is a subset of \(\mathcal P=\mathcal E\), as claimed. ◻

Proof. Near an interior zero set, \(B,u\) have positive lower bounds. Contracting \(F=0\) with \(X\) in Equation (265) gives \[|\,\mathrm d_T\log N|\leq B^{-2}|X|\,|\,\mathrm dX^\flat|\leq C.\] Together with Equation (262), this yields \[ |\,\mathrm d\log N|_h^2 =|\,\mathrm d_T\log N|^2+\frac{|\,\mathrm d_eN|^2}{u^2N}\leq C \tag{269}\] on compact neighborhoods of the interior zero set. A finite-length path approaching that set would make \(\log N\to-\infty\), contradicting this bound.

Every possible limit of \(\mathcal E\) in the compact part of the original closed exterior either belongs to \(\mathcal E\), lies in \(\{N=0\}\), or belongs to \(S\). At the latter two types of points \(\lambda\to0\), while \(\lambda\to1\) at the AF end. If \(\lambda>1\) somewhere, a positive superlevel set bounded away from one is contained in a compact subset of \(\mathcal E\). The harmonic function would attain an interior maximum, forcing it constant, contrary to its limit at infinity. Thus \(\lambda\leq1\). If equality holds at one interior point, the strong maximum principle gives \(\lambda\equiv1\) on \(\mathcal E\). That component then has no interior boundary, since \(N\) remains one at any such boundary point. It is open and closed in connected \(\Omega\), hence equals \(\Omega\), contradicting the timelike collar where \(\lambda\to0\) at \(S\). Therefore \(0<\lambda<1\).

For boundary regularity fix a component \(S_i\) and write \(S\) for it through the following local calculation. Suppose first that \(u|_S>0\). By Equation (256), use \((N,y^1,y^2)\) as original smooth collar coordinates. Smooth division gives \[X^\flat=a(N,y)\,\mathrm dN+N\beta_A(N,y)\,\mathrm dy^A,\qquad a(0,y)=\frac1{2\kappa}.\] Writing \(N=\lambda^2\), the coefficients of \(h\) are \[ \begin{aligned} h_{\lambda\lambda}&=4\lambda^2g_{NN}+4a^2,\\ h_{\lambda A}&=2\lambda(g_{NA}+a\beta_A),\\ h_{AB}&=g_{AB}+\lambda^2\beta_A\beta_B, \end{aligned} \tag{270}\] with all right-side coefficients evaluated at \((\lambda^2,y)\). They extend smoothly to negative \(\lambda\). The map \((\lambda,y)\mapsto(-\lambda,y)\) is an isometry: the diagonal blocks are even and the mixed block is odd. At zero, \(h_{\lambda\lambda}=\kappa^{-2}\), \(h_{\lambda A}=0\), and \(h_{AB}=g_{AB}|_S\). The fixed hypersurface of this reflection is totally geodesic, and all assertions in Equation (268) follow.

If \(u|_S=0\), put \(D=a^2-s^2|Y|^2\) in Equation (257). Then \[h=g+s^2D^{-1}Y^\flat\otimes Y^\flat,\qquad \lambda=s\sqrt D,\qquad D|_S=\kappa^2.\] These are smooth in the original one-sided collar. The static equations extend by continuity to give \(\mathop{\mathrm{Hess}}_h\lambda=0\) at \(S\). Its tangential components are \(\kappa\,\mathrm{II}_h\), so \(\mathrm{II}_h=0\). In \(h\)-Gaussian coordinates \(\rho\geq0\), write \(h=\,\mathrm d\rho^2+h_\rho\). Then \(\partial_\rho h_\rho|_0=0\), and \(\partial_\rho^2\lambda|_0=0\). Even reflection of \(h_\rho\) and odd reflection of \(\lambda\) are therefore \(C^2\).

These constructions apply on the finitely many disjoint collars, with no compatibility condition between different lapse alternatives. They attach the entire original \(S\), by Lemma 62. Let \(\overline{\mathcal E}\) denote this attached base and let \(j:\overline{\mathcal E}\to M_{\mathrm{ext}}\) be its identity map on the interior and on boundary points. It is a homeomorphism onto the original exterior with any interior zero set removed, but need not be a boundary diffeomorphism: on a positive-lapse collar the original coordinate is \(N=\lambda^2\). The identity \(h=g+N^{-1}X^\flat\otimes X^\flat\) gives \(h\geq j^*g\) in the interior and, by continuity in each regular collar, on the attached boundary as a nonnegative quadratic-form inequality.

It remains to check completeness without classifying the missing sets. The original exterior is complete as a metric space with boundary: a finite-length Cauchy path is Cauchy in the complete ambient initial-data manifold and its limit remains in the closed exterior. The inequality \(h\geq j^*g\) makes the image of a finite \(h\)-length escaping path have such an original limit. An interior limit with \(N>0\) is an ordinary smooth base limit. An interior zero is impossible by Equation (269). A limit in \(S\) is accounted for by the regular collar just constructed. The AF end has infinite distance. These possibilities exhaust finite-length escapes, proving completeness of \(\overline{\mathcal E}\) with its boundary included.

Take two copies of this base and identify corresponding points of each \(S_i\) by the collar reflections just constructed. The resulting double is connected because the base is connected and its boundary is nonempty. Give the second copy the opposite orientation; the identifications yield an orientable manifold without boundary. There is no assertion that this double is simply connected. The reflected metric \(h_{\mathrm d}\) and odd reflected lapse are \(C^2\) across every join, and smooth at the positive-lapse joins. The static scalar and harmonic equations hold continuously across the joins, since the indicated second jets match.

For completeness, fold the double onto \(\overline{\mathcal E}\) by identifying the two copies. The fold is distance nonincreasing, because lengths on either side are the base lengths. A Cauchy sequence in the double projects to a Cauchy sequence in the complete base. The base is a complete locally compact length space, hence is proper, so the projected sequence lies in a compact base set. The preimage of that set is a quotient of two compact copies and is compact. The original sequence consequently has a convergent subsequence and, being Cauchy, converges. This proves completeness of the double. ◻

Proof. The three-dimensional conformal formulas applied to Equation (267) give \[ \mathop{\mathrm{Ric}}_\gamma=2\,\mathrm dU\otimes\,\mathrm dU,\qquad \Delta_\gamma U=0. \tag{272}\] Initially \(\gamma-\delta,U=O_2(r^{-q_0})\). We give the coordinate construction with this derivative count.

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

Put \(s_0=1-q_0\in(0,1/2)\). For a source with weighted scaled \(C^{0,\alpha}\) size \(O(r^{s_0-2})\), the Euclidean inverse normalized to vanish at zero is \[ Pf(x)=-\frac1{4\pi}\int_{\mathbb R^3} \left(\frac1{|x-y|}-\frac1{|y|}\right)f(y)\,\mathrm dy. \tag{273}\] It maps that weighted norm boundedly to scaled \(C^{2,\alpha}\) size \(O(r^{s_0})\). To verify the zeroth-order bound, split at \(|y|=2|x|\). On the far part, the kernel difference is bounded by \(C|x||y|^{-2}\), and its integral is \(O(|x|^{s_0})\) because \(s_0<1\). On the near part the separate kernels have the same bound because \(s_0>0\). Interior Poisson estimates on rescaled annuli give the derivative and Hölder bounds. The ordinary local estimates also apply on the fixed inner ball.

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

There is a derivative issue in changing metric coordinates which must be addressed before using a classical Ricci equation. The original assumptions bound \(a,c\), respectively, in scaled \(W^{2,\infty}\) and \(W^{1,\infty}\). Differentiate the equation for \(v\) once and apply local \(W^{2,p}\) estimates to \(Dv\) on rescaled annuli. All differentiated coefficients and forcing are controlled by these bounds and the preceding \(C^{2,\alpha}\) estimate. For every finite \(p\) this gives \[v=O_{W^{3,p}}(r^{1-q_0}).\] Consequently the transformed \(\gamma-\delta,U\) have scaled \(W^{2,p}\) size \(O(r^{-q_0})\). Take \(p>3\); Morrey embedding gives scaled \(C^{1,\alpha}\) control for some \(\alpha>0\). In the harmonic chart Equation (272) has the elliptic form \[-\tfrac12\gamma^{ab}\partial_{ab}\gamma_{ij} +Q_{ij}(\gamma^{-1},D\gamma)=2U_iU_j,\qquad \gamma^{ab}\partial_{ab}U=0,\] where \(Q_{ij}\) is quadratic in \(D\gamma\). Schauder estimates first give scaled \(C^{2,\alpha}\) control; differentiating this elliptic system then gives \(\gamma-\delta,U=O_k(r^{-q_0})\) for each finite \(k\). No higher original-coordinate derivative bound has been assumed.

We use two elementary exterior Poisson estimates. If \(w=o(1)\), \(\Delta_\delta w=f\), and \(f=O_k(r^{-2-t})\), \(1<t<2\), then \[ w=\frac ar+O_k(r^{-t}). \tag{274}\] To prove this, cut \(w\) off on a fixed inner annulus and extend it by zero; this changes \(f\) only on a compact set. The extended solution equals the Newton potential of its Laplacian, since their difference is an entire decaying harmonic function. The coefficient is \(a=-(4\pi)^{-1}\int f\). For \(|y|<r/2\), subtracting \(1/r\) from the kernel costs \[Cr^{-2}\int_{|y|<r/2}|y||f(y)|\,\mathrm dy=O(r^{-t}).\] The remaining region, including the integrable kernel singularity, has the same bound. Rescaled interior estimates prove the derivative version. If instead \(f=O_k(r^{-4-\epsilon})\), \(0<\epsilon<1\), its first moments converge and the same reasoning gives \[ w=\frac ar+\frac{d\cdot x}{r^3}+O_k(r^{-2-\epsilon}). \tag{275}\] Here subtract \(1/r+(x\cdot y)/r^3\) in the near region. The remainder costs \(Cr^{-3}\int_{|y|<r/2}|y|^2|f(y)|\,\mathrm dy =O(r^{-2-\epsilon})\); the far region has that order as well.

Set \(\epsilon=2q_0-1\in(0,1)\). The harmonic-coordinate equations and initial symbol bounds yield \[\Delta_\delta(\gamma_{ij}-\delta_{ij}), \ \Delta_\delta U=O_k(r^{-2-2q_0}).\] Equation (274) gives \[\gamma_{ij}=\delta_{ij}+\frac{A_{ij}}r +O_k(r^{-1-\epsilon}),\qquad U=\frac ar+O_k(r^{-1-\epsilon}).\] The harmonic-coordinate condition is \(\partial_j(\sqrt{\det\gamma}\,\gamma^{ij})=0\). Expanding it at the displayed order gives \[\left(A_{ij}-\tfrac12(\mathop{\mathrm{tr}}A)\delta_{ij}\right) \frac{x^j}{r^3}=O(r^{-2-\epsilon}).\] Taking a radial limit shows \(A=\tfrac12(\mathop{\mathrm{tr}}A)I\). Its trace in dimension three is zero, hence \(A=0\). Since \(D^2U=O_k(r^{-3})\), the scalar equation now improves to \[\Delta_\delta U =(\delta^{ab}-\gamma^{ab})\partial_{ab}U =O_k(r^{-4-\epsilon}).\] Equation (275) proves Equation (271), initially with a real \(M\).

Its positivity uses the lapse range from Lemma 63. The function \(-U>0\) is \(\gamma\)-harmonic. For \(0<\eta<1\) and a fixed positive \(C\), the function \[w=r^{-1}+Cr^{-1-\eta}\] is positive and strictly \(\gamma\)-subharmonic sufficiently far out: \(\Delta_\gamma r^{-1}=O(r^{-4-\epsilon})\), while the leading Laplacian of \(r^{-1-\eta}\) is \(\eta(1+\eta)r^{-3-\eta}\). Choose \(c>0\) so small that \(-U\geq cw\) on a large inner sphere. The difference tends to zero and is superharmonic, so the exterior minimum principle gives \(-U\geq cw\) throughout that end. Taking the radial limit proves \(M\geq c>0\). ◻

Proof. Scalar flatness follows from Equation (267) and the three-dimensional conformal scalar-curvature formula, since \(1\pm\lambda\) are harmonic. In terms of the preceding variables, \[ h_+=\cosh^4(U/2)\gamma,\qquad h_-=\sinh^4(U/2)\gamma. \tag{276}\] The first factor is \(1+O_k(r^{-2})\), proving the assertion for \(h_+\). Its ADM flux is \(O(r^{-\epsilon})\), so its mass is zero.

For the minus end invert \(x=y/|y|^2\), write \(\rho=|y|\), and put \(R(y)=I-2y\otimes y/\rho^2\). The inversion differential is \(\rho^{-2}R(y)\), with \(R(y)^TR(y)=I\). Equation (271) gives \[I^*\gamma=\rho^{-4} \bigl(I+O_k(\rho^{1+\epsilon})\bigr), \qquad U(I(y))=-M\rho+(b_1\cdot y)\rho +O_k(\rho^{2+\epsilon}).\] Derivatives of the angular matrix \(R(y)\) cost the corresponding powers of \(\rho^{-1}\), so the displayed metric estimate retains the symbol bounds. Dividing the expansion of \(\sinh(U/2)\) by \(\rho\), and using \(\epsilon<1\), gives \[ I^*h_-=\left(\frac M2\right)^4 \left[ \left(1-\frac{4b_1\cdot y}{M}\right)I +O_k(\rho^{1+\epsilon}) \right]. \tag{277}\] In particular, its only first-order term is linear and scalar.

For the remainder \(E(y)\), the bounds \[|E|\leq C\rho^{1+\epsilon},\quad |DE|\leq C\rho^\epsilon,\quad |D^2E|\leq C\rho^{\epsilon-1}\] show that \(E(0)=DE(0)=0\) extends it as \(C^{1,\epsilon}\). To see the Hölder estimate between two points, use the gradient bound when their separation is comparable to their radii; otherwise integrate the Hessian along the short segment in a common annulus. Moreover choose \[3<p<\frac3{1-\epsilon}.\] Then \(D^2E\in L^p\). Integration by parts across small coordinate spheres has an error bounded by a constant times their area, because the first derivatives are bounded. It follows that the punctured classical derivatives are the weak derivatives of the extension. The Christoffel symbols are continuous with weak derivatives in \(L^p\), so scalar curvature is its usual \(L^p\) expression in \(\partial\Gamma\) and \(\Gamma\Gamma\). It vanishes off the point and therefore also as a distribution on the whole ball. There is no point curvature term. ◻

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

The weak Lichnerowicz formula has the following integrated form for every compactly supported \(H^1\) spinor \(\phi\): \[ \|D\phi\|_2^2=\|\nabla^S\phi\|_2^2. \tag{278}\] Here and below norms and integrals are for \(\mathfrak g\). To check the formula at the indicated regularity, first use a smooth compactly supported spinor. Approximate the metric on its support in \(C^1\) and \(W^{2,p}\) by smooth metrics, using the same background-bundle identification. The smooth Lichnerowicz formula passes to the limit: connection coefficients converge uniformly and scalar curvatures converge in \(L^p\). The limiting scalar term is zero by the assumed distributional scalar flatness and the \(L^p\) curvature expression. Density then proves Equation (278) for compact \(H^1\) spinors. Thus no integration surface is retained around the compactified point.

We next establish the coercivity needed to complete compact spinors. For a compactly supported scalar function \(f\) on the AE end, one-dimensional integration by parts along coordinate rays gives \[ \int_{r_0}^\infty f^2\,\mathrm dr \leq4\int_{r_0}^\infty r^2|\partial_rf|^2\,\mathrm dr. \tag{279}\] The inner boundary term has the favorable sign. Apply this to \(f=|\phi|\) and integrate over angles. Metric comparability and \(|\,\mathrm d|\phi||\leq|\nabla^S\phi|\) give \[ \int_{\mathrm{AE}}r^{-2}|\phi|^2\,\mathrm dV_{\mathfrak g} \leq C\|\nabla^S\phi\|_2^2. \tag{280}\] For any fixed compact \(K\subset W\), connect it to a fixed end annulus by a relatively compact connected domain. Poincaré’s inequality with the norm on that annulus retained gives \[\||\phi|\|_{L^2(K)} \leq C_K\left( \|\,\mathrm d|\phi|\|_{L^2(\text{domain})} +\|\phi\|_{L^2(\text{annulus})}\right).\] For example, this version follows from the ordinary Poincaré inequality after subtracting the mean; the observed annulus controls that mean because it has positive measure. Equation (280) controls its last term. We have proved \[ \|\phi\|_{L^2(K)}\leq C_K\|\nabla^S\phi\|_2. \tag{281}\] Connectedness is the only global input in this propagation.

Let \(\mathcal H\) be the completion of compactly supported smooth spinors in the norm \(\|\nabla^S\phi\|_2\). Equation (281) embeds \(\mathcal H\) in \(L^2_{\mathrm{loc}}\), and the locally bounded connection then embeds it in \(H^1_{\mathrm{loc}}\). Both \(\nabla^S:\mathcal H\to L^2\) and \(D:\mathcal H\to L^2\) are well-defined, and Equation (278) makes the latter an isometry. Its range is therefore closed. Equation (280) also passes to this completion.

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

Completeness supplies compactly supported Lipschitz distance cutoffs \(\chi_R\), equal to one on the radius-\(R\) ball and with \(|\,\mathrm d\chi_R|\leq C/R\). Indeed a complete locally compact Riemannian length space is proper, and a piecewise linear cutoff of its distance has these properties. Equation (278) applies to \(\chi_Rw\), and the weak equation gives \[\|\nabla^S(\chi_Rw)\|_2^2 =\|D(\chi_Rw)\|_2^2 =\|c(\,\mathrm d\chi_R)w\|_2^2 \leq \frac C{R^2}\|w\|_2^2.\] On every fixed compact set the cutoff eventually equals one. It follows that \(\nabla^S w=0\). A parallel spinor has constant length on the connected manifold; the distinguished end has infinite volume, so \(w\in L^2\) forces \(w=0\). No condition at another end entered this argument.

It remains to show that \(\psi\), rather than just \(D\psi\), is parallel. Define the continuous sesquilinear form \[\mathcal B(a,b)= \int\left(\langle\nabla^Sa,\nabla^Sb\rangle -\langle Da,Db\rangle\right)\,\mathrm dV_{\mathfrak g}\] on fields for which the indicated derivatives are in \(L^2\). For compact \(\phi\), the polarized Lichnerowicz identity, with one slot compactly supported, gives \(\mathcal B(\psi_0,\phi)=0\). Approximate \(\eta\) in \(\mathcal H\) by compact spinors. Continuity and Equation (278) imply \[\mathcal B(\psi_0,\eta)=0,\qquad \mathcal B(\eta,\eta)=0,\qquad \mathcal B(\psi,\psi)=\mathcal B(\psi_0,\psi_0).\] The last expression is zero. To see this directly, integrate the Lichnerowicz identity for \(\psi_0\) up to a large end sphere. There is no contribution at another end because \(\psi_0\) vanishes there. On that sphere \(\psi_0=O(1)\) and \(\nabla^S\psi_0,D\psi_0=O(r^{-2-\epsilon})\); the boundary integral is \(O(r^{-\epsilon})\) and tends to zero. This is also the zero ADM boundary term of the spin proof. Since \(D\psi=w=0\), we conclude that \(\|\nabla^S\psi\|_2^2=0\).

For \(z\ne0\) the resulting parallel spinor cannot be zero. Otherwise \(\eta=-\psi_0\), contradicting Equation (280), because a nonzero constant spinor has infinite \(r^{-2}\)-weighted \(L^2\) norm on the end. The construction is linear in \(z\), so it produces a full complex basis of parallel spinors at each point. Their parallel equations first imply \(C^{1,\beta}\) regularity locally for some \(\beta>0\): from \(H^1\subset L^6\) and the bounded connection one obtains \(W^{1,6}\), hence local Hölder continuity. The connection coefficients are locally Hölder by the metric regularity, so the parallel equations make the first derivatives locally Hölder as well. Their curvature equations hold weakly. The spin representation is faithful on the Lie algebra \(\mathfrak{spin}(3)\), so a full parallel spinor frame makes the Riemann curvature zero. Equivalently, its spinor bilinears give a global parallel orthonormal tangent frame. The dual coframe is closed by torsion-freeness and integrates locally to coordinates in which the metric is exactly Euclidean. This also removes any apparent metric singularity at the compactified point.

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

Proof. The complete base of Lemma 63 reaches every component of \(S\). Take its complete two-copy double, use \(h_+\) on the plus copy and \(h_-\) on the minus copy, and identify corresponding boundary points. The signed reflected lapse expresses the joined metric as \[q=\left(\frac{1+\lambda_{\mathrm{signed}}}{2}\right)^4h_{\mathrm d}.\] It is smooth at each positive-lapse join and \(C^2\) at each zero-lapse join. The matched second jets make its scalar curvature zero across every join, as well as on the two open sides. Compactify the unique AF end of the minus copy by Lemma 65.

We verify completeness of this conformal manifold without placing any restriction on the number or regularity of the possible internal-null ends. On the plus copy the factor \((1+\lambda)/2\) is at least \(1/2\). Fix a sufficiently large original AF tail. On its complement the continuous function \(\sqrt N\) is defined on a compact subset of the original closed exterior, equals zero on \(S\) and the internal zero set, and is strictly less than one elsewhere by Lemma 63. Its maximum on this compact set is therefore some \(a_0<1\). Thus on the entire corresponding minus remainder \((1-\lambda)/2\geq(1-a_0)/2>0\) uniformly. On the double away from the minus AF tail the conformal metric is consequently bounded below by a fixed positive multiple of the complete doubled base metric. All joins are already attached. A finite-length escape can therefore only run down the minus AF tail, and its proved nondegenerate one-point compactification supplies its limit. More explicitly, truncate that tail at a large sphere. The other side of the sphere is complete with compact boundary by the uniform lower bound, and the compactified tail is a compact metric region with that same boundary; their gluing is complete.

The conformal manifold is connected, orientable, and without boundary. It has local \(C^{1,\alpha}\cap W^{2,p}\) regularity for some \(\alpha>0\) and \(p>3\): the finitely many joins are at least \(C^2\), and the sole compactification point has the regularity established in Lemma 65. Its scalar curvature is zero distributionally, including at that point, and its distinguished plus end has the fast decay required by Lemma 66. That lemma makes this entire manifold Euclidean space.

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

It is a nonempty open and closed subset of connected \(\Omega\), and hence equals \(\Omega\). Vacuum now follows from Proposition 50, since \(N>0\) everywhere in \(\Omega\). ◻

Proof. The completed conformal double in the proof of Lemma 67 is Euclidean. In particular its plus metric \(q=h_+\) is flat up to every original boundary component. Put \(\phi=(1+\lambda)/2\). On each \(S_i\), Lemma 63 gives \[\phi=\tfrac12,\qquad \partial_{\nu_h}\log\phi=\kappa,\qquad \mathrm{II}_h=0,\qquad h|_{TS_i}=g|_{TS_i}.\] The conformal second-fundamental-form formula, with normal pointing into the plus side, is therefore \[\mathrm{II}_q =\phi^2\bigl(\mathrm{II}_h+ 2\partial_{\nu_h}\log\phi\,h|_{TS_i}\bigr) =8\kappa\,q|_{TS_i}, \qquad q|_{TS_i}=\tfrac1{16}g|_{TS_i}.\] Let \(F_i\) be the Euclidean position vector on \(S_i\) and \(\nu_i\) its unit normal into the plus side. Its shape operator is \(8\kappa I\), so the tangential differential of \(F_i-a\nu_i\) is zero for \(a=(8\kappa)^{-1}\). Connectedness makes \(F_i-a\nu_i\) a constant point \(c_i\). Thus the image lies on the sphere of radius \(a\) centered at \(c_i\), with \(\nu_i\) its outward radial normal. The image is open in that sphere by local invertibility and closed by compactness, hence is the whole sphere. The Euclidean embedding therefore identifies \(S_i\) diffeomorphically with it. Since \(q|_{TS_i}=g|_{TS_i}/16\), its area satisfies \[ |S_i|_g=16|S_i|_q=64\pi a^2 =\frac{\pi}{\kappa^2}=16\pi m^2=|S|_g. \tag{282}\] Summing over the finite family yields \(|S|_g=\sum_{i=1}^{\ell}|S_i|_g=\ell|S|_g\). The boundary is nonempty and \(|S|_g>0\), hence \(\ell=1\). In particular no connectedness or spherical topology assumption was needed before this step. ◻

Proof. Lemma 68 identifies the boundary in the Euclidean conformal double as one round sphere of radius \(a=(8\kappa)^{-1}\). The plus side is its outside: its normal is the outward radial normal, its entire boundary is this sphere, and it contains the distinguished unbounded end. Center Euclidean coordinates at this sphere and write \(\rho\) for their radius.

In the resulting Euclidean coordinates, write \(\psi=2/(1+\lambda)\), so \(h=\psi^4\delta\). The scalar-flat conformal equation gives \(\Delta_\delta\psi=0\) outside the sphere, with \(\psi=2\) on it and \(\psi\to1\) at infinity. The function \(1+a/\rho\) has the same data. Their difference vanishes by the exterior maximum principle. This proves Equation (283). Its Schwarzschild mass is \(2a=(4\kappa)^{-1}=m\), since \(\kappa=1/(4m)\). In particular \(a=m/2\).

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

Proof of Proposition 57. Lemma 68 proves \(\ell=1\) and the spherical topology of \(S\). Lemma 67 proves \(N>0\) on \(\Omega\), and Lemma 69 gives the global static base with mass \(m\). For the reconstruction write \(M=m\), so \(\kappa=1/(4M)\). Simple connectivity and Lemma 61 give a global smooth function \(H\) with \[ \,\mathrm dH=-A. \tag{284}\] In static Schwarzschild coordinates use time \(T=t+H(x)\). Equations (254) and (284) give the exact identity \[ -N\,\mathrm dT^2+h =-u^2\,\mathrm dt^2+g_{ij}(\,\mathrm dx^i+X^i\,\mathrm dt) (\,\mathrm dx^j+X^j\,\mathrm dt). \tag{285}\] The graph \(T=H(x)\), corresponding to \(t=0\), therefore induces the original metric \(g\). The stationary future unit normal is \(n=(\partial_t-X)/u\). With the second-fundamental-form convention of the theorem, \[ K_{\mathrm{graph}} =\frac1{2u}\bigl(\partial_tg-\mathcal L_Xg\bigr) =-\frac1{2u}\mathcal L_Xg =K. \tag{286}\] This calculation recovers the original \(K\), including a non-time-symmetric slice.

We next establish smoothness in the original boundary structure. If \(u|_S=0\), Equation (257) gives \[A=\frac{Y^\flat}{a^2-s^2|Y|_g^2}.\] This is smooth in the original collar. The primitive \(H\) therefore extends smoothly and finitely to \(S\). For instance, fix its values on one interior collar section and integrate its smooth normal derivative to the boundary. Tangential derivatives of this extension agree with those prescribed by \(\,\mathrm dH=-A\), by continuity from the interior.

If \(u|_S>0\), the numerator \(X^\flat-(2\kappa)^{-1}\,\mathrm dN\) vanishes as a one-form on \(S\), by Equation (256). Smooth division by the defining function \(N\) gives \[ A=\frac1{2\kappa}\,\mathrm d\log N+\beta,\qquad H=-\frac1{2\kappa}\log N+B, \tag{287}\] where \(\beta\) and \(B\) are smooth in the original collar. The smoothness of \(B\) follows by applying the same normal-integration argument to its differential \(-\beta\). In this case \(H\to+\infty\) at the boundary, so static time itself is not a regular attachment coordinate.

Let \(r\) be the Schwarzschild area radius. On the whole exterior \[N=1-\frac{2M}{r},\qquad r=\frac{2M}{1-N}.\] Normalize the tortoise coordinate and Kruskal coordinates by \[ \begin{aligned} r_*&=r+2M\log\left(\frac r{2M}-1\right),\\ \mathsf U&=-e^{-\kappa(T-r_*)},& \mathsf V&= e^{\kappa(T+r_*)}. \end{aligned} \tag{288}\] On the right exterior \(\mathsf U<0,\mathsf V>0\), and the Schwarzschild metric extends as \[\mathbf g_M =-\frac{32M^3}{r}e^{-r/(2M)} \,\mathrm d\mathsf U\,\,\mathrm d\mathsf V+r^2\,\mathrm d\omega^2.\] An exact useful form of Equation (288) is \[ \mathsf U=-Q(N)\sqrt N\,e^{-\kappa T},\qquad \mathsf V= Q(N)\sqrt N\,e^{\kappa T},\qquad Q(N)=\frac{\exp\!\bigl(1/[2(1-N)]\bigr)}{\sqrt{1-N}}. \tag{289}\] The function \(Q\) is smooth and positive near zero, with \(Q(0)=\sqrt e\).

If \(u|_S=0\), both \(H\) and \(\sqrt N=s\sqrt{a^2-s^2|Y|^2}\) are smooth. Substitution into Equation (289) gives \(\mathsf U=\mathsf V=0\) on \(S\) and \[\partial_s\mathsf U|_S =-\sqrt e\,\kappa e^{-\kappa H|_S}<0,\qquad \partial_s\mathsf V|_S =\sqrt e\,\kappa e^{\kappa H|_S}>0.\] This is a smooth attachment at the bifurcation sphere. If \(u|_S>0\), Equation (287) instead gives \[ \mathsf U=-Q(N)N e^{-\kappa B},\qquad \mathsf V=Q(N)e^{\kappa B}. \tag{290}\] These are smooth in the original collar; \(\mathsf U\) vanishes simply and \(\mathsf V|_S=\sqrt e\,e^{\kappa B|_S}>0\). Thus the attachment is on the future horizon, with each generator met once after the angular identification.

The angular regularity requires a further check when \(u|_S>0\), because the regular base coordinate was \(\lambda\), not the original \(N\). On the smooth reflected collar in Equation (270), normal projection onto its fixed boundary is invariant under \(\lambda\mapsto-\lambda\). This follows from uniqueness of the normal geodesics and reflection invariance of the metric. Schwarzschild angular coordinates are constant along these normal geodesics, and their boundary values are the smooth round-sphere identification from Lemma 69. They are therefore smooth even functions of \(\lambda\). A smooth even function, with smooth tangential parameters, is a smooth function of \(N=\lambda^2\) for \(N\geq0\): all odd Taylor coefficients vanish, and repeated application of \((2\lambda)^{-1}\partial_\lambda\) to the Taylor remainder gives continuous derivatives of every order at zero. This proves the required angular smoothness in the original collar. In the zero boundary-lapse case the original collar is already a smooth base collar, so its completed base isometry supplies the angular extension directly.

We have constructed a smooth map \(\iota:M_{\mathrm{ext}}\to\) Schwarzschild. Equation (285) extends to \(S\) by continuity, hence \(\iota^*\mathbf g_M=g\) there as well. The positive definiteness of \(g\) proves that \(\,\mathrm d\iota\) is injective at every boundary point. Interior injectivity follows from the global base isometry; boundary injectivity follows from the sphere identification. The boundary image, at \(r=2M\), is disjoint from the open image, where \(r>2M\). The map is proper: a compact subset of Schwarzschild has bounded area radius, and its inverse image is closed in the compact base region corresponding to \([2M,R]\times S^2\). Thus the injective immersion is an embedding.

The future normal extends smoothly even where \(u=0\). To avoid taking a limit of the expression involving \(u^{-1}\), choose the smooth future timelike Kruskal vector \(Z=\partial_{\mathsf U}+\partial_{\mathsf V}\) near the boundary. Let \(Z^\top\) be its tangential projection along the embedded spacelike hypersurface and set \[\widehat n= \frac{Z-Z^\top} {\sqrt{-\mathbf g_M(Z-Z^\top,Z-Z^\top)}}.\] The denominator is positive, and this is a smooth future unit normal. On the open exterior it agrees with the normal used in Equation (286). Its second fundamental form is smooth and equals \(K\) in the interior, so equality extends to \(S\).

There is also a smooth spacelike extension of the image through its boundary. In the positive boundary-lapse case, use \((\mathsf U,\omega)\) as hypersurface coordinates; its normal derivative is nonzero by Equation (290). The remaining null coordinate is a smooth graph function and can be extended across \(\mathsf U=0\). In the zero boundary-lapse case use \((\chi,\omega)\), where \(\chi=(\mathsf V-\mathsf U)/2\); the displayed normal derivatives give \(\partial_s\chi>0\). Again extend the remaining graph function across zero. Smooth one-sided graph functions extend across a smooth boundary, and compactness of \(S\), together with openness of the spacelike condition, preserves spacelikeness on a sufficiently short collar. This extension makes no assertion about the original data behind \(S\).

Finally, the exact inverse-metric identity gives \[ |\,\mathrm dH|_h^2 =\left|\frac{X^\flat}{N}\right|_C^2 =\frac{|X|_g^2}{u^2N} \longrightarrow\frac{|b|^2}{(b^0)^2}<1. \tag{291}\] Choose a constant \(a_0<1\) larger than the limiting slope norm. Along radial rays in the Schwarzschild base, uniformly in angle, \[|H(r,\omega)| \leq C+a_0\int_R^r \frac{\,\mathrm d\rho}{\sqrt{1-2M/\rho}} =a_0r+O(\log r).\] Since \(r_*=r+O(\log r)\), it follows that \[H-r_*\longrightarrow-\infty,\qquad H+r_*\longrightarrow+\infty.\] This is approach to the chosen Schwarzschild spatial infinity. It allows a nonzero asymptotic tilt and does not impose \(P=0\) or \(K=0\). All conclusions of Proposition 57 follow. ◻

Proof. After forgetting \(K\) and the fields, Definition 70 gives an exterior in Definition 1, with the same boundary \(S\). Lemma 45 therefore gives \(0<a_g(S)<\infty\).

For the perimeter assertion, apply Lemma 5 to the constant sequence \(g_j=g\). Local convergence is immediate. The falloff in Equation (293) gives uniform end bounds \(c\delta\le g\le C\delta\) and \(r_x|\partial g|\le C\), and a fixed large coordinate sphere is a compact enclosing competitor of finite area. The lemma consequently identifies the smooth and perimeter infima. Its competitors are the same bounded filled inner sets used here: they retain the entire obstacle, bounded complementary pockets are filled, and the full frontier, including contact with \(S\), is counted. Its outward smoothing preserves enclosure and proves the claimed equality of infima. The auxiliary filling in that perimeter argument requires no extension of the physical data or energy condition. ◻

Proof. Apply Theorem 2 with its exterior \(\Omega=N\). Definition 70, after the field requirements are omitted, uses the same smoothness, orientation, one-endedness and completeness convention as Definition 1. Equations (294), (297) and (298) are exactly its constraint and ADM conventions. The decay, integrability, weak expansion and future timelike hypotheses in the proposition are precisely those of that theorem. The class of filled enclosing cuts is also identical: all boundary components are enclosed, bounded pockets are filled, and coincident obstacle frontier is counted. Hence its enclosing infimum is \(a_g(S)\), giving the displayed inequality. The theorem is one-sided and requires no extension across \(S\). ◻

Proof. The conformal scalar-curvature formula and the scaling of the tensor terms in (294) give \[16\pi u^4\mu'=16\pi\mu-8u^{-1}\Delta_gu.\] For \(f=\log u\), the difference of the two connections is \[\nabla'_AB-\nabla_AB =2\bigl(df(A)B+df(B)A-g(A,B)\nabla f\bigr).\] Taking the divergence of the rescaled trace reversal \(u^2(K-(\mathop{\mathrm{tr}}_gK)g)\) therefore yields \[8\pi u^2J'=8\pi J+4K(\nabla\log u,\cdot).\] The new field vectors are \(X_i'=u^{-6}X_i\), so their components in orthonormal frames scale by \(u^{-4}\) and \(I_{g'}(u^{-2}Y)=u^{-8}I_g(Y)\). Consequently \[u^4D_{g'}(u^{-2}Y) =D_g(Y)+(1-u^{-4})I_g(Y) +4u^{-1}\bigl(-\Delta_gu+K(\nabla u,Y)\bigr).\] Dropping the nonnegative second term and estimating the last tensor contraction proves (306). Finally, \(H'=u^{-2}(H+4\partial_\nu\log u)\) and \(\mathop{\mathrm{tr}}_\Sigma^{g'}K'=u^{-2}\mathop{\mathrm{tr}}_\Sigma^gK\), which give (307). ◻

Proof. The physical matter condition and Equation (305) give \[\mu-|J|_g\ge\mu_m-|J_m|_g\ge0.\] These are the actual total constraint densities of \((g,K)\). Definition 70 supplies smoothness, orientation, one-endedness, completeness with the nonempty compact boundary included, the required metric and tensor falloff, integrable total densities, finite ADM limits, and weak future trapping with the normal toward the end. The total constraint and ADM conventions are exactly those of Equations (1), (4) and (5). All hypotheses of Corollary 46 are therefore satisfied. Applying it to the unchanged data \((N,g,K)\) yields \(E>|P|\) for their original ADM vector. ◻

Proof. Lemma 76 gives \(E>|P|\). The constraint and ADM conventions in Equations (294), (297), and (298) agree with those in Part I. Lemma 6 therefore gives the required reference end, including the stated differentiated decay and the future second-form convention.

For the later bend, retain its static isotropic coordinates: \[ \mathbf g=-\left(\frac{1-m/(2r_y)}{1+m/(2r_y)}\right)^2dT^2 +\left(1+\frac m{2r_y}\right)^4|dy|^2. \tag{309}\] The reference plane is \(T=v\cdot y\), with \(|v|<1\) chosen to give \((E,P)\). Its asymptotically Euclidean coordinates are \(y=Ax\), where \(A=(I-vv^\top)^{-1/2}\), since \(A^\top(I-vv^\top)A=I\). ◻

Proof. The original data satisfy neutral DEC by Equation (305). Definition 70 gives the smooth one-ended exterior, completeness with its compact boundary included, decay \(g-\delta=O_2(r_x^{-q})\) and \(K=O_1(r_x^{-1-q})\) with \(q>1/2\), integrable total constraint densities, and finite ADM limits. The total constraint and ADM normalizations agree with those used in Proposition 9. The reference in Lemma 77 has matching charges in the common \(x\) coordinates, and \(E>|P|\) by Lemma 76. That proposition applies and gives \[16\pi(\widetilde\mu_R- |\widetilde J_R|_{\widetilde g_R}) \ge -\eta_RR^{-3},\qquad \eta_R\longrightarrow0,\] with the stated exact agreement on both sides of the annulus and with quantitative estimates using only the original two metric derivatives and one tensor derivative. Absorbing the fixed normalization into \(\eta_R\) gives the assertion. The scaled construction in Equation (30) and the compact corrections in that proposition’s proof also give \[\|\widetilde g_R(R\,\cdot)-I\|_{C^2(\{1<|z|<4\})} +\|R\widetilde K_R(R\,\cdot)\|_{C^1(\{1<|z|<4\})} \le C R^{-\beta}.\] The two closed flux forms are retained throughout; their contribution to the physical matter deficit is estimated next. ◻

Proof. Set \(\xi=r_y/R\) and seek \[u_R=1+\frac{e_R}{R}F_R(\xi),\qquad F_R(\xi)=\int_\xi^\infty z_R(t)\,dt, \qquad z_R\ge0,\] where \(e_R\to0\) will be chosen below. On \([a,b+1]\) initially solve \(z_R'=C_0z_R+\omega\), \(z_R(a)=0\), where \(\omega\ge0\) is smooth, zero near \(a\), and at least one on \([\xi_0,b+1]\); extend \(z_R=0\) below \(a\). Equation (311) yields \[\begin{align*} -\Delta_{g_R}u_R-|K_R|_{g_R}|du_R|_{g_R} &=\frac{e_R}{R^3} \{z_R'|dr_y|^2+Rz_R\Delta r_y-Rz_R|K_R||dr_y|\}\\ &\ge\frac{e_R}{R^3}(c z_R'-Cz_R). \tag{313}\end{align*}\] Choose \(C_0\) large. This is nonnegative on \([a,b]\) and at least \(c'e_R R^{-3}\) on \([\xi_0,b]\).

On the horizontal region put \(A_R(t)=(t+m/(2R))^2z_R(t)\). Its derivative from the preceding prescription is positive near \(b\). Continue that derivative smoothly by a convex blend with \(t^{-2}\) over \([b,b+1]\), and integrate from \(A_R(b)\). Thus \(A_R'\ge c''t^{-2}\) for \(t\ge b\) and \(A_R'=t^{-2}\) eventually. Defining \(z_R=A_R(t)/(t+m/(2R))^2\) there makes every join smooth. For fixed \(L_0\), \(A_R^\infty=\lim_{t\to\infty}A_R(t)\) and \(F_R\) are bounded uniformly in \(R\). The exact radial Laplacian on the horizontal slice is \[ -\Delta_{g_R}u_R =\frac{e_R}{R^3} \left(1+\frac m{2R\xi}\right)^{-6}\xi^{-2}A_R'(\xi) \ge c'''e_R R^{-3}\xi^{-4}. \tag{314}\] Choose, for example, \(e_R=C_1(\eta_R+R^{-1})\), with a sufficiently large fixed \(C_1\). For large \(R\), \(1\le u_R\le2\). Equations (313)–(314) then dominate \(u_R/4\) times the deficit in Equation (312). Equation (306) proves the physical matter condition. The factor is constant near \(S\), so Equation (307) preserves its weak expansion.

At infinity, for each fixed \(R\), \[u_R=1+\frac{e_RA_R^\infty}{r_y}+O_k(r_y^{-2}) \quad\hbox{for every }k.\] The ADM energy is therefore \(m+2e_RA_R^\infty\to m\) and the momentum vanishes because \(K_R\) has compact support. The differentiated radial formula gives scalar curvature \(O_k(r_y^{-4})\). This also verifies integrability of the repaired constraints. Completeness follows from the comparability of the intermediate metric and \(u_R\ge1\). ◻

Proof. Choose a smooth nonnegative compactly supported majorant \(b_0\ge|K_0|_{g_0}\) and a smooth positive extension \(w_0\) of the specified tail. We solve the regularized equation \[ -\Delta_{g_0}\phi=b_0\sqrt{1+|d\phi|_{g_0}^2}+w_0, \qquad\partial_\nu\phi=-1,\qquad\phi\longrightarrow0. \tag{316}\] First truncate at \(r_y=T\) and impose outer Dirichlet value zero. Multiply \(b_0\) by \(t\in[0,1]\) for continuation. At \(t=0\) the mixed Laplacian is invertible, by the Dirichlet Poincaré inequality and elliptic regularity. Every linearization is a uniformly elliptic Laplacian with drift of norm at most \(b_0\), with homogeneous Neumann data on \(S\) and Dirichlet data on the outer sphere. These faces are disjoint. The maximum and Hopf principles give zero kernel; as a compact lower-order perturbation of the mixed Laplacian, it is Fredholm of index zero and is invertible.

We supply the estimates needed for closedness. On this fixed truncation, the \(W^{2,p}\) estimate and gradient interpolation give, uniformly in \(t\), \[\|\phi\|_{W^{2,p}} \le C(1+\|\phi\|_{L^p}+\|d\phi\|_{L^p}) \le\tfrac12\|\phi\|_{W^{2,p}}+C'(1+\|\phi\|_\infty).\] If the supremum norms were unbounded, divide by them and take \(p>3\). A subsequence converges in \(C^1\) to a nonzero solution \(v\) with homogeneous mixed data and \(-\Delta_{g_0}v=t_*b_0|dv|\). Indeed the normalized right side is \(t b_0\sqrt{M^{-2}+|dv|^2}+w_0/M\) when \(M=\|\phi\|_\infty\). The limit is a homogeneous bounded-drift equation, taking drift \(t_*b_0\nabla v/|dv|\) off the critical set and zero on it. The strong and Hopf maximum principles exclude such a nonzero \(v\). The resulting supremum and \(W^{2,p}\) bounds, followed by Schauder estimates for the smooth regularized equation, close continuation. Thus every sufficiently large truncation has a smooth solution.

All solutions are nonnegative: a negative minimum cannot be interior since the source is positive, or on \(S\) since the outward-domain derivative there is \(-\partial_\nu\phi=1\), contradicting the Hopf sign at a minimum. Fix a core containing \(\mathop{\mathrm{supp}}b_0\). On its exterior \(f=r_y^{-1}-r_y^{-1-\delta}\) is positive and \[-\Delta_{g_0}f =\delta(1+\delta)r_y^{-3-\delta}+O(r_y^{-4}) \ge c w_0\] after enlarging the fixed core. Comparison with a multiple of \(f\) gives, independently of \(T\), \[ 0\le\phi\le C(1+M_T)r_y^{-1}\quad\hbox{outside the core}, \qquad M_T=\sup_{\mathrm{core}}\phi. \tag{317}\] If \(M_T\to\infty\) along an exhaustion, divide by \(M_T\). Local \(W^{2,p}\) estimates, including the fixed inner Neumann boundary, and Equation (317) give a limit \(v\ge0\) with maximum one on the core, homogeneous inner Neumann condition, \(-\Delta v=b_0|dv|\), and \(v\to0\) at infinity. Its positive global maximum is attained, contradicting the strong or Hopf principle. Hence the core bounds are uniform in \(T\). Diagonal compactness and bootstrap give a smooth global solution of Equation (316), positive also on \(S\) by the same minimum argument.

On the end its equation is the linear Poisson equation \(-\Delta_{g_0}\phi_0=w_0\). On a scaled annulus the function \(\rho\phi_0(\rho\,\cdot)\) is uniformly bounded; the scaled metric has all-order bounds and the scaled source is \(O(\rho^{-\delta})\) with all its derivatives. Interior Poisson estimates of each order therefore give \(\phi_0=O_k(r_y^{-1})\). The full right side of Equation (316) is integrable. The divergence theorem then gives a finite metric gradient flux, and its difference from the Euclidean flux is \(O(r_y^{-1})\). This proves every assertion. ◻

Completion of the proof of Proposition 75. It remains to check area and organize the parameters. On the vacuum plane and bend, write \(dT_R=\psi a+b\), where \(a=v\cdot dy\) and \(|b|\le C L_0^{-1}|dy|\). Since \(0\le\psi\le1\), \(dT_R^2\le a^2+C L_0^{-1}|dy|^2\) as quadratic forms. The spatial factor and lapse in Equation (309) consequently give \[ g_R\ge |dy|^2-dT_R^2 \ge|dy|^2-(v\cdot dy)^2-\frac C{L_0}|dy|^2 \ge\left(1-\frac{C'}{L_0}\right)|dx|^2. \tag{320}\] The constants here depend on the fixed boost, not on \(R\) or large \(L_0\). In the gluing annulus \(g_R=I+O(R^{-\beta})\), whereas the original metric is uniformly \(I+o(1)\) throughout the receding end. On the unchanged core the metrics agree. Both conformal changes increase the metric. Thus first choosing \(L_0\to\infty\) and then \(R\to\infty\) sufficiently fast gives the global comparison \(g_j\ge(1-\varepsilon_j)g\) with \(\varepsilon_j\to0\).

For each fixed \(L_0\) the energy of the repaired data tends to \(m\) by Lemma 79. Select \(R\) to make this energy error, the constant core conformal change, and the preceding comparison error arbitrarily small. Finally choose the strictification parameter \(s_j\) to make Equation (319) tend to zero and to control the first \(j\) smooth norms on the first \(j\) sets of a compact exhaustion. This gives local smooth convergence and all the analytic assertions of the proposition. Each metric is complete up to \(S\), by the global positive comparison and smoothness. The fixed linear relation \(y=Ax\) and the all-order metric asymptotics preserve the \(O_1(r_y^{-2})\) field bounds, since the forms themselves are fixed.

For every smooth filled enclosing cut \(\Gamma\), the quadratic-form comparison gives \[\mathop{\mathrm{Area}}_{g_j}(\Gamma)\ge(1-\varepsilon_j)\mathop{\mathrm{Area}}_g(\Gamma).\] Hence \(\liminf_j a_{g_j}(S)\ge a_g(S)\). Conversely fix any such cut with area within \(\zeta>0\) of \(a_g(S)\). Local convergence on this compact cut gives \(\limsup_j a_{g_j}(S)\le\mathop{\mathrm{Area}}_g(\Gamma)\le a_g(S)+\zeta\); let \(\zeta\downarrow0\). This proves the area limit directly for the smooth-cut infimum, including obstacle-coincident components. Finally each enclosing cut has the same flux of each fixed closed form by Stokes’ Theorem, so both charges are preserved exactly. ◻

Proof. Let \(\Sigma\) be a connected component of a full maximal frontier at threshold \(c\), for \(T=K\) or \(-K\), with its outward normal. For normal graphs define \(\Phi(v)=\Theta_T(\Sigma(v))-c\) as a map \(C^{2,\alpha}(\Sigma)\to C^{0,\alpha}(\Sigma)\). Its linearization \(\mathcal L\) is the stability operator of \(T-(c/2)g\). It has principal eigenvalue \(\lambda\geq0\) and a positive eigenfunction \(\varphi\). If \(\lambda>0\), the graphs \(v=s\varphi\) have \(\Phi(s\varphi)=s\lambda\varphi+O(s^2)>0\) for small \(s>0\).

If \(\lambda=0\), the kernels of \(\mathcal L\) and its adjoint are one-dimensional, with positive generators \(\varphi\) and \(\varphi^*\). The bordered linear map \[ (v,a)\longmapsto \left(\mathcal L v-a,\int_\Sigma v\,dA_g\right) \tag{325}\] is invertible. In fact, solving \(\mathcal L v-a=f\) first determines \(a=-\int_\Sigma\varphi^*f\,dA_g/\int_\Sigma\varphi^*\,dA_g\) by Fredholm compatibility, and adding a unique multiple of \(\varphi\) fixes the prescribed integral of \(v\). Elliptic estimates give a bounded inverse. The implicit-function theorem applied to \((\Phi(v)-a,\int v)=(0,s)\) now yields constant-expansion leaves with \(a'(0)=0\) and \(v'(0)=\varphi/\int_\Sigma\varphi\,dA_g>0\). They foliate a short outward collar. If \(a(s)\leq0\) for any positive small \(s\), adjoining this collar and replacing just this whole frontier component by its new leaf would strictly enlarge the full maximal bounding region while preserving expansion \(\leq c\) on every free component. The collar avoids all other faces and designated barriers, so this is an admissible competitor, contradicting maximality. Thus \(a(s)>0\).

Smooth elliptic regularity gives smooth leaves; positive graph velocity gives the two bounds on \(|\nabla s|\). There are finitely many faces. Shortening the collars of those adjacent to \(X\) makes them disjoint and contained in \(X\). ◻

Proof. Take a fixed compact neighborhood \(\mathcal K\) containing the uniform confinement range and \(\mathop{\mathrm{supp}}K\) in its interior, and put \(\eta_*=\min_{\mathcal K}M_0>0\) and \(T_*=\sup_{\mathcal K}|\mathop{\mathrm{tr}}_gK|\). Restrict both negative thresholds in advance to an interval so close to zero that \(\max\{|c_b|,|c_w|\}T_*<\eta_*/8\), while maintaining the strict black inner-barrier inequality. The countable-exception argument permits the successive right-continuous choices there. Construct the regions and fix their disjoint collars inside \(\mathcal K\).

On each collar choose a smooth cutoff \(\zeta_B(s)\) in \([0,1]\), equal to one near zero and zero near the far end. Fix \(\kappa_B>0\) with \(1-\kappa_Bs>0\) throughout the collar. For a common amplitude \(a_*>0\) define the black contribution to \(C\) as \(a_*\zeta_B(s)(1-\kappa_Bs)\) and the white contribution as its negative. Extend by zero and sum on the disjoint collars. Then \(C_b=a_*\), \(C_w=-a_*\), \(k_B=a_*\kappa_B\), and \(\|C\|_{C^1}=O(a_*)\) with constants depending only on the fixed collars and profiles. For every assigned height, \[|h+C|\leq\max\{|c_b|,|c_w|\}+2a_*.\] Choose \(a_*\) small enough that \(2a_*T_*+\|\nabla C\|_\infty<\eta_*/8\). The first two terms of Equation (328) are then at most \(\eta_*/4\leq M_0/4\) on \(\mathcal K\), and vanish outside it. Finally scale a fixed smooth positive weight, equal to \(r_y^{-4}\) far out, so that \(\rho\leq M_0/4\) everywhere. This is possible by compact positivity and \(r_y^{-4}/r_y^{-3-\delta_*}=r_y^{-1+\delta_*}\to0\). This proves the asserted uniform inequality for all \(h\) and the parameter order. ◻

Proof. Proposition 75 supplies the reduced geometry, integrability and end estimates; Proposition 83 supplies the full-region preparation. Equation (305) gives \(M_N\ge M_0\), so the strict physical margin and its end lower bound imply their neutral counterparts. The stronger bound (328) implies (331). These are the actual total constraint densities of \((g,K)\), as required by the neutral construction. ◻

Proof. We first identify the input class. The reduced properties in Definition 84 are the properties (i)–(ii) of Proposition 12; that proposition explicitly designates them as the class for the subsequent construction. In particular, provenance as a particular member of its approximation sequence is not an additional hypothesis. The full black and white regions, their right-continuity conventions including emptiness, the adjacent complete boundary components, and the disjoint outward collars are exactly those of Proposition 19. Equation (331) is its offset inequality with \(M_N=8\pi(\mu-|J|)\). Thus its preparation, including the possibility of either missing face color, applies to every datum in the stated class.

Next identify the system. The neutral construction uses the capillarity parameter denoted here by \(\tau_0=\ell^{3/2}\), to distinguish it from \(\operatorname{tr}K\). Its functions \(l,L\), logarithmic profile derivative \(p=l'/l\), graph quantities \(D,d,w,a,v\), matrices \(A_d,A_\chi\), and unknowns \((f,Z)\) are exactly Equations (332)–(333). The notation \(p_L=L'/L\) in the neutral construction therefore means \(p_L=p+2\). Its tensor \(S\) and axial component \(q\) are denoted here by \(S_f\) and \(k\), respectively. Its quadratic form is Equation (334), with no change of sign in the momentum constraint. The physical endpoint of Proposition 23 is its first continuation stage at \(s=0\), with drift coefficient \(\lambda=1\): its trace equation is \(F=\tau_0f+C\), and its divergence source is \(\delta_0\mathcal T+\rho+\delta_0\tau_0a\sigma-m_0(t)\mathcal P\). These are the two interior equations of Equation (335). The inner normal in both formulations points toward the end. The componentwise assigned heights and the identities \(F=P_B+H\) on black faces and \(F=P_B-H\) on white faces give exactly the same inner flux condition; the outer values are \(f=Z=0\) in both systems. All auxiliary continuation data are chosen by that construction after the fixed preparation, as stipulated above.

Lemma 27 and Proposition 28 choose a fixed \(\delta_0>0\) and a polynomial \(C_n\). Theorem 38 then supplies item 1, including its sufficiently large truncation radius and the strict floor; enlarging the lower bounds on \(n\) and \(R\) imposes the additional displayed numerical restrictions. At the physical endpoint, Lemma 30 gives item 2. Its slope constants are independent of \(n,R\), its height interval is asserted on the fixed compact regions used in the curvature argument, and its face conditions are componentwise. No boundary connectedness is required.

For fixed \(\epsilon,n\), Proposition 36 and the exhaustion following Theorem 38 give the local smooth estimates and subsequential limit in item 3, including every fixed inner face. These estimates allow arbitrary fixed-\(n\) constants and do not change the polynomial penalty already chosen. Lemma 39 supplies the stated normalized end solution: the exponential decay of \(f\) and the graph errors, the \(O_2(r_y^{-1})\) bounds for \(t,Z\), and the finite flux with \(V=4\nabla_gt+O(r_y^{-3})\). These are exactly Equations (338)–(339). All three conclusions follow from the proved neutral construction. ◻

Proof. Apply Proposition 20. The variables \(l,L,p,D,d,w,a,v,A_d,A_\chi,Z,u,H^f,F,V\) in Equations (332)–(333) are those of that proposition. Its tensor \(S\) is our \(S_f\), its axial component \(q\) is our \(k\), and its transverse and axial derivatives of \(t\) are \(y_t,x_t\). Its logarithmic derivative \(p_L=L'/L\) is \(p+2\). In particular the graph and conformal metrics, momentum convention, and full flux agree. The identity holds for arbitrary smooth \(f,t\); no elliptic equation or constraint beyond the definitions of \(\mu,J\) is assumed in its proof.

Equation (84) is exactly Equation (334) under this correspondence. It proves nonnegativity, since \(p\ge0\) and \(0<d\le1\). The invariant formula (81) proves smoothness at \(\nabla f=0\), where its value is \[\mathcal T=\tfrac12\bigl(|S_f|^2+(\mathop{\mathrm{tr}}S_f)^2\bigr) +4(p+1)|\nabla t|^2.\] Thus the shared identity gives (341), with the momentum term \(+8\pi J(w)\). ◻

Proof. Use the square completion (84), identified above with (334). Its last coefficient is at least \(2d/[3(1+p)^2]\) and at least \(2\chi^2/3\). Its transverse-gradient coefficient is at least \(3(p+1)\). Comparing these terms and the shifted squares gives \[\begin{gathered} dk^2\le\tfrac32(1+p)^2\mathcal T,\qquad \chi^2k^2\le\tfrac32\mathcal T,\qquad |y_t|^2\le\frac{\mathcal T}{3(p+1)},\\ F^2\le3\mathcal T,\qquad |M|^2\le\tfrac83(p+1)\mathcal T,\qquad |T|^2\le4\mathcal T. \end{gathered}\] Here the shifted \(F\) and \(M\) estimates use \(|A+B|^2\le2|A|^2+2|B|^2\); for \(T\) write \(\mathop{\mathrm{tr}}T=(F-\chi k/3)-2\chi k/3\). The axial-gradient square gives \[dx_t^2\le\frac{\mathcal T}{2(p+1)}+\frac{\chi^2vk^2}{8d} \le\left(\frac1{2(p+1)}+\frac16\right)\mathcal T \le\tfrac23\mathcal T,\] because the ratio of its second coefficient to the last square coefficient is \(3v/[2(9-d)]\le1/6\). Summing and using \(0\le p\le n\) proves (342). At zero slope the same estimates hold for every axial decomposition of the direction-independent value of \(\mathcal T\). ◻

Proof. The assigned height is constant on each face, and the normal into \(X\) is the normal used in Lemma 22. Its flux, trace \(F\), and tangential trace \(P_B\) agree with ours by the correspondence in Lemma 87. Applying Lemma 22 gives (343) and \[H_{\hat g}=e^{-2t}\sqrt d\,(H+4\partial_\nu t).\] The boundary equation in (335) now implies \(d(H+4\partial_\nu t)=-N_Bd\). Since \(d>0\), substitution gives (344); the last equality is the mean-curvature scaling under \(\widetilde g=e^{4\epsilon}\hat g\). ◻

Proof. Substituting Equation (335) into Equation (341), and using \(w(h)=\tau_0a\sigma\), gives \[\begin{align*} \frac12e^{4t}R_{\hat g} ={}&8\pi(\mu+J(w))+(1-\delta_0)\mathcal T +(1-\delta_0)\tau_0a\sigma\\ &+w(C)-F\mathop{\mathrm{tr}}K-\rho+m_0(t)\mathcal P. \tag{346}\end{align*}\] The physical matter condition says \[8\pi(\mu+J(w))-I_g(w) =8\pi(\mu_m+J_m(w))\ge M_0,\] since \(|w|<1\). The compact height bound in Proposition 86 and Equation (328) imply \(|w(C)|+|F\mathop{\mathrm{tr}}K|+\rho\le M_0/2\); outside the supports of \(K\) and \(C\) only the already controlled term \(\rho\) remains. All other terms in Equation (346) are nonnegative. Consequently \[ \frac12e^{4t}R_{\hat g}\ge I_g(w). \tag{347}\]

For \(Q>0\), take the constant rotation \[O=\frac1Q\begin{pmatrix}Q_E&Q_B\\-Q_B&Q_E\end{pmatrix}\in SO(2), \qquad \binom{X_e}{X_o}=O\binom{\mathcal E}{\mathcal B};\] for \(Q=0\), use \(O=I\). Rotate the flux forms by the same matrix. Their charges become \((Q,0)\), while \(I_g\) is unchanged: the sum of squared norms is invariant and \(X_e\times X_o=\mathcal E\times\mathcal B\). Let \(\alpha=\iota_{X_e}dV_g\). Completing one vector square gives \[\begin{align*} I_g(w)&=|X_o+w\times X_e|^2+|X_e|^2-|w\times X_e|^2\\ &\ge|X_e|^2-|w\times X_e|^2 =|\alpha|_{\bar g}^2. \tag{348}\end{align*}\] The last equality is a two-form norm identity. In a positively oriented \(g\)-orthonormal frame with first vector parallel to \(w\), the inverse graph metric has eigenvalues \((d,1,1)\). Writing \(X_e=(x_1,x_2,x_3)\) gives \[\alpha=x_1 e^2\wedge e^3+x_2 e^3\wedge e^1+x_3 e^1\wedge e^2, \qquad |\alpha|_{\bar g}^2=x_1^2+d(x_2^2+x_3^2) =|X_e|^2-|w\times X_e|^2.\] At \(w=0\) this formula is independent of the frame. This is the electric graph transport of (Disconzi and Khuri 2012, sec. 2 and Appendix A), expressed through its flux form.

Retain the rotated pair \((\alpha,0)\) and define the new pair by the inverse rotation \[ \binom{\widetilde\alpha_1}{\widetilde\alpha_2} =O^{\mathsf T}\binom{\alpha}{0}. \tag{349}\] This discards a form of zero total charge. Since \(O\) is constant, the new forms are closed, have separately the original total charges, and satisfy \(|\widetilde\alpha_1|_h^2+|\widetilde\alpha_2|_h^2=|\alpha|_h^2\) for every metric \(h\). No assertion about the individual component charges is needed. Flux conservation across every enclosing cut follows from Stokes’ Theorem.

Finally set \(s_0=t+\epsilon\ge0\), using the floor in Proposition 86. Scalar curvature under constant scaling and the conformal norm of a two-form give \[R_{\widetilde g}=e^{-4\epsilon}R_{\hat g},\qquad |\alpha|_{\widetilde g}^2=e^{-8s_0}|\alpha|_{\bar g}^2.\] Equations (347) and (348) therefore imply \[R_{\widetilde g}\ge2e^{-4s_0}|\alpha|_{\bar g}^2 \ge2e^{-8s_0}|\alpha|_{\bar g}^2 =2\sum_{i=1}^2|\widetilde X_i|_{\widetilde g}^2.\] The last equality uses the isometry \(X\mapsto\iota_XdV_{\widetilde g}\) between vectors and two-forms in dimension three. Closedness gives the divergence equations because \(d(\iota_XdV_{\widetilde g})=(\mathop{\mathrm{div}}_{\widetilde g}X)dV_{\widetilde g}\). Also \(\widetilde g=e^{4s_0}(g+l^2df\otimes df)\ge g\). A \(\widetilde g\)-Cauchy sequence is therefore \(g\)-Cauchy; its limit lies in \(X\) with \(B\) included, and local smooth comparison of the two metrics gives convergence for \(\widetilde g\) as well. Completeness follows, and the boundary sign is Lemma 89. ◻

Proof. For \(k=0\), this is Corollary 94. Suppose \(k\ge1\). In each unwanted end \(\mathscr E_j\), choose a coordinate sphere \(S_{R_j}^j\) sufficiently far out, and remove its open outside tail. Denote the retained manifold by \(\Omega_{\mathbf R}\). The radii can be chosen beyond a connected compact core: fix paths in \(\Omega\) connecting the finitely many inner end collars and the components of \(S\), and put these paths inside that core before increasing the radii. Each retained end annulus is connected and attached to the core. Thus \(\Omega_{\mathbf R}\) is connected, has exactly the designated end, and has nonempty compact smooth boundary \[S_{\mathbf R}=S\ \sqcup\ \bigsqcup_{j=1}^kS_{R_j}^j.\] It is a closed subset of the original complete manifold. A Cauchy sequence for its intrinsic length metric therefore has a limit in that subset; smooth boundary half-ball charts give convergence in the intrinsic metric as well. This proves completeness with its boundary included.

On an added sphere, the normal pointing into the retained exterior is the negative of the normal pointing toward that unwanted infinity. The asymptotic estimates give \[H=-\frac2{R_j}+O(R_j^{-1-q}),\qquad \operatorname{tr}_{S_{R_j}^j}K=O(R_j^{-1-q}).\] Hence its future outward expansion, in the retained-exterior orientation, is strictly negative for large \(R_j\). The original boundary retains its weakly trapped sign. All fields are merely restricted, so the charged DEC and divergence equation are unchanged. No extension of the fields over a cap is involved. The designated end and its ADM energy and momentum are also unchanged.

Put \(\mathcal F=\iota_{\mathcal E}dV_g\). The equation \(d\mathcal F=0\) and Stokes’ theorem on the region between \(S_{\mathbf R}\) and a distant sphere of the designated end give \[ \frac1{4\pi}\int_{S_{\mathbf R}}\mathcal F=Q_0, \tag{361}\] where every inner component is oriented into the retained exterior. In particular, if \(Q_j\) is the outward charge of the \(j\)th unwanted end and \(Q(S)\) is the charge of the original boundary with normal into \(\Omega\), then \(Q_0=Q(S)-\sum_{j=1}^kQ_j\). The charge in (360) is therefore the designated end charge, not generally \(Q(S)\).

Let \(a_{\mathbf R}=a_g(S_{\mathbf R})\) be the minimum enclosing area in the retained one-ended manifold. Every smooth filled cut of \(S_{\mathbf R}\) also gives a designated-end cut in \(\Omega\): keep its chosen-end exterior, and assign phase zero to all removed tails. This introduces no new frontier because the original cut already encloses every added inner sphere. If the cut coincides with any added sphere, that sphere is the same smooth surface in the original end and its full area remains counted. The same observation applies to coincident portions of \(S\). Selecting the component toward the designated end and filling bounded pockets keeps the whole-boundary convention and can only remove boundary components. Consequently \[ a_{\mathbf R}\ge a_0, \qquad r_{\mathbf R}:=\sqrt{\frac{a_{\mathbf R}}{4\pi}} \ge r_0\ge|Q_0|. \tag{362}\] This argument compares the smooth full-cut classes directly; it does not require attainment of either infimum or a limiting statement as \(\mathbf R\to\infty\).

All hypotheses of Corollary 94 now hold on \(\Omega_{\mathbf R}\), with charge \(Q_0\) by (361). Hence \[m_0\ge\frac12\left(r_{\mathbf R} +\frac{Q_0^2}{r_{\mathbf R}}\right).\] The function \(r\mapsto(r+Q_0^2/r)/2\) is nondecreasing for \(r\ge|Q_0|\) (and \(r>0\)). Combining this fact with (362) proves (360), including the endpoint \(r_0=|Q_0|\). ◻

Proof. A common outer truncation. On the coordinate end, let \(Y=\nabla r/|\nabla r|_g\) be the unit normal pointing toward increasing radius. The differentiated asymptotic decay gives \[ \operatorname{div}_gY=\frac{3}{r}+O(r^{-1-q})>0 \qquad (r\ge R_0) \tag{372}\] after increasing \(R_0\). For a competitor \(E\) and almost every \(R>R_0\), put \(E_R=E\setminus\{r>R\}\). Gauss–Green on \(V_R=E\cap\{r>R\}\) gives \[\int_{V_R}\operatorname{div}_gY\,dV_g =\int_{\partial^*E\cap\{r>R\}} g(Y,\nu_E)\,dA_g -\int_{\{r=R\}}\mathbf 1_E\,dA_g.\] The trace on the slicing sphere is understood at a good slicing radius; \(\nu_E\) is the measure-theoretic outward normal of the filled set. Consequently \[ P_g(E_R)-P_g(E) \le -\int_{V_R}\operatorname{div}_gY\,dV_g\le0, \tag{373}\] and the inequality is strict if \(V_R\) has positive volume.

Fix \(R_1>R_0\). For each competitor separately, choose a good radius in \((R_0,R_1)\) and use Equation (373). The infimum therefore equals the infimum in the class with \(E\setminus O\) confined to the same fixed truncation \(\{r\le R_1\}\) together with the compact core. BV compactness on a slightly larger compact region and lower semicontinuity give a minimizer. The constraints of containing \(O\) and being empty in the prescribed exterior tail are closed under this convergence. The competitor \(O\) gives a finite upper bound.

The resulting set is also a global minimizer in \(\mathcal A\). Applied to any global minimizer, Equation (373) forbids positive filled volume beyond a good radius above \(R_0\). Choosing such a radius below a fixed number \(R_2\) with \(R_0<R_2<R_1\) shows that all minimizers have their filled exterior parts in the same compact truncation at \(R_2\). This also keeps their frontiers away from the artificial boundary used in the compactness argument. There is no requirement that one slicing radius be good for every BV set.

Regularity at the obstacle. A global minimizer is locally perimeter-minimizing among competitors that contain the smooth open obstacle \(O^\circ\). The relevant form of the Riemannian obstacle regularity theorem is the following: in a smooth Riemannian manifold of dimension less than eight, a local perimeter minimizer constrained to contain an open set with \(C^2\) boundary has a \(C^{1,1}\) frontier and is smooth away from contact. This is precisely Regularity Theorem 1.3(iii) of Huisken–Ilmanen (Huisken and Ilmanen 2001, 368–69), applied with zero bulk term. Its hypotheses are local, so the arbitrary geometry of the attached inner side and its possible noncompactness are irrelevant. Here the ambient dimension is four. The result describes the entire frontier of the regular representative, not just the reduced boundary outside an unaccounted exceptional set.

For clarity, the obstacle also gives the usual two-sided almost-minimality estimate needed for density and compactness estimates. Choose a smooth compactly supported vector field \(Z\) on \(\widetilde X\) with \(|Z|_g\le1\) and \(Z=\nu_O\) on \(B\). Such a field is obtained from a tubular neighborhood of the compact smooth boundary. For a local replacement \(F\), with \(F\triangle E\) compactly supported, \(F\cup O\) is admissible. Perimeter submodularity and Gauss–Green imply \[\begin{align*} P_g(E) &\le P_g(F\cup O) \le P_g(F)+P_g(O)-P_g(F\cap O),\\ P_g(O)-P_g(F\cap O) &\le \int_{O\setminus F}\operatorname{div}_gZ\,dV_g \le C\mathop{\mathrm{Vol}}_g(E\triangle F). \tag{374}\end{align*}\] Thus local replacements satisfy the standard perimeter almost-minimality inequality. In particular, the usual perimeter and volume density estimates apply, including at contact (Maggi 2012). The \(C^{1,1}\) conclusion above directly gives the asserted Sobolev regularity in graph charts. Off the obstacle, arbitrary local replacements are allowed; first variation gives zero mean curvature and interior regularity gives smoothness.

Take the representative that is the closure of its measure-theoretic interior. Its frontier is the compact \(C^{1,1}\) hypersurface just described, and the filled and exterior sides occupy the two local sides of every frontier chart. This choice discards all irrelevant null-set slits. The frontier is nonempty: the filled set contains the open inner side, whereas its complement contains an exterior tail. On the connected extended manifold a BV indicator with zero perimeter is constant almost everywhere, which is impossible here. The nonempty regular frontier consequently has positive area.

Connectedness of the exterior. The complement of the regular filled set has a component containing the entire sufficiently distant end. Any other component is relatively compact, because the frontier is compact and \(X\) has one end. If such a component existed, adjoining it to the filled set would remove its nonempty boundary and introduce no new perimeter. More explicitly, near each frontier point there is only one exterior side; the boundary of a complement component is therefore a union of connected components of the compact embedded frontier. Filling a bounded component removes a positive amount of full perimeter, including any portion on \(B\). This contradicts minimality. Thus the exterior is connected.

Equality with the smooth-cut infimum. Equation (371) gives \(\min_{\mathcal A}P_g\le a_g(B)\). To prove the reverse inequality, let \(T\) be a minimizing frontier with its outward coorientation. Choose a smooth compactly supported vector field \(V\) with \(g(V,\nu_T)\ge c_0>0\) on \(T\); approximate its continuous unit normal in local charts and use a partition of unity. If \(\Phi_t\) is its flow, compactness and transversality give a \(C^1\) flow collar \[\Psi:T\times(-\delta,\delta)\longrightarrow\widetilde X, \qquad \Psi(p,t)=\Phi_t(p),\] in which the filled side is \(\{t\le0\}\). For sufficiently small \(\varepsilon>0\) the displaced filled set satisfies \[E_\varepsilon:=\Phi_\varepsilon(E) =E\cup\Psi\bigl(T\times(0,\varepsilon]\bigr).\] This inclusion is global: a point can change its membership only when its flow line crosses \(T\), and every such short crossing is outward in the chosen collar. Thus \(O\subset E\subset\operatorname{int}E_\varepsilon\). The displaced frontier \(T_\varepsilon=\partial E_\varepsilon\) is disjoint from \(O\) and lies in \(\operatorname{int}X\). Its distance from the compact original boundary is positive.

Choose a smaller cooriented collar of \(T_\varepsilon\) with compact closure disjoint from \(O\), and a \(C^1\) defining function there whose derivative along \(V\) is positive. Smooth this defining function in \(C^1\). A sufficiently close approximation still has positive \(V\)-derivative, and its zero set meets each collar fiber exactly once. It is consequently a smooth hypersurface given by a small graph over \(T_\varepsilon\) along those fibers. The graph displacement extends to a collar isotopy: in collar coordinates use \((p,t)\mapsto(p,t+\tau\chi(t)w(p))\), where \(w\) is the small graph height, \(\chi\) equals one near zero and vanishes near the collar ends, and \(0\le\tau\le1\). Smallness of \(w\) makes the derivative in \(t\) positive. This isotopy is the identity on \(O\) and in the distant end. It therefore preserves enclosure and connectedness of the exterior. Its final smooth frontier bounds a connected smooth exterior domain, closed in \(X\), and is exactly that domain’s full intrinsic boundary.

The resulting cuts are admissible. Their areas tend first to the area of \(T_\varepsilon\) by \(C^1\) approximation and then to \(\mathop{\mathrm{Area}}_g(T)\) as \(\varepsilon\downarrow0\). Hence \(a_g(B)\le P_g(E)\), proving equality of the two minimization problems and the claimed approximation. This also proves that the minimum is independent of the chosen inner extension.

A strict inner barrier. Suppose \(H_B<0\) for the normal into \(X\). Compactness of \(B\) gives an exterior signed-distance collar \(0\le t\le\varepsilon\) whose unit normal \(Y=\partial_t\) satisfies \[\operatorname{div}_gY\le-\beta<0\] for fixed \(\varepsilon,\beta>0\). For almost every \(s\in(0,\varepsilon)\), let \(O_s=O\cup\{0<t<s\}\) and \(W_s=O_s\setminus E\). Extend the collar field smoothly across \(B\) when applying Gauss–Green. The boundary flux identity on \(W_s\) gives \[\int_{W_s}\operatorname{div}_gY\,dV_g =\int_{\{t=s\}}\mathbf 1_{E^c}\,dA_g -\int_{\partial^*E\cap\{0\le t<s\}} g(Y,\nu_E)\,dA_g.\] The second integral includes the frontier at \(t=0\). Therefore \[ P_g(E\cup O_s)-P_g(E) \le\int_{W_s}\operatorname{div}_gY\,dV_g \le-\beta\mathop{\mathrm{Vol}}_g(W_s). \tag{375}\] Minimality forces \(\mathop{\mathrm{Vol}}_g(W_s)=0\). For each minimizer choose a good \(s\in(\varepsilon/2,\varepsilon)\). Its regular representative then contains the whole open collar \(0\le t<\varepsilon/2\) as interior points in \(\widetilde X\). Thus the same collar is included by all minimizers, despite allowing their good slicing levels to differ.

The frontier is now disjoint from \(B\), hence is smooth and minimal. Its closed exterior \(D\) is a smooth submanifold with boundary \(T\), closed as a subset of \(X\), with connected interior and the same distant end. It is complete in its intrinsic length metric. Indeed an intrinsic Cauchy sequence is Cauchy for the original metric distance and converges to a point of the closed set \(D\) by completeness of \(X\). Interior charts and smooth boundary half-ball charts identify the two local notions of convergence, with locally comparable path lengths, so the sequence converges also intrinsically in \(D\).

Finally, a smooth enclosing cut in \(D\) determines a filled competitor containing the minimizing set \(E\), hence also containing \(O\). Its full area is at least \(P_g(E)=\mathop{\mathrm{Area}}_g(T)\) by global minimality. This is the required outer area-minimizing property, with all components and any frontier on \(T\) retained. ◻

Proof. The proof of the outer barrier in Proposition 102 is uniform for this family: all minimizing frontiers lie in one compact set \(L\subset X\). The metrics can be extended smoothly through the fixed boundary in a common collar. On that collar and on \(L\), their smooth dependence on \(s\) supplies uniform local bounds and convergence to the fixed extended metric. The regularity and density results used above therefore remain available with uniform local bounds where needed.

First consider a sequence of minimizers for \(g\). BV compactness gives \(\mathbf 1_{E_j}\to\mathbf 1_E\) locally in \(L^1\) along a subsequence, with \(E\) satisfying the same obstacle and exterior-tail constraints. Lower semicontinuity and global minimality imply \[a_g(B)\le P_g(E)\le\liminf_jP_g(E_j)=a_g(B).\] Thus \(E\) is another minimizer, and the perimeters converge as well.

This strict BV convergence also gives the claimed tangent-plane convergence, including mass carried by the original boundary. We give the localization argument to keep track of the metric norm. Set \(\sigma_j=|D\mathbf 1_{E_j}|_g\) and \(\sigma=|D\mathbf 1_E|_g\), as scalar perimeter measures on \(\widetilde X\). Every subsequential weak limit of \(\sigma_j\) dominates \(\sigma\): this follows from lower semicontinuity of perimeter weighted by an arbitrary nonnegative continuous function. Their total masses converge and their supports lie in one compact set, so this domination must be equality. Hence \(\sigma_j\) converges weakly to \(\sigma\). This conclusion uses only weighted lower semicontinuity, before any continuity theorem for polar directions is applied.

Now form the vector-valued measures \(\mu_j=\nu_{E_j}\sigma_j\) and \(\mu=\nu_E\sigma\). Gauss–Green and local \(L^1\) convergence give \(\mu_j\rightharpoonup\mu\). Choose finitely many smooth coordinate charts covering the common compact support and a nonnegative smooth partition of unity \(\{\rho_\alpha\}\) subordinate to them. In each chart choose a smooth matrix \(A_\alpha(x)\) such that \(|A_\alpha(x)v|_\delta=|v|_g\). The coordinate vector measures \[\lambda_{\alpha,j}=\rho_\alpha A_\alpha\mu_j \quad\hbox{satisfy}\quad |\lambda_{\alpha,j}|_\delta=\rho_\alpha\sigma_j.\] They converge weakly, and their Euclidean total variation masses converge by the already established weak convergence of \(\sigma_j\). The vector-measure continuity theorem therefore applies in each chart (Spector 2011, Theorem 1.3). Use its bounded continuous polar integrand \[(x,z)\longmapsto \Phi\bigl(x,(A_\alpha(x)^{-1}z)^{\perp_g}\bigr), \qquad |z|_\delta=1,\] and sum over the partition. This proves weak convergence of \(V_{T_j}\) to \(V_T\). Charts crossing \(B\) are ordinary interior charts of \(\widetilde X\), so no contact mass is removed in this localization. The compact base region and its Grassmann bundle are metrizable, so the indicated \(L^1\) and weak-measure topology is metrizable. The subsequence argument proves compactness. Taking \(\Phi(x,\Pi)=\tfrac12\mathop{\mathrm{tr}}_\Pi h\) proves the asserted continuity.

For the varying metrics, uniform comparison with \(g_s\to g\) on \(L\) first gives \[ a(s)\longrightarrow a(0),\qquad P_g(E_s)\longrightarrow a(0) \tag{379}\] for any choice of a minimizer \(E_s\) for \(g_s\). Indeed testing on a fixed \(g\)-minimizer gives the upper bound for \(a(s)\); comparison of perimeters gives the lower bound, since \(P_g(E_s)\ge a(0)\). Consequently every BV limit of such \(E_s\) is a \(g\)-minimizer and the same strict-convergence argument gives tangent-plane convergence. This proves the last assertion of the proposition.

It remains to calculate the derivative. The area element of a fixed three-plane has the uniform Taylor expansion \[ dA_{g_s}|_\Pi =\left(1+\frac{s}{2}\mathop{\mathrm{tr}}_\Pi h+O(s^2)\right)dA_g|_\Pi. \tag{380}\] The remainder is uniform over \(x\in L\) and all three-planes \(\Pi\), because of the parameter-derivative bounds and uniform metric positivity. Integrating over the fixed reduced boundary of a competitor gives \[P_{g_s}(E)=P_g(E)+s\,\frac12 \int_{\partial^*E}\mathop{\mathrm{tr}}_{T_x\partial^*E}h\,dA_g +O(s^2)P_g(E).\] For each \(T\in\mathcal T\), using its fixed filled set as a competitor therefore yields \[\limsup_{s\downarrow0}\frac{a(s)-a(0)}s \le a'_T(h).\] The continuous function in Equation (377) has a minimum on \(\mathcal T\), so this gives the required upper bound by that minimum.