Proof. Choose a small boundary disk in one component of \(S\). A smooth embedded arc beginning transversely at that disk can be continued into the coordinate end and then along a straight coordinate ray. By shrinking a tubular neighborhood, obtain a proper embedded tube \[[0,\infty)\times B^{n-1}_\delta\longrightarrow\Omega\] whose initial face lies in the chosen boundary disk. The transverse coordinates have fixed small coordinate width sufficiently far along the ray. Choose a nonnegative function \(\zeta\in C_c^\infty(B^{n-1}_\delta)\) with integral one. The form \[\zeta(z)\,\mathrm dz^1\wedge\cdots\wedge\mathrm dz^{n-1}\] on the tube is closed. Extend it by zero across the lateral boundary; call the resulting smooth closed form \(\beta\). Its comass is bounded: on the compact part this follows from smoothness, and on the end from asymptotic Euclidean comparability and the fixed transverse width. Orient the initial disk so that \(\int_S\beta=1\).

Every full cut \(\Gamma\) is homologous to \(S\) with the orientation toward the end. This can be seen by truncating both exteriors at one distant coordinate sphere and applying Stokes to their difference; portions of \(S\) shared by the two boundaries cancel with their full multiplicity. Equivalently, integration over the compact difference of the two codimension-zero sets gives the same identity. Thus \[1=\int_\Gamma\beta \leq \|\beta\|_{\mathrm{comass},\infty}\mathop{\mathrm{Area}}_g(\Gamma).\] Taking the infimum proves the lower bound. The admissible choice \(\Gamma=S\) gives the upper bound. In particular, this argument uses the entire intrinsic boundary and never replaces it by an empty relative frontier. ◻

Proof. The scalar-curvature and hypersurface conformal laws give \[e^{2s}R_{g_s}=R_g-2k\Delta_gs-k(n-2)|\mathrm ds|_g^2, \qquad e^sH_s=H+k\partial_\nu s.\] Also \(\mathop{\mathrm{tr}}_{g_s}K_s=e^{-s}\mathop{\mathrm{tr}}_gK\). If \(\Pi=K-(\mathop{\mathrm{tr}}_gK)g\), then \(\Pi_s=e^s\Pi\). Contracting the connection difference \[(\nabla^s-\nabla)^l{}_{ij} =s_i\delta_j^l+s_j\delta_i^l-s^lg_{ij}\] gives \(J_s=e^{-s}(J+kK(\nabla s,\cdot))\). These identities prove (26)–(27). Finally, \(\mathop{\mathrm{vol}}_{g_s}=e^{ns}\mathop{\mathrm{vol}}_g\) and \(i_{E_s}\mathop{\mathrm{vol}}_{g_s}=\alpha\), proving the assertion about \(E_s\) and its norm. ◻

Proof. For \(f(r)=r^{2-n}(1-r^{-\beta})\), asymptotic differentiation gives \[-\Delta_g f =\beta(n-2+\beta)r^{-n-\beta}+O(r^{-n-q}), \qquad |K|_g|\mathrm df|_g=O(r^{-n-q}).\] Thus the desired strict inequality holds on a sufficiently distant coordinate region. On that region \(-f'>0\). Multiply \(-f'\) by a smooth nondecreasing cutoff that changes from zero to one, entirely in this valid region, and integrate from infinity. If \(\chi\) is the cutoff, the resulting radial function satisfies \[-\Delta_g\phi_0-|K|_g|\mathrm d\phi_0|_g =\chi\bigl(-\Delta_g f-|K|_g|\mathrm df|_g\bigr) +\chi'(-f')|\mathrm dr|_g^2\geq0.\] It is positive and constant on the remaining core. Equations (29) and (28) preserve charged DEC; the expansion is only rescaled on \(S\). The asymptotic constraints remain integrable, since the new Laplacian term is \(O(r^{-n-\beta})\), the drift term is \(O(r^{-n-q})\), and the gradient-square term is integrable. All other end bounds persist.

The leading change of the metric is \(4t r^{2-n}\delta/(n-2)\), which has ADM energy \(2t\) in the stated normalization. The change in the momentum integrand has sphere integral \(O(r^{-q})\), and hence zero limit. Charge is unchanged because its flux form is unchanged. Finally \(U_t\geq1\), so every full-cut area increases. ◻

Proof. We first describe a compact scalar divergence inverse. On a ball a regularized Bogovskii operator, with a fixed \(\rho\in C_c^\infty\) of integral one, is given by \[(\mathcal B_\rho h)(x) =\int h(y)(x-y)\int_1^\infty \rho(y+t(x-y))t^{n-1}\,\mathrm dt\,\mathrm dy.\] It has divergence \(h-\rho\int h\) and gains one Sobolev derivative; see (Costabel and McIntosh 2010). Its support is contained in the convex hull of the supports of \(h\) and \(\rho\). The derivative bounds needed here also follow directly from the first-derivative estimate and \[\partial_j\mathcal B_\rho h =\mathcal B_\rho(\partial_jh) +\mathcal B_{\partial_j\rho}h,\] first for compactly supported smooth \(h\). The kernel estimate does not require the new kernel \(\partial_j\rho\) to have integral one.

To work in a connected annulus, cover the prescribed source support by finitely many balls compactly contained in a larger subannulus, adding balls along paths so that the intersection graph is connected. Choose a partition of unity and a spanning tree of that graph. In each parent–child overlap fix a unit-integral smooth bump. Transfer the integral of a child source to its parent using that bump, beginning at the leaves. This writes a source of total integral zero as a sum of mean-zero sources on the balls. The transfer coefficients are bounded by the original source norm. Summing the ball operators gives an operator \(\mathcal B\) with \[ \operatorname{div}\mathcal B h=h, \qquad \|\mathcal B h\|_{W^{j+1,p}} \leq C_j\|h\|_{W^{j,p}},\qquad j=0,1,2, \tag{31}\] and support in a fixed larger compact subset of the annulus.

For (i), set \(u=\mathcal B f\). Its component means vanish, since compact support and integration by parts give \[\int u_i=-\int z_i\partial_j u_j=-\int z_i f=0.\] Apply \(\mathcal B\) to each \(u_i\) to obtain \(T_{ij}\) with \(\partial_jT_{ij}=u_i\). The symmetric part \(H=(T+T^{\mathsf T})/2\) has the same double divergence as \(T\). Two applications of (31) give its \(W^{3,p}\) bound.

For (ii), translation moments first give a row primitive \(B^0\) with \(\partial_jB^0_{ij}=F_i\) and a \(W^{2,p}\) bound. Write \(A_{ij}=(B^0_{ij}-B^0_{ji})/2\). Rotation moments imply \[2\int A_{ij}=-\int(z_jF_i-z_iF_j)=0.\] Solve \(\partial_lD_{ijl}=-A_{ij}\) using (31), choosing \(D_{ijl}=-D_{jil}\). Thus \(D\) has a \(W^{3,p}\) bound. Define \[C_{ij}=\partial_l(D_{ijl}-D_{ilj}-D_{jli}).\] Then \[C_{ij}-C_{ji}=-2A_{ij},\qquad \partial_jC_{ij}=0.\] For the divergence identity the first two terms cancel after exchanging \(j,l\), and the third vanishes by antisymmetry in those indices. Consequently \(B=B^0+C\) is symmetric, has the required divergence, and has a \(W^{2,p}\) bound. None of these differentiations enlarges support.

For completeness, the moment bumps can be fixed without a choice depending on the source. Let \(\eta\) be a smooth nonnegative function supported in the innermost subannulus and positive on a ball. For the affine basis \(\varphi_a\in\{1,z_1,\ldots,z_n\}\), the matrix \(G_{ab}=\int\eta\varphi_a\varphi_b\) is positive definite. The functions \(\eta\sum_b(G^{-1})_{ab}\varphi_b\) are dual moment bumps. For vector sources use instead a basis of Euclidean Killing fields and its Gram matrix for the \(\eta\)-weighted Euclidean inner product. A nonzero affine function or Killing field cannot vanish on a ball, so both Gram matrices are invertible. These projections have all the asserted norm and support bounds. Sobolev embedding finishes the proof. ◻

Proof. Write \(a_0=r^{-k}\) and \(\rho_0=r^{-n-\beta}\). Truncate at a distant coordinate sphere \(S_R\), and continue in \(\lambda\in[0,1]\) the problem \[-\Delta_g\phi_R =\lambda B\sqrt{|\mathrm d\phi_R|_g^2+a_0^2}+\rho_0, \qquad \partial_\nu\phi_R=-1\text{ on }S, \qquad \phi_R=0\text{ on }S_R.\] The inner Neumann and outer Dirichlet faces are disjoint. The Poisson problem at \(\lambda=0\) has a unique smooth solution. The linearization at any solution is \[-\Delta_g h-\lambda B \frac{\langle\mathrm d\phi_R,\mathrm dh\rangle_g} {\sqrt{|\mathrm d\phi_R|_g^2+a_0^2}}, \qquad \partial_\nu h=0,\quad h|_{S_R}=0.\] Its drift is bounded by \(B\); the maximum principle and boundary Hopf lemma give a trivial kernel. Fredholm theory for this scalar mixed boundary problem, or continuation from the mixed Laplacian, then gives invertibility. We use the usual scalar local and boundary estimates in this argument; see (Gilbarg and Trudinger 2001). There is no meeting edge between Dirichlet and Neumann conditions.

Every solution is nonnegative. An interior negative minimum contradicts the positive right side, and a minimum on \(S\) contradicts the inward derivative \(-1\). Given a supremum bound, local and boundary \(W^{2,p}\) estimates, interpolation, and the at-most-linear gradient growth of the equation give uniform \(W^{2,p}\) bounds on a fixed truncation. More explicitly, for \(p>n\), \[\|\phi_R\|_{W^{2,p}} \leq C_R(1+\|\mathrm d\phi_R\|_{L^p}+\|\phi_R\|_{L^p}) \leq\tfrac12\|\phi_R\|_{W^{2,p}} +C_R'(1+\|\phi_R\|_\infty).\] The resulting \(C^{1,\gamma}\) bound makes the right side \(C^{0,\gamma}\), and Schauder estimates and differentiation give smooth bounds. Positivity of \(a_0\) makes the square root a smooth function of the gradient on every compact set.

We next obtain the required supremum bounds. A normalized blow-up on a fixed truncation, if those bounds failed, would give a nonzero nonnegative solution \(v\) of \[-\Delta_g v=\lambda B|\mathrm dv|, \qquad \partial_\nu v=0, \qquad v|_{S_R}=0.\] The preceding estimates apply to the normalized solutions as well. The limiting equation is homogeneous with bounded drift: where \(\mathrm dv\ne0\) use \(\lambda B\nabla v/|\mathrm dv|\), and set the drift to zero elsewhere. The maximum principle and Hopf lemma exclude such a \(v\). This closes the continuation on every fixed truncation.

For uniformity as \(R\to\infty\), use \(f_0(r)=r^{2-n}(1-r^{-\beta})\) on a sufficiently distant fixed region. As in Lemma 12, \[-\Delta_g f_0-B|\mathrm df_0|\geq c_0\rho_0, \qquad Ba_0=O(r^{-n-q})\leq C_0\rho_0.\] For \(A\) larger than a fixed constant and chosen also to dominate the values on a fixed inner sphere \(S_{R_0}\), \(Af_0\) is a supersolution, because \[\lambda B\sqrt{A^2|\mathrm df_0|^2+a_0^2}+\rho_0 \leq AB|\mathrm df_0|+Ba_0+\rho_0 \leq-\Delta_g(Af_0).\] Subtraction of the two equations produces a bounded-drift inequality, so comparison is valid without a gradient monotonicity assumption. The outer Dirichlet inequality is automatic. It follows that \[ 0\leq\phi_R(x) \leq C\bigl(1+\sup_{S_{R_0}}\phi_R\bigr)r^{2-n}, \qquad R_0\leq r\leq R, \tag{34}\] with \(C\) independent of \(R\) and \(\lambda\).

If the suprema on an exhaustion were unbounded, divide by those suprema. Estimate (34) forces their unit maxima into a fixed compact set and bounds their tails by a fixed multiple of \(r^{2-n}\). Local estimates, including the fixed inner Neumann charts, yield a nonzero decaying limit solving the same homogeneous bounded-drift equation with zero Neumann data on \(S\). Its positive maximum is attained either in the interior or on \(S\); the strong maximum principle and Hopf lemma exclude both alternatives. This proves a uniform supremum bound. Exhaustion and local compactness now produce a smooth solution of (32). It is strictly positive by the same minimum argument. The difference of two decaying solutions solves a homogeneous equation with bounded drift and zero Neumann data, giving uniqueness.

To check the asymptotic derivatives using only the weak hypotheses, rescale on fixed end annuli by \(\phi_R^*(z)=R^{n-2}\phi(Rz)\) and \(g_R(z)=g(Rz)\). The equation becomes \[-\Delta_{g_R}\phi_R^* =RB(Rz)\sqrt{|\mathrm d\phi_R^*|_{g_R}^2+|z|^{-2k}} +R^{-\beta}|z|^{-n-\beta}.\] Its solutions have uniformly bounded suprema by (34). The two controlled derivatives of \(g\) give uniformly Lipschitz first-order coefficients and uniformly Hölder principal coefficients on these rescaled annuli. \(W^{2,p}\) estimates followed by \(C^{1,\gamma}\) and Schauder estimates therefore give uniform second-derivative bounds. This proves \(\phi=O_2(r^{2-n})\) without a bound on a third metric derivative. If all end derivatives of \(g\) are controlled and \(B=0\) there, the same argument differentiates repeatedly. The equation and these bounds also give \[B\sqrt{|\mathrm d\phi|^2+r^{-2k}}=O(r^{-n-q}),\] which proves the ratio limit in (33). The source is integrable. The divergence theorem, remembering that the outward domain normal on \(S\) is \(-\nu\), gives \[ \int_{S_R}\partial_{\nu_R}\phi\,\mathrm dA_g =-\mathop{\mathrm{Area}}_g(S)-\int_{\Omega\cap\{r\leq R\}} \bigl(B\sqrt{|\mathrm d\phi|^2+r^{-2k}}+\rho_0\bigr)\,\mathrm dV_g. \tag{35}\] The flux thus has a finite negative limit. Its difference from the Euclidean derivative flux is \(O(R^{-q})\).

For the final strictification, apply (26) with \(s=\delta\phi\). The new linear bulk term obeys \[k\delta\bigl(-\Delta_g\phi+K(\nabla\phi,v)\bigr) \geq k\delta\rho_0.\] Moreover \[|\mathrm d\phi|^2/\rho_0=O(r^{2-n+\beta})\] is globally bounded. Thus for sufficiently small \(\delta\) the quadratic term costs at most \(k\delta\rho_0/2\). In the purely electric case the conformal electromagnetic cost decreases. On a fixed magnetic support any existing positive margin absorbs the small change in (24); outside that support the purely electric argument applies. This proves a positive multiple of \(\delta\rho_0\) as a charged margin. On every original boundary component, exactly, \[H_\delta+\epsilon_a\mathop{\mathrm{tr}}_{S_a,g_\delta}K_\delta =e^{-\delta\phi} \bigl(H+\epsilon_a\mathop{\mathrm{tr}}_{S_a,g}K-k\delta\bigr)<0.\] The energy change is \[ \mathcal E_{\mathrm{ADM}}(g_\delta,K_\delta) -\mathcal E_{\mathrm{ADM}}(g,K) =-\frac\delta\omega \lim_{R\to\infty}\int_{S_R}\partial_{\nu_R}\phi\,\mathrm dA_g. \tag{36}\] Terms quadratic in the decaying conformal factor have vanishing ADM flux. The momentum change again has zero limit, since its integrand has one extra factor \(\phi=O(r^{2-n})\). Charge is fixed by \(\alpha\). Finally \(\phi\) is positive and bounded, giving both the area monotonicity and the uniform relative metric convergence. On a prepared end \(K=B=0\), and the scalar conformal formula with \(-\Delta\phi=\rho_0\) also proves the asserted prepared scalar decay \(O(r^{-n-\beta})\). ◻

Proof. A smooth radial majorant on the end can be patched to a sufficiently large smooth function on the core; it need not reproduce higher derivatives of \(K\). Differentiate (26) and (28). The derivative of the overall factor \(e^{-2\delta\phi}\) multiplies the active value zero. The drift term is \(O(r^{-n-q})\), and the electric term divided by \(\rho_0\) tends to zero. The assertions now follow from (33), (27), and (36). ◻

Proof. The proof has four stages. The last, a small strictification, is supplied by Lemma 15. We first construct data with nonnegative charged margin, compactly supported second tensor, and energy approaching \(m\).

Proof. In collar coordinates with \(s\geq0\) on the original side, write \(\alpha=a(s)+\mathrm ds\wedge b(s)\), where \(a(s)\) and \(b(s)\) are tangential forms. Closedness says \(\mathrm d_Sa(s)=0\) and \(\partial_sa(s)=\mathrm d_Sb(s)\). Extend the smooth family \(b(s)\) across zero and define on the interior collar \[a(s)=a(0)+\int_0^s\mathrm d_Sb(t)\,\mathrm dt.\] This gives a closed smooth extension across \(S\), with exactly the original jets. Multiply it by a cutoff on the interior side that is one on a smaller collar and zero before the end of the fixed collar, then extend by zero farther into the filling. The derivative of the result is supported where that cutoff changes, a positive collar distance behind \(S\). The same fixed extension and cutoff work before every added neck, proving uniformity. A globally closed extension is neither asserted nor needed; for nonzero total charge it would in general contradict Stokes on a one-ended filling. ◻

Proof. First fix a future-timelike original data set and a number \(s\in(0,1)\). Apply Proposition 17. The prepared energies converge to its invariant mass and are eventually positive; charges are identical and the full-cut infima have the stated lower limit. Passing to the limit in (43) gives \[ m\geq s|Q|+\frac{1-s^2}{2}X_A, \qquad 0<s<1. \tag{44}\] Only after this timelike result has been established do we apply Remark 13; Lemma 10 supplies its strictly positive lower bound. It proves \(\mathcal E_{\mathrm{ADM}}>|\mathbf P_{\mathrm{ADM}}|\) for every original purely electric data set, including the initially possible past or null cases. Thus (44) applies to all of them.

Letting \(s\uparrow1\) gives \(m\geq|Q|\). If \(X_A>|Q|\), the maximum of the right side of (44) is attained at \(s=|Q|/X_A\) when \(Q\ne0\), and by \(s\downarrow0\) when \(Q=0\); its value is \((X_A+Q^2/X_A)/2\). If \(X_A\leq|Q|\), the endpoint \(s\uparrow1\) gives precisely the required branch. These alternatives are equivalent to \(X_A\leq m+\sqrt{m^2-Q^2}\) together with \(m\geq|Q|\). The argument does not assert the polynomial inequality on the unprescribed branch. For the auxiliary magnetic class the preparation already assumes strictness and timelikeness; the first limiting argument proves its required extension without making a magnetic endpoint claim. ◻

Proof. Cut the original exterior at a large coordinate sphere. A second copy of the resulting compact manifold, with its orientation reversed and its spherical boundary capped by a ball, provides a compact filling of the entire boundary \(S\). This construction does not require the individual components of \(S\) to bound separately. Smoothly extend the original metric and tensor a short distance behind \(S\). Strictness of each expansion persists on a sufficiently short collar. In its continuation, interpolate the metric to a fixed product metric. During that interpolation add \(-\epsilon_a Lg\) to \(K\), with a nonnegative cutoff that becomes one before the original strict collar is left. Choose \(L\) sufficiently large to dominate the bounded mean curvatures of all intermediate leaves and the remaining tangential trace. One can then interpolate \(K\) to \(-\epsilon_a L_0g\) while preserving the strict inequality. This only uses a fixed transition with bounded geometry.

Insert the product cylinder between this transition and the corresponding boundary collar of the compact filling. Prescribe the same pure trace on the latter collar, and extend \(K\) smoothly and arbitrarily farther into the filling. In particular, the prescriptions for opposite signs are made on disjoint collars, and are not imposed on the common interior of the filling. There is no energy-condition requirement in that interior. The fixed pieces have finite uniform atlases; translates of finitely many product charts cover the cylinders. This proves the last assertion and the volume bound. Increasing \(C_0,C_1\) later to accommodate a prescribed ramp does not change these local bounds. ◻

Proof. We supply a tensorial derivation to retain all background-curvature terms. For symmetric tensors set \(\mathsf B(A,B)=\langle A,B\rangle-(\mathop{\mathrm{tr}}A)(\mathop{\mathrm{tr}}B)\) and \(P_A=A-(\mathop{\mathrm{tr}}A)g\). When such a tensor acts as an endomorphism, its first index is raised with \(g\). Consider on \(\mathbb R_z\times\mathcal M_N\) the stationary Lorentz metric \[\mathbf g=-l^2(\mathrm dz-\mathrm df)^2+\bar g.\] The metric induced on \(z=\mathrm{constant}\) is \(g\). Its lapse is \(U=l\sqrt D\) and its shift is \(Uw=l^2\nabla f\). With the future-normal second-form convention of the theorem, its second form is \[ b=-U^{-1}\mathop{\mathrm{sym}}\nabla(Uw)^\flat =-H^f-p(\mathrm dt\otimes w^\flat+w^\flat\otimes\mathrm dt). \tag{51}\] Let \(\mathbf n=U^{-1}\partial_z-w\). Contracted Gauss and the normal Ricci equation give, respectively, \[\mathbf R=R_g+(\mathop{\mathrm{tr}}b)^2-|b|^2-2\mathbf{Ric}(\mathbf n,\mathbf n), \qquad \mathbf{Ric}(\mathbf n,\mathbf n) =-\mathbf n(\mathop{\mathrm{tr}}b)-|b|^2+U^{-1}\Delta U.\] Because the coefficients are stationary, \(\mathbf n(\mathop{\mathrm{tr}}b)=-w(\mathop{\mathrm{tr}}b)\) and \(\mathop{\mathrm{div}}(Uw)=-U\mathop{\mathrm{tr}}b\). Substitution proves \[ \mathbf R=R_g+\mathsf B(b,b) -2U^{-1}\mathop{\mathrm{div}}\bigl(\nabla U+(\mathop{\mathrm{tr}}b)Uw\bigr). \tag{52}\] Alternatively, the coordinate \(z-f\) displays \(\mathbf g\) as a static warped product, so that \(\mathbf R=R_{\bar g}-2l^{-1}\Delta_{\bar g}l\). Since \(\mathrm dV_{\bar g}=\sqrt D\,\mathrm dV_g\) and \(\bar g^{-1}=A_d\), \[l^{-1}\Delta_{\bar g}l =U^{-1}\mathop{\mathrm{div}}\bigl((U/l)A_d\mathrm dl\bigr),\qquad \nabla U-(U/l)A_d\mathrm dl=-U\bigl(b(w,\cdot)\bigr)^\sharp.\] The last equality follows by differentiating \(U=l\sqrt D\) and using (51); its components include all derivatives of \(l\). Subtracting the two scalar formulas gives \[\tfrac12R_{\bar g}=\tfrac12R_g+\tfrac12\mathsf B(b,b) +U^{-1}\mathop{\mathrm{div}}(UP_bw).\] Put \(Q=K-b\). The product rule and \(\mathop{\mathrm{sym}}\nabla(Uw)^\flat=-Ub\) imply \[J(w)-U^{-1}\mathop{\mathrm{div}}(UP_Kw)=\mathsf B(K,b).\] Expanding \(\mathsf B(Q,Q)\) therefore yields \[ \tfrac12R_{\bar g} =\mu+J(w)+\tfrac12\mathsf B(Q,Q)-U^{-1}\mathop{\mathrm{div}}(UP_Qw). \tag{53}\]

The conformal scalar law, with dimension \(n=k+1\), subtracts \(k\Delta_{\bar g}t+k(k-1)|\mathrm dt|_{A_d}^2/2\) from \(R_{\bar g}/2\). Moreover, \[\Delta_{\bar g}t =U^{-1}\mathop{\mathrm{div}}(UA_d\mathrm dt)-p|\mathrm dt|_{A_d}^2, \qquad u=e^{(k-1)t}U.\] Changing the divergence weight from \(U\) to \(u\) in (53) consequently gives \[ \begin{split} \tfrac12e^{2t}R_{\widehat g}={}&\mu+J(w) +\tfrac12\mathsf B(Q,Q)+(k-1)P_Q(w,\nabla t) +ks_p|\mathrm dt|_{A_d}^2\\ &-u^{-1}\mathop{\mathrm{div}}\bigl(u(P_Qw+kA_d\mathrm dt)\bigr). \end{split} \tag{54}\] Equation (48), or its transverse and axial components, gives \[ u(P_Qw+kA_d\mathrm dt)=V-uFw, \qquad u^{-1}\mathop{\mathrm{div}}(uw)=F+(1-\chi)j-\tau+2s_pa_wx. \tag{55}\] For completeness, the transverse and axial components of \(P_Qw+kA_d\mathrm dt\) are \(a_wM+(k+pv)y\) and \(-a_w\mathop{\mathrm{tr}}T+kdx\), respectively; those of \(V/u-Fw\) are exactly the same. The second equality follows by taking the trace of (51) and then differentiating the weight \(e^{(k-1)t}\).

It remains to expand the algebraic terms. The blocks of \(Q\) are \(T\), \(M+pa_wy\), and \(j+2pa_wx\). Thus \[\tfrac12\mathsf B(Q,Q) =\tfrac12|T^0|^2-\frac{k-1}{2k}(\mathop{\mathrm{tr}}T)^2 +|M+pa_wy|^2-(\mathop{\mathrm{tr}}T)(j+2pa_wx),\] and \(P_Q(w,\nabla t)=a_w(M+pa_wy)\cdot y-a_w(\mathop{\mathrm{tr}}T)x\). Insert these formulas and (55) into (54), and use \(\mathop{\mathrm{tr}}T=F-\chi j\). The mixed transverse terms complete to \(|M+s_pa_wy|^2-(k-1)^2v|y|^2/4\); the remaining terms are precisely (50). This proves (49). Formula (54) and (55) also express \(\mathcal T\) by smooth tensorial operations, proving the assertion at zero slope. ◻

Proof. Completing first the \(x\) square and then the \(F\) square gives the exact formula \[ \begin{split} \mathcal T={}&\tfrac12|T^0|^2+|M+s_pa_wy|^2 +\left(ks_p-\tfrac{(k-1)^2}{4}v\right)|y|^2\\ &+ks_pd\left(x+\frac{\chi a_w}{kd}j\right)^2 +\frac{k+1}{2k}\left(F-\frac{\chi}{k+1}j\right)^2 +\frac{\chi\,[k(k+2)+v]}{2(k+1)(k+pv)}j^2 . \end{split} \tag{59}\] Indeed the last coefficient, before simplification, is \[\chi-\frac{k\chi^2}{2(k+1)} -\frac{s_p\chi^2v}{kd} =\chi\frac{k-\frac{k^2}{2(k+1)}d-\frac{k-1}{2}v}{k+pv},\] and the numerator in the final fraction equals \([k(k+2)+v]/[2(k+1)]\).

The coefficient of \(|y|^2\) is at least \((k^2-1)/4+kp\). The coefficient of \(j^2\), divided by \(d\), is at least \(c_k(1+N)^{-2}\). In the variables \(X=\sqrt d\,x\) and \(J_0=\sqrt d\,j\), the \(x\) square is \(ks_p(X+a_wJ_0/(k+pv))^2\); its shear coefficient is bounded by \(1/k\). The shear from the \(F\) square has coefficient \(\chi/((k+1)\sqrt d)\leq1/(k+1)\). These observations control \(X^2,J_0^2,F^2\) with at most \(C_k(1+N)^2\) loss. Recovering \(M\) costs at most \(C_k(1+N)\), since \(s_p^2/(ks_p-(k-1)^2v/4)\leq C_k(1+N)\). Finally \(|T|^2=|T^0|^2+(F-\chi j)^2/k\). This proves (56).

When \(\mathrm dt=0\), the \(F,j\) part is \[\frac{k+1}{2k}F^2-\frac{\chi}{k}Fj +\left(\chi-\frac{k-1}{2k}\chi^2\right)j^2.\] Put \(z=\sqrt\chi\,j\) and bound the mixed term by \((F^2+z^2)/(2k)\). Since \(0<\chi\leq1\), the claimed lower and upper bounds follow, together with the separate \(T^0,M\) terms.

For the sharper gradient bound, minimize over \(M,F,j\), keeping \(x,y\) fixed. The transverse coefficient remaining is \(ks_p-(k-1)^2v/4\geq(k^2-1)/4\). Put \[B_v=\frac{k(k+2)+v}{2(k+1)},\qquad D_v=k-\frac{k^2}{2(k+1)}d.\] The axial coefficient divided by \(d\) is \[C(p,v)=\frac{ks_pB_v}{D_v+pv}.\] As \(D_v-\frac{k-1}{2}v=B_v\), one has \(\partial_pC=kB_v^2/(D_v+pv)^2>0\). At \(p=0\) the inequality \(C(0,v)\geq(k^2-1)/4\) is equivalent to \(D_v\geq kv\), which follows from \(D_v-kv=(k-k^2/(2(k+1)))d\geq0\). This proves (58). ◻

Proof. In terms of \(h\) the trace equation reads \[ \frac{A_\chi^{ij}\nabla_i\nabla_jh} {\sqrt{(\eta/l)^2+|\mathrm dh|^2}}+\mathop{\mathrm{tr}}_{A_\chi}(sK)=h. \tag{63}\] At a positive maximum, the first term is nonpositive and the second is bounded by \(n\sup|K|\); a negative minimum is treated with the opposite sign. The supremum of \(|K|\) on the filling is uniform in \(N\). Taking \(C_h>n\sup|K|\) proves the first assertion. The same argument with \(K=0\) proves \(h=0\) on the last homotopy part.

Let \(B_R=C(r^{-b_1}-R^{-b_1})\). At a contact with \(h-B_R\), the axis of \(A_\chi\) is the radial-gradient direction, so that \[\mathop{\mathrm{tr}}_{A_\chi}\nabla^2B_R =Cb_1r^{-b_1-2} \bigl((b_1+1)\chi-k+O(r^{2-n})\bigr)<0.\] The error is uniform for \(0<\chi\leq1\); choose \(r_0\) large enough using \(b_1<k-1\). Choose \(C\) so that \(B_R\geq C_h\) on \(r=r_0\) for every \(R\geq2r_0\). At a positive interior maximum of \(h-B_R\), the Hessian term in (63) would be negative while \(h>B_R\geq0\), a contradiction. The lower barrier is \(-B_R\). Their one-sided boundary derivatives give the last estimate in (62); tangential derivatives there vanish.

On \(r=R\), \(Z=0\) and \(l\leq e\). Hence \[t=-\frac{1}{2k}\log(1+l^2|\mathrm df|^2) \geq-\frac{1}{2k}\log\bigl(1+C\eta^{-2}R^{-2b_1-2}\bigr).\] For example, in addition to \(R\geq2r_0\), require \(C\eta^{-2}R^{-2b_1-2}<e^{k\varepsilon}-1\). This gives \(t>-\varepsilon/2\) on the outer sphere and defines a permissible \(R_{\min}(N)\). ◻

Proof. First put \(B=H^f+K\). Its blocks give \[\mathcal S(B)=\frac{k-1}{2k}(F-\chi j)^2 -\tfrac12|T^0|^2+dj(F-\chi j)-d|M|^2.\] Subtracting this expression from (50) at \(\mathrm dt=0\) yields \[ \begin{split} \mathcal T-\mathcal S(B)={}&|T^0|^2+(1+d)|M|^2+F^2/k\\ &+\left(\frac{k-2}{k}\chi-d\right)Fj +\left(\chi(1+d)-\frac{k-1}{k}\chi^2\right)j^2. \end{split} \tag{65}\] The last coefficient is at least \(\chi\), while the absolute value of the mixed coefficient is at most \(d\). Fix a small dimension-dependent \(\delta_k>0\). If \((1+p)v\geq\delta_k\), Young’s inequality in the variables \(F,\sqrt\chi j\), together with \[\frac{d^2}{\chi}=\frac{d(k+pv)}k\leq1+N/k, \qquad v^{-1}\leq(1+N)/\delta_k,\] gives \(\mathcal T-\mathcal S(B)\geq\tfrac12E_0 -C_k(1+N)^2vF^2\). If \((1+p)v<\delta_k\), both \(d\) and \(\chi\) differ from one by at most \(C_k\delta_k\). The quadratic form in (65) then differs, by at most \(C_k\delta_k(|M|^2+F^2+j^2)\), from \[|T^0|^2+2|M|^2+(F-j)^2/k+j^2.\] The least eigenvalue of the last \(F,j\) form is at least \(1/(k+2)\). Choose \(\delta_k\) small enough to preserve half this lower bound. Combining the cases proves a fixed positive \(E_0\) lower bound with the stated error.

Finally, polarization of \(\mathcal S\) and \(d^2/\chi\leq1+N/k\) give \[|\mathcal S(B)-\mathcal S(H^f)| \leq C_k\sqrt{1+N}\sqrt{E_0}|K|+C_k|K|^2.\] Apply Young’s inequality, using half the positive \(E_0\) lower bound, and then (57). This proves the result. All steps are unchanged on replacing \(K\) by \(sK\). ◻

Proof. Lemma 23 excludes a boundary minimum at this value. At an interior minimum \(\mathrm dt=0\), \(\nabla^2t\geq0\). View \(\bar g\) as the metric of the graph \(z=f(x)\) in the Riemannian warped product \(g+l^2\mathrm dz^2\). At this point its second form is \(H^f\). The ambient scalar curvature is \(R_g-2p\Delta t\). With vertical unit vector \(e_z=l^{-1}\partial_z\) the graph unit normal is \(\sqrt d\,e_z-a_we\), and the ambient normal Ricci curvature equals \[v\mathop{\mathrm{Ric}}_g(e,e)-vp\nabla^2t(e,e)-dp\Delta t.\] The Gauss and conformal formulas therefore give the exact equality \[ \tfrac12 e^{2t}R_{\widehat g} =\tfrac12R_g-v\mathop{\mathrm{Ric}}_g(e,e)+\mathcal S(H^f) -vp\mathop{\mathrm{tr}}_{e^\perp}\nabla^2t-k\mathop{\mathrm{tr}}_{A_d}\nabla^2t. \tag{66}\] The last two terms are nonpositive. Compare this upper bound with (49), use \(F=h\) and \(w(F)=\eta a_w|\nabla f|\), and apply Lemma 24. It follows that \[ u^{-1}\mathop{\mathrm{div}}V \geq c_1\mathcal T+\eta a_w|\nabla f| -P_N(\rho_0+v\rho_1) \tag{67}\] for some polynomial \(P_N\) independent of \(R\) and the homotopy parameter. Here are all the error estimates used in this conclusion. The difference \(\mu-R_g/2=(\tau^2-|K|^2)/2\), \(J(w)\), and \(F\tau\) is uniformly bounded on the enlarged core and vanishes on the end, so is bounded by \(C\rho_0\). On the end \(|\mathop{\mathrm{Ric}}_g|=O(r^{-n})\leq C\rho_1\) because \(2b_1<n\). The height barrier gives \(vF^2\leq Cv\rho_1\) there; on the core its bound is absorbed by \(C\rho_0\). The same bounds hold for \(sK\), including its momentum density.

At \(t=-\varepsilon\), \(m_0=1\). Choose \(\delta_0\) as in the statement, and then take, for instance, \(C_N\geq P_N+c_*+1\). Subtracting the right side of (61) from (67) gives a strictly positive quantity, contradicting that equation. This choice is polynomial in \(N\).

In the last homotopy part, the height comparison gives \(K=f=0\), \(t=Z\), \(w=0\), and \(\mathcal T=ks_p|\mathrm dt|^2\). At a floor minimum the left side is \(k\Delta t\geq0\), whereas for \(\lambda>0\) the right side is \(\lambda(\rho-C_N\rho_0)<0\). For \(\lambda=0\) the equation is \(\mathop{\mathrm{div}}(ku(t)\nabla t)=0\) with zero boundary data. Testing by \(t\) gives \(\nabla t=0\), hence \(t=0\). This finishes the homotopy check. ◻

Proof. We give the full high-level argument for the first homotopy part and write \(K\) for \(sK\) throughout it. Put \(q_D=\sqrt D\) and \(a(t)=e^{(k-1)t}l\), so that \[ u=a(t)q_D=l e^{kZ-t},\qquad \eta a_w|\nabla f|=\frac\eta l(q_D-q_D^{-1}),\qquad q_D=e^{k(Z-t)}. \tag{69}\] The penalty can be nonzero only on \(-\varepsilon<t<-\varepsilon+1/N\); its absolute size is at most \(\mathcal P_N\). Equations (45) and (69) show that a positive threshold \(Z_0=\mathcal P_N\) can be chosen such that on \(Z>Z_0\) \[ \Xi\geq\delta_0\mathcal T+\rho +\tfrac12\delta_0\eta a_w|\nabla f|. \tag{70}\] For example, on the penalty band it suffices to make \((\eta/e)(e^{kZ}-1)\) larger than \(2C_N\sup(\rho_0+\rho_1)/\delta_0\); the logarithm of this requirement is bounded by \(C(1+N)+C\log(1+C_N)\). Off the penalty band the assertion is immediate.

Let \(\zeta\) be smooth, nondecreasing, zero up to \(Z_0\), and one above \(Z_0+1\), with bounded derivative. Testing the divergence equation by \(\zeta(Z)\) is legitimate since \(Z=0\) at the outer boundary. Writing \(A=A_\chi\) and \(B_K=K(w,\cdot)\), we obtain \[\int\zeta(Z)u\Xi\,\mathrm dV_g =-\int\zeta'(Z)u \bigl(k|\mathrm dZ|_A^2+\langle\mathrm dZ,B_K\rangle_A\bigr)\,\mathrm dV_g \leq C\int_{\{Z_0<Z<Z_0+1\}\cap\mathop{\mathrm{supp}}K}u|K|^2\,\mathrm dV_g.\] On the last set \(u\leq e\exp(k(Z_0+1)+\varepsilon)\). The support has volume \(O(1+N)\). Using (70) proves \[ \int_{\{Z>Z_0+1\}}u \bigl(\rho+\eta a_w|\nabla f|+\mathcal T\bigr)\,\mathrm dV_g \leq\exp(\mathcal P_N). \tag{71}\]

Let \(\mathcal K_N\) be the enlarged core and a fixed additional annulus containing \(\mathop{\mathrm{supp}}K\) in its interior. On this set \(\rho\geq c>0\), uniformly. Since \(l\geq\ell\geq\eta\) and \(q_D\geq1\), \[u\bigl(\rho+\eta a_w|\nabla f|\bigr) =q_De^{(k-1)t}\bigl(l\rho+\eta(q_D-q_D^{-1})\bigr) \geq c'\eta e^{(k-1)t}q_D^2.\] For any fixed \(0<\alpha\leq k-1\), \(t\geq-\varepsilon\) implies \(e^{\alpha Z}q_D=e^{\alpha t}q_D^{1+\alpha/k} \leq C e^{(k-1)t}q_D^2\). On the complementary level set \(Z\leq Z_0+1\), \(q_D\leq\exp(k(Z_0+1+\varepsilon))\). Thus (71) and the core volume bound give, in particular, the fixed-exponent seed \[ \int_{\mathcal K_N}e^Z\sqrt D\,\mathrm dV_g \leq\exp(\mathcal P_N). \tag{72}\] We chose \(\alpha=1\); this exponent does not deteriorate as \(N\) grows.

We next prove the local Sobolev inequality needed to turn (72) into a supremum estimate. Use the graph of \(f\) in the ordinary product \((\mathcal M_N\times\mathbb R,g+\mathrm dz^2)\). Over a base patch its measure and inverse metric are \[\mathrm dV_\Gamma=S_f\,\mathrm dV_g,\qquad S_f=\sqrt{1+|\nabla f|^2},\qquad A_\Gamma=g^{-1}-(1-S_f^{-2})e\otimes e.\] Both \(\mathrm dV_\Gamma\) and \(A_\Gamma\) are comparable to \(\mathrm d\mu_D=\sqrt D\,\mathrm dV_g\) and \(A_\chi\), respectively, with \(\exp(\mathcal P_N)\) factors. In fact \[\min\{1,l\}\leq q_D/S_f\leq\max\{1,l\},\qquad \frac{S_f^{-2}}\chi =\frac{1+l^2|\nabla f|^2}{1+|\nabla f|^2}\frac{k+pv}{k}.\] The second ratio lies between \(\ell^2\) and \(e^2(1+N/k)\).

The scalar mean curvature of this graph in the product is \[H_\Gamma=\frac{q_D}{lS_f} \left(\mathop{\mathrm{tr}}(T-K|_{e^\perp})+S_f^{-2}(j-K(e,e))\right).\] The coefficients satisfy \[\frac{q_D}{lS_f}\leq\max\{1,l^{-1}\},\qquad \frac{q_D}{lS_f}\frac{S_f^{-2}}{\sqrt d} =\frac{q_D^2}{lS_f^3}\leq\max\{l,l^{-1}\}.\] Consequently (56) gives \[ |H_\Gamma|\leq\exp(\mathcal P_N)(1+\sqrt{\mathcal T}). \tag{73}\] Notice that this uses only \(\sqrt d\,j\) and the transverse Hessian block, not an unweighted bound for the whole Hessian.

For clarity concerning the ambient geometry in this step, each of the finitely many fixed base models used in Lemma 20 can be extended beyond the patch to a closed smooth manifold. Choose a smooth Euclidean isometric embedding of that fixed manifold (Nash 1956), and take its product with the line. The bound on its second fundamental form is fixed; it is the same on all translated copies along a cylinder. Its contribution to the Euclidean mean curvature vector of the graph is at most \(n\) times that bound, because the horizontal projection of a graph unit vector has length at most one. Thus (73) holds also for the Euclidean mean curvature vector. A compactly supported base test lifts to a compactly supported graph test for each individual smooth \(f\).

The Michael–Simon inequality (Michael and Simon 1973), applied to \(|\varphi|^{2(n-1)/(n-2)}\) and followed by Hölder’s inequality, therefore gives \[ \|\varphi\|_{L^{2n/(n-2)}(\mathrm d\mu_D)}^2 \leq\exp(\mathcal P_N) \int\bigl(|\mathrm d\varphi|_{A_\chi}^2 +(1+\mathcal T)\varphi^2\bigr)\,\mathrm d\mu_D \tag{74}\] for compactly supported tests in each patch. The Euclidean theorem has no slope or graph-area hypothesis. The comparisons just established check precisely the metric and mean-curvature terms used here.

It remains to supply a compatible energy estimate. Globally \(\Xi\geq\delta_0\mathcal T-\mathcal P_N\). Also \[\mathrm d\log a=(k-1+p)\mathrm dt,\qquad |\mathrm d\log a|_{A_\chi}\leq\mathcal P_N\sqrt{\mathcal T}\] by (56). Test the divergence equation by \(\psi^2e^{2sZ}/a\), where \(\psi\) is a compactly supported base cutoff. The weight \(a\) cancels from \(u=a\sqrt D\). After moving the positive principal term to the left, this gives \[\begin{align*} &\int\psi^2e^{2sZ} (\delta_0\mathcal T+2ks|\mathrm dZ|_A^2)\,\mathrm d\mu_D\\ &\quad\leq\mathcal P_N\int\psi^2e^{2sZ}\,\mathrm d\mu_D +\int e^{2sZ}\psi^2\langle\mathrm d\log a,k\mathrm dZ+B_K\rangle_A\,\mathrm d\mu_D\\ &\qquad-2s\int e^{2sZ}\psi^2\langle\mathrm dZ,B_K\rangle_A\,\mathrm d\mu_D -2\int e^{2sZ}\psi\langle\mathrm d\psi,k\mathrm dZ+B_K\rangle_A\,\mathrm d\mu_D. \end{align*}\] The logarithmic-weight terms cost at most \(\delta_0\psi^2\mathcal T/2+ \mathcal P_N\psi^2(|\mathrm dZ|_A^2+|K|^2)\) by Young’s inequality. The term multiplied by \(s\) costs a fixed fraction of \(ks\psi^2|\mathrm dZ|_A^2\) plus \(Cs\psi^2|K|^2\). The cutoff terms cost another such fraction plus \(C(1+s^{-1})|\mathrm d\psi|_g^2+C\psi^2|K|^2\). It follows, on choosing a polynomial \(s_*(N)\) sufficiently large, that for every \(s\geq s_*(N)\), \[ \int\psi^2e^{2sZ} (\mathcal T+s|\mathrm dZ|_{A_\chi}^2)\,\mathrm d\mu_D \leq\mathcal P_N(1+s) \int e^{2sZ}(\psi^2+|\mathrm d\psi|_g^2)\,\mathrm d\mu_D. \tag{75}\] All absorption thresholds are polynomial in \(N\). No derivative of \(A_\chi\) is estimated in this test.

Set \(W=e^Z\), \(\kappa_*=n/(n-2)\), and \(p_0=2s_*(N)>1\). Insert \(\varphi=\psi e^{sZ}\) into (74) and use (75). For two nested base patches whose gap is \(b\in(0,1]\), the resulting estimate, with \(p=2s\geq p_0\), is \[ \|W\|_{L^{p\kappa_*}(\mathrm{inner},\mathrm d\mu_D)} \leq\bigl[\exp(\mathcal P_N)(1+p)^3b^{-2}\bigr]^{1/p} \|W\|_{L^p(\mathrm{outer},\mathrm d\mu_D)}. \tag{76}\] Iteration with \(p_i=p_0\kappa_*^i\) and gaps proportional to \(b2^{-i}\) is legitimate: both \(\sum_i1/p_i\) and \(\sum_i i/p_i\) converge. It gives \[ \sup_{\mathrm{inner}}W \leq A_Nb^{-C_n}\|W\|_{L^{p_0}(\mathrm{outer},\mathrm d\mu_D)}, \qquad A_N\leq\exp(\mathcal P_N). \tag{77}\]

We record explicitly why starting at the polynomially large \(p_0\) does not spoil the bound obtained from (72). Let \(J_*\geq1\) bound that integral and take increasing patches with successive gaps \(b_j=b_*2^{-j-1}\), all inside a fixed larger patch. If \(m_j=\log\max\{1,\sup W\}\) on patch \(j\), interpolation gives \[\|W\|_{p_0}\leq(\sup W)^{1-1/p_0}J_*^{1/p_0},\qquad m_j\leq a+cj+(1-1/p_0)m_{j+1},\] where \(a=\log A_N+(\log J_*)/p_0+C_n\log(2/b_*)\) and \(c=C_n\log2\) can be increased to be positive. The supremum on the largest compact patch is finite for each individual smooth solution. Iterating the recurrence and letting its length tend to infinity thus gives \[m_0\leq p_0a+c(p_0^2-p_0)\leq\mathcal P_N.\] This last estimate is polynomial even though \(1-1/p_0\) tends to one. The patches covering \(\mathcal K_N\) have uniform sizes and geometry, so the conclusion is uniform on that whole set.

Outside \(\mathcal K_N\) one has \(K=0\) on a neighborhood. A larger interior maximum of \(Z\) above \(Z_0\) would have \(\mathop{\mathrm{div}}V=ku\mathop{\mathrm{tr}}_{A_\chi}\nabla^2Z\leq0\), contrary to (70); the outer boundary has \(Z=0\). This proves the global upper bound for \(Z\) in the first homotopy part. In the last part \(K=f=0\), \(t=Z\), and the same high-maximum argument applies whenever \(\lambda>0\). At \(\lambda=0\) the solution is zero, as already proved.

Finally \(t\leq Z\) follows from the implicit relation, \(q_D=e^{k(Z-t)}\leq\exp(\mathcal P_N)\) follows from the floor, and \(|\nabla f|\leq q_D/l\leq q_D/\ell\). Lemma 23 gives \(|f|\leq C_h/\eta\). These are the remaining assertions of (68). ◻

Proof. We give the boundary reduction and both improvement arguments.

Proof. Choose a fixed large \(L\) and set \(Q_\pm=e^{\pm LZ}\). The distributional chain rule gives \[\begin{align*} -\operatorname{div}(A\,DQ_\pm) ={}&\operatorname{div}(\pm Le^{\pm LZ}B)\\ &+e^{\pm LZ}\big(\pm LF-L^2 A\,DZ\cdot DZ-L^2B\cdot DZ\big). \end{align*}\] If the lower ellipticity bound is \(\lambda_0\), choose \(L>2a_0/\lambda_0\). Young’s inequality absorbs the bounded-flux cross term into the remaining negative gradient square. The boundedness of \(Z\) gives \[ -\operatorname{div}(A\,DQ_\pm) \le\operatorname{div}B_\pm+C(\mathrm dx+\mu), \qquad \|B_\pm\|_\infty\le C. \tag{91}\] The matrix on the left is the same for both signs.

On \(B_r\), let \(w_\pm\) be the zero-Dirichlet potential of the right side of (91). Scalar Dirichlet Green estimates for bounded measurable uniformly elliptic coefficients give \[ \|w_\pm\|_\infty\le C(r+r^{\alpha_0}). \tag{92}\] Here is the dependence on the source hypotheses. The bound \(\mathcal G(x,y)\le C|x-y|^{2-n}\) and dyadic annuli give \(Cr^{\alpha_0}\) for the measure source. The exterior-pole gradient energy bound \(\int_{\{|x-y|>\rho\}}|D_y\mathcal G(x,y)|^2 \le C\rho^{2-n}\) gives \(L^1\) gradient mass \(C\rho\) on an annulus of scale \(\rho\). Summing gives \(Cr\) for bounded divergence data. These scalar Green bounds hold in every \(n\ge3\); see (Kang and Kim 2010, Corollary 4.1 and Theorem 3.6). One can first use the individually bounded densities here and then pass to their uniform growth bounds.

Put \(\epsilon_r=C(r+r^{\alpha_0})\), and define \[U_\pm=\sup_{B_r}Q_\pm+\epsilon_r-Q_\pm+w_\pm.\] They are nonnegative supersolutions. Let \(M=\sup_{B_r}Z\) and \(m=\inf_{B_r}Z\). On at least half of an intermediate ball either \(Z\le(M+m)/2\) or \(Z\ge(M+m)/2\). In the first case the uncorrected plus deficit is bounded below by a fixed multiple of \(M-m\); in the second case the minus deficit is. The multiplicative constant is positive because \(Z\) is bounded. Weak Harnack for the corresponding \(U_\pm\) therefore lowers the supremum, or raises the infimum, on \(B_{r/4}\), up to \(C\epsilon_r\). Thus \[\operatorname{osc}_{B_{r/4}}Z \le(1-c_0)\operatorname{osc}_{B_r}Z+C(r+r^{\alpha_0}).\] Iteration gives a Hölder modulus. The same proof applies after the stated reflection. ◻

Proof. Lemma 25 and Proposition 26 give \[-\varepsilon<t\le Z\le\mathcal P_N,\qquad \ell\le l\le e,\qquad |f|+|Df|\le e^{\mathcal P_N}.\] At fixed \(N\), the eigenvalue \(\chi=kd/(k+pv)\) is consequently bounded away from zero. The normalized trace equation is \[ A_\chi:\nabla^2f =\frac{\sqrt D}{l} \big(\eta f-\operatorname{tr}_{A_\chi}K\big). \tag{93}\] Its right side is bounded. Lemma 27 therefore bounds \(Df\) in \(C^\alpha\), including at the outer face, and gives (79) for \(f\).

Write the other equation in the form of Lemma 28, with \[A=kuA_\chi,\qquad B=uA_\chi K(w,\cdot).\] The sign of the scalar source is immaterial for its absolute bound. The implicit relation for \(t\) gives \[|Dt|\le C_N(1+|DZ|+|D^2f|).\] The explicit quadratic formula for \(\mathcal T\) then bounds the absolute scalar source by \[C_N(1+|DZ|^2)+C_N|D^2f|^2.\] All lower-order fluxes are bounded. The Hessian estimate supplies \[\int_{B_r}|D^2f|^2\le C_Nr^{n-2+2\alpha};\] we may decrease the exponent when applying Lemma 28. This proves a uniform Hölder modulus for \(Z\).

At the outer face extend \(Z\) oddly and the divergence matrix by \(A^-=RA^+R\). Extend the total flux by \(-R\), and the scalar source oddly. The normal flux is continuous, so its distributional divergence has no plane mass. Boundedness and the source-growth estimates are preserved. This justifies the use of the boundary case of Lemma 28.

The coefficients and right side of (93) are now Hölder, since they are smooth functions of \(Z,Df\) on the bounded parameter range. Interior and Dirichlet Schauder estimates give \(f\) through second derivatives. Expanding the divergence equation for \(Z\), its principal matrix is Hölder and its remaining source is bounded by \(C_N(1+|DZ|^2)\). For finite \(s>n\), use the local interpolation inequality \[ \|DZ\|_{L^{2s}(Q)}^2 \le C\,\operatorname{osc}_{Q'}Z\,\|D^2Z\|_{L^s(Q')} +C_{Q,Q'}(\operatorname{osc}_{Q'}Z)^2 \tag{94}\] on concentric ball or half-ball patches \(Q\Subset Q'\). It follows by scaled derivative interpolation after subtraction of a constant. The already proved Hölder estimate makes its Hessian coefficient uniformly small on sufficiently small fixed patches. Local nondivergence \(W^{2,s}\) estimates therefore absorb the quadratic term. The leading estimate constants stay bounded as patches shrink, since the principal matrices have a uniform Hölder modulus. Lower-order constants may depend on the selected scale. Use bounded overlap of patch enlargements and the supremum of their local Hessian norms to carry out the absorption; these norms are individually finite on every truncation. This proves \(W^{2,s}\) bounds for \(Z\).

Schauder estimates and differentiation now bootstrap both functions. At each differentiation the \(f\) equation is solved first with the known regularity of \(Z\), and then the expanded \(Z\) equation gains the next derivative. These are the usual local and Dirichlet elliptic estimates (Gilbarg and Trudinger 2001). All charts, including uniformly sized end charts and outer boundary charts, have controlled geometry. Hence the constants on fixed compact sets do not depend on the truncation radius. ◻

Proof. For any prescribed \(Z,Df\), the defining relation for \(t\) is strictly increasing in \(t\), so the coefficients are smooth functions of these variables. Continue the normalized equation (93) from \(\Delta f=f\), by straight interpolation of the two operators. Its height coefficient is strictly negative when all terms are placed on the left. At a positive maximum the gradient vanishes, \(A_\chi=I\), and the maximum test gives \(f\le\|\operatorname{tr}K\|_\infty/\eta\); the analogous bound holds at a minimum. The same height bound works throughout the interpolation.

We supply the gradient bound, since ellipticity must not be assumed before it. Put \(\sigma=|Df|\). For bounded trial \(Z\), as \(\sigma\to\infty\), the implicit relation has \(t<0\) and hence \(l=e^{Nt}\), so \[t=-\frac{\log\sigma}{k+N}+O(1),\qquad d\asymp\sigma^{-2k/(k+N)}.\] For \(N>k\) this gives \[ \chi\ge\frac{c_N}{1+\sigma},\qquad \frac{\sqrt D}{l}\le C_N(1+\sigma). \tag{95}\] The constants are uniform on a bounded set of trials. Compact \(\sigma\) ranges are included by positivity and continuity. The normalized right side, using the height bound, is at most \(C_N(1+\sigma)\) in absolute value. Straight operator interpolation has the same properties: its axial eigenvalue is at least \(\chi\).

Choose a short distance interval \([0,s_0]\) and a modulus \[\psi(0)=0,\qquad \psi''=-K_0(\psi')^2,\qquad \psi'>0.\] Choose \(K_0\) large first, then \(s_0\) sufficiently small for the distance geometry and so that \(\psi'\ge1\) can hold throughout. Finally choose the initial slope sufficiently large that \(\psi(s_0)\) exceeds twice the height bound. This is possible from \[\psi(s)=K_0^{-1}\log(1+K_0\psi'(0)s).\] On a boundary collar, \(\pm\psi(\operatorname{dist}(\cdot,\partial))\) are upper and lower barriers. The negative axial term \(\chi\psi''\), by (95), has size at least a fixed multiple of \(K_0\psi'\), and dominates the linear-growth right side and the bounded second derivatives of the distance.

For the interior estimate consider a positive maximum of \[f(x)-f(y)-\psi(\operatorname{dist}(x,y))\] with distance less than \(s_0\), using the smooth ambient metric extension across the outer boundary. Contacts at a boundary endpoint are excluded by the preceding barriers; contacts at distance \(s_0\) are excluded by the height bound. At an interior contact the two gradients are parallel along the short minimizing geodesic and have length \(\sigma=\psi'\). Parallel transverse endpoint variations bound the difference of the transverse Hessian traces by \(C\operatorname{dist}(x,y)\sigma\). The separate axial variations give \[\nabla^2f_x(e,e)\le\psi'',\qquad -\nabla^2f_y(e,e)\le\psi''.\] Thus the difference of the two operators is at most \[C s_0\sigma+(\chi_x+\chi_y)\psi'' \le C s_0\sigma-c_NK_0\sigma.\] For large \(K_0\) this contradicts the absolute linear-growth bound for the difference of their right sides. The same argument with \(x,y\) exchanged proves the Lipschitz bound. It is valid throughout the operator interpolation.

The gradient bound gives uniform ellipticity. Apply Lemma 27, then Schauder to obtain \(C^{2,\beta}\) for some \(\beta>0\). These bounds make \(Df\) Lipschitz, so the original \(C^{1,\alpha}\) trial and the smooth coefficient functions give \(C^\alpha\) coefficients; Schauder then yields \(C^{2,\alpha}\) and, by one differentiation, \(C^{3,\alpha}\). The Dirichlet linearization has a strictly negative zero-order term. The maximum principle and the usual elliptic Fredholm alternative therefore make it invertible. The same maximum principle, applied to the linear equation for the difference, gives uniqueness. Continuation is open by this invertibility and closed by the uniform estimates. The implicit-function theorem and uniqueness also give continuous dependence in the stated spaces. ◻

Proof. First truncate at a coordinate sphere of radius \(R\), with \(R\ge R_{\min}(N)\) as in the floor estimate. For a trial \(Z\in C^{1,\alpha}\) with zero outer trace solve the trace equation by Lemma 30. Freeze this pair in the divergence equation, except for the gradient of the new \(Z\) in its principal flux. The resulting linear Dirichlet solve defines a map \(T\) into \(C^{2,\alpha}\). Indeed its principal coefficients are \(C^{1,\alpha}\), its drift flux is \(C^{1,\alpha}\), and its scalar source is \(C^\alpha\). Thus \(T\) is a continuous compact map on \(C^{1,\alpha}\). Fixed points bootstrap to smoothness.

Use the homotopy described with (61), in reverse order: first take \(K=0\) and scale the full scalar source from zero to one, then scale \(K\) from zero to its specified value with the full source. When \(K=0\), uniqueness of the trace equation gives \(f=0\) for every trial. At zero scalar scaling the map is identically zero.

For a homotopy parameter \(\lambda\), write \(f_\lambda[Z]\) for the trace solve. Apply degree on the relatively open moving domain \[\left\{(\lambda,Z): \min t[Z,Df_\lambda[Z]]>-\varepsilon,\quad \|Z\|_{C^{1,\alpha}}<M_{N,R}\right\}.\] The dependence is continuous by Lemma 30. Choose \(M_{N,R}\) larger than the fixed-point bound supplied by Lemma 29. There is no norm-boundary fixed point. There is no floor-boundary fixed point by the floor exclusion, including its outer-sphere estimate and final \(K=f=0\) stage. The a priori estimates also hold with a non-strict floor inequality when taking a closure, before that exclusion is applied. The map has relatively compact image on the bounded trial closure. Leray–Schauder degree for a bounded relatively open subset of the product with the parameter interval therefore applies (Leray and Schauder 1934, secs. 12–13). Its degree at the initial zero map is one, by excision around \(Z=0\); it stays one throughout the homotopy. This gives a smooth solution above the floor on each sufficiently large truncation.

Lemma 29 gives smooth local compactness uniform in \(R\). To retain the boundary condition at infinity, we establish uniform end estimates before exhausting. On the end \(K=0\), and (93) becomes \[A_\chi:\nabla^2f=\frac{\eta\sqrt D}{l}f,\qquad \frac{\eta\sqrt D}{l}\ge\eta/e.\] For fixed \(N\), sufficiently small \(\kappa_N>0\), and a sufficiently large fixed inner radius, the functions \(C_Ne^{-\kappa_N(r-r_0)}\) are barriers for both signs of \(f\). The positive zeroth-order coefficient dominates their radial second derivatives; enlarge \(r_0\) to control the radial Hessian error. They dominate at the inner sphere and at the outer zero boundary. Consequently \(f\) decays exponentially, uniformly in \(R\). Local and outer-face estimates give the same decay for every derivative. The implicit relation then gives exponentially small \(t-Z\), with all derivatives.

Up to exponentially decaying errors, the end equation is \[ k\Delta_gZ+ k\big[(k-1)+p(Z)-\delta_0s_{p(Z)}\big]|dZ|_g^2 =\rho-\vartheta(N(Z+\varepsilon))C_N\rho_0, \qquad s_p=p+(k-1)/2. \tag{97}\] Set \(a_N(z)=(k-1)+p(z)-\delta_0s_{p(z)}\), and choose \(\Phi_N(0)=0\), \(\Phi_N'(0)=1\), with \(\Phi_N''/\Phi_N'=a_N\). On the bounded range of \(Z\), \(\Phi_N'\) has positive lower and upper bounds depending on fixed \(N\). Thus (97) implies \[\Delta_g\Phi_N(Z)=O_N(r^{-n-\beta})+O_N(e^{-\kappa_Nr}).\] Comparison with multiples of \(r^{2-n}(1-r^{-\beta/2})\), on each truncation, gives \(\Phi_N(Z)=O_N(r^{2-n})\), hence \(Z=O_N(r^{2-n})\). These barriers have the required signed Laplacian for large \(r\); their \(r^{-n-\beta/2}\) term dominates both the source and the asymptotically flat metric errors.

Scaled Poisson \(W^{2,s}\) estimates first give weighted first-derivative Hölder bounds; the source is a smooth function of \(Z\), the radius weights and exponentially small terms. Scaled Schauder and differentiation then give \(Z=O_j(r^{2-n})\) for every \(j\). The same follows for \(t\). Its Laplacian has the improved bound in (96), since \(|dZ|^2=O(r^{2-2n})\) and \(\beta<1\). Exhaustion now supplies the asserted solution with zero limit at infinity. If the limit touched \(-\varepsilon\), that point would be an interior minimum, and the identical floor calculation of Lemma 25 would exclude it. All height bounds pass to the limit. ◻

Proof. All terms in \(u\Xi\) are integrable. On the end this follows from the exponential decay of \(f\), \(\mathcal T=O(r^{2-2n})\), and the integrable radius weight \(\rho_0=r^{-n-\beta}\); on the filled compact part it is ordinary smooth integrability. The divergence theorem on the manifold without an inner boundary proves the flux limit in (98). At infinity \(V=k\nabla t+o(r^{1-n})\) in the flux sense, since \(u=1+O_N(r^{2-n})\) and all graph terms decay exponentially. The metric perturbation \(2t\delta_{ij}\) contributes \(-2k\,\partial_i t\) to the Euclidean ADM integrand. The normalization \(1/(2k\omega)\) therefore gives the second identity in (98); all mixed error fluxes vanish.

Only the penalty band can contribute negatively to the integral. There \(m_0\ne0\) implies \(-\varepsilon<t<-\varepsilon+1/N\), and hence \(\ell\le l\le e\ell\) for sufficiently large \(N\). Writing \(\sigma=|\nabla f|_g\), the definitions give \[ u\le C(\ell+\ell^2\sigma),\qquad uv\le C\ell^2\sigma,\qquad ua_w\sigma\ge c\ell^2\sigma^2. \tag{101}\] For example the last expression equals \(e^{(k-1)t}l^2\sigma^2\) exactly. The negative penalty is bounded by \[C\mathcal P_N\ell\rho_0 +C\mathcal P_N\ell^2\sigma(\rho_0+\rho_1).\] Absorb its second term into the positive \(\delta_0\eta ua_w\sigma\), using (101). The remaining negative part is bounded by \[C\mathcal P_N\ell\rho_0 +C\mathcal P_N\frac{\ell^2}{\eta}(\rho_0+\rho_1)^2.\] The integrals of \(\rho_0,\rho_0^2,\rho_1^2\) grow at most polynomially in \(N\): the core volume is \(O(1+N)\), and on the end \(4b_1>n\). This proves (99). Notice that no fixed-\(N\) Schauder constant enters this estimate.

A constant metric dilation by \(e^{2\varepsilon}\), followed by the corresponding coordinate dilation, multiplies ADM energy by \(e^{(n-2)\varepsilon}\), proving (100). All-order metric decay follows from Proposition 31 and the prepared end. The conformal scalar law, the prepared scalar tail, and \(\Delta_gt=O(r^{-n-\beta})\) give the stated scalar decay. ◻

Proof. The last assertion follows immediately from the uniform height tail in Lemma 23. We prove the first assertion by a trace supersolution on each long product and its prescribed collars. Only the trace equation is used in this comparison, so no sign condition in the distant cap is needed.

Treat a plus component. Choose a small fixed \(\delta>0\) below the strict negative leaf-expansion margin on the prescribed collars. On the middle of its long product use height \(-\delta\). There \(K=-L_0g\), with \(L_0\) the fixed large number used when constructing the filling. Towards the original boundary let the slope increase from zero to a small positive fixed value \(a\). It can be chosen as a smoothly truncated exponential: start with size of order \(\eta\), increase exponentially, and stop smoothly at \(a\), with \[|(\hbox{slope})'|\le C(\hbox{slope}+\eta).\] Its length is \(O(1+|\log\eta|)=O(1+N)\), and its total height increase is as small as desired by choosing \(a\) small. The initial smoothing from slope zero has the same displayed bound.

For a height test \(H\), the Hessian multiplier in the trace equation is \[\frac{1}{\sqrt{(\eta/l)^2+|dH|^2}}.\] Thus the axial Hessian contribution along this ramp is uniformly bounded, since \(l\le e\) and \[\frac{\hbox{slope}+\eta} {\sqrt{(\eta/l)^2+\hbox{slope}^2}}\le C.\] The transverse contribution of the product metric is zero. The trace of \(-L_0g\) is at most \(-kL_0\). Taking \(L_0\) large compared with the fixed ramp bound makes the ramp a strict supersolution while its height is at least \(-\delta\). This choice is part of the fixed filling geometry, before \(N\).

Continue at small positive slope through the fixed transitions to \(S\), keeping the height there below \(-\delta/2\). Then, on a fixed strict trapped collar on the original side, increase the slope smoothly until the height exceeds the uniform coarse height bound. All these slopes are bounded below by \(a>0\); the slopes and their derivatives on this fixed portion are fixed finite numbers. At a contact with the solution, \(\eta/l\le\ell^{1/2}\), and the axial eigenvalue satisfies \[\chi\le d =\frac{(\eta/l)^2}{(\eta/l)^2+|dH|^2}.\] Consequently the trace operator on the test converges uniformly, on that fixed portion, to the expansion of its coordinate leaf, \(H_{\mathrm{leaf}}+\operatorname{tr}_{\mathrm{leaf}}K\). It is strictly negative by construction. The error from the axial second derivative tends to zero as \(N\to\infty\). With the original choice of \(\delta\), this trace is below the test height, including while the height increases to dominate the coarse bound.

On the side of the long product facing the cap, mirror the ramp and continue farther into product geometry at fixed slope, with height increasing until it also dominates the coarse bound. This adds only a fixed amount of length after the \(O(N)\) ramp. It closes the comparison region without assuming anything about the values of \(h\) in the cap, even when opposite-sign products join there.

The test therefore dominates \(h\) on both ends of the comparison region and is a strict supersolution within it. At a positive interior maximum of \(h-H\), the two gradients agree. The implicit \(t\), \(l\), and \(A_\chi\) consequently agree when evaluating the test with the actual \(Z\), while the Hessian of \(h-H\) is nonpositive. The trace equation would give \(h\le F(H)<H\), a contradiction. Thus \(h\le H<-\delta/2\) on \(S\). For a minus component apply the same argument to \(-h,-K\): \(D,t,l,\chi\) are unchanged under the reversal, and the trace equation reverses sign. Taking the minimum over finitely many components gives \(b_2\). These barriers concern the final system; the homotopy stages need not satisfy this comparison. ◻

Proof. The strict margin and the choice of \(\rho\) in (61) give \(\mu_m-|J_m|-\rho>0\). Since \(\tau\) is compactly supported, we may choose \(b_3<b_2\), independent of \(N\), so that \[\mu_m-|J_m|-\rho-b_3|\tau| \ge\tfrac12(\mu_m-|J_m|-\rho)>0\] everywhere on the original exterior. The scalar identity (49), the two equations, and \(w(F)=\eta a_w|\nabla f|\) give \[\begin{align*} e^{2t}\frac{R_{\widehat g}}2 ={}&\mu+J(w)-\rho-h\tau+(1-\delta_0)\mathcal T\\ &+(1-\delta_0)\eta a_w|\nabla f| +m_0C_N(\rho_0+v\rho_1). \end{align*}\] The DEC in the retained electromagnetic variables implies \(\mu+J(w)\ge I_g(w)+\mu_m-|J_m|\). All displayed terms on the second line are nonnegative. This proves a positive matter remainder and the retained \((1-\delta_0)\mathcal T\) throughout the low region. On the stated band its lower bound is uniform: by Proposition 33 the band is contained in a fixed compact set. This proves (103), since \(R_{g_b}=e^{-2\varepsilon}R_{\widehat g}\).

The inverse of \(\bar g\) is \(A_d\). Hence \[|dh|_{\bar g}^2 =\eta^2d|\nabla f|_g^2 =(\eta/l)^2v\le(\eta/l)^2.\] For the \((n-1)\)-form \(\alpha=\iota_EdV_g\), the volume factor \(\sqrt D\) gives the exact norm identity \[ |\alpha|_{\bar g}^2 =d|E|_g^2+\langle w,E\rangle_g^2 =|E|_g^2-|w\wedge E|_g^2. \tag{104}\] The electromagnetic square completion gives \(I_g(w)\ge c_n^2|\alpha|_{\bar g}^2\). Set \(z=t+\varepsilon>0\). Scalar curvature scales by \(e^{-2z}\) in the preceding inequality, whereas the squared norm of an \((n-1)\)-form scales by \(e^{-2kz}\). Since \(k\ge1\), \[\frac{R_{g_b}}2 \ge c_n^2e^{-2z}|\alpha|_{\bar g}^2 \ge c_n^2e^{-2kz}|\alpha|_{\bar g}^2 =c_n^2|\alpha|_{g_b}^2.\] Finally \(e^{2z}\ge1\) and \(\bar g\ge g\), proving metric dominance. ◻

Proof. The inverse graph metric is \(A_d\). Hence (114) is exactly (58) in the Coercivity Lemma (Lemma 22). Combining that estimate with \(1-\delta_0\geq1/2\) in (103), the electromagnetic square completion above, and (113) gives (115). ◻

Proof. Let \(a_0=k-1\) and \(c=1-s^2\). On \((y_0,\infty)\) consider \[ L_{\mathrm{opt}}(y)^2= \frac{1-c e^{-a_0y}}{s^2}. \tag{121}\] This is positive and increasing, and \(L_{\mathrm{opt}}(0)=1\). Direct differentiation gives \[ k\sqrt{a_0(a_0+2L_{\mathrm{opt}}'/L_{\mathrm{opt}})} e^{a_0y}L_{\mathrm{opt}}= \frac{ka_0}{s}e^{a_0y}. \tag{122}\] Its integral from \(y_0\) to zero is \(ks\).

Here is a positive smooth replacement at the lower endpoint. Take a small \(\sigma>0\) and a smooth nondecreasing cutoff \(\chi_\sigma\) which is zero below \(y_0+\sigma/3\) and one above \(y_0+2\sigma/3\). For \(y\leq b_\sigma:=y_0+\sigma\) define \[ L(y)=L_{\mathrm{opt}}(b_\sigma) -\int_y^{b_\sigma} \chi_\sigma(v)L_{\mathrm{opt}}'(v)\,\mathrm dv, \tag{123}\] where the integrand is zero below \(y_0+\sigma/3\). For \(y\geq b_\sigma\) use \(L=L_{\mathrm{opt}}\). The definitions agree with all derivatives near \(b_\sigma\). The lower constant value is positive, since it is at least \(L_{\mathrm{opt}}(y_0+\sigma/3)\). Thus \(L\) is smooth, positive and nondecreasing on the whole line, and still has \(L(0)=1\).

Choose \(0\leq\zeta_\sigma\leq1\) smooth with compact support in \((y_0+\sigma,-\sigma)\), tending pointwise to one on \((y_0,0)\) as \(\sigma\downarrow0\), and put \[ H_0'(y)=\zeta_\sigma(y)\frac{ka_0}{s}e^{a_0y}, \qquad H_0(y)=\int_{-\infty}^y H_0'(v)\,\mathrm dv. \tag{124}\] On the support of \(H_0'\) the function \(L\) is exactly \(L_{\mathrm{opt}}\), so (118) follows from (122). Elsewhere its left side is zero. Dominated convergence gives \(H_0(0)\to ks\) from below, proving (119) after decreasing \(\sigma\). Finally, \(B_0'>0\) and, for \(y\geq0\), \(L(y)\geq1\), so \[ B_0(y)\geq\frac{e^{(k-1)y}-1}{k-1}. \tag{125}\] This proves the range assertion. ◻

Proof. We give the estimates on expanding smooth coordinate-sphere truncations of \(\mathcal M_N\), with zero outer Dirichlet value. There is no inner boundary.

Proof. Where the extended flux form is closed, expand the equation as \[ k\Delta_{g_b}y= \frac{H_0'}{B_0'}E_b(y) -k\left(k-1+\frac{L'}L\right)|\mathrm dy|_{g_b}^2. \tag{146}\] The conformal scalar formula gives \[\begin{align*} e^{2y}R_\gamma/2 &=R_{g_b}/2-k\Delta_{g_b}y -\frac{k(k-1)}2|\mathrm dy|_{g_b}^2\\ &\geq c_n^2|E_b|_{g_b}^2 -\frac{H_0'}{e^{(k-1)y}L}E_b(y) +k\left(\frac{k-1}{2}+\frac{L'}L\right)|\mathrm dy|_{g_b}^2. \tag{147}\end{align*}\] The last expression is nonnegative: the square of its mixed coefficient is at most \[ 4\frac{k(k-1)}2\, k\left(\frac{k-1}{2}+\frac{L'}L\right) =k^2(k-1)\left(k-1+2\frac{L'}L\right), \tag{148}\] exactly by (118). Completing the square in \(E_b\) proves the assertion on \(\mathcal U_N\). The positive terms beyond the electric term in (115) have not been used in this completion. If \(y\geq0\), then \(H_0'=0\) and \(L'\geq0\), so the entire additional term \(k(k-1)|\mathrm dy|_{g_b}^2/2\) can also be retained. Multiplication by \(e^{2z}\) proves (145). The metric comparison follows from \(z\geq0\), \(y\geq y_0\), and \(\bar g\geq g\).

In coordinates normalized for \(g_b\), the conformal ADM calculation is \[ \mathcal E(e^{2y}g_b)-\mathcal E(g_b) =-\frac1\omega\lim_{R\to\infty} \int_{S_R}\partial_{\nu_b}y\,\mathrm dA_{g_b}. \tag{149}\] To check the normalization, the leading perturbation is \(2y\delta_{ij}\); its Euclidean ADM integrand is \(-2k\partial_i y\). The normalizing factor is \(1/(2k\omega)\). The difference between geometric and Euclidean surface fluxes, and every quadratic error, is \(O(R^{2-n})\). Equation (128) now proves (144). The end scalar curvature remains integrable: outside a compact set \(Y\) is harmonic, so (146) has no mixed term and \(\Delta_{g_b}y=O(r^{2-2n})\), while \(R_{g_b}=O(r^{-n-\beta})\). ◻

Proof. Choose \(b'\) smooth, zero up to zero and equal to one eventually, and integrate from zero. Then \[ 0<C_-\leq C(y):=e^{y-b(y)}\leq C_+<\infty \quad(y\geq y_0), \tag{152}\] with constants independent of \(N\). For a pure-trace tensor \(-\mathfrak a\gamma\) the neutral densities are \[ \mu_{\mathrm{new}}=R_\gamma/2+\frac{nk}{2}\mathfrak a^2, \qquad J_{\mathrm{new}}=k\,\mathrm d\mathfrak a. \tag{153}\] On the band, multiply the required inequality by \(e^{2(z+y)}\). The new quadratic energy is \(nk C^2\mathcal H^2/2\), while (107) bounds the momentum by \[ kC\left(\mathcal H|\mathrm dt+b'\mathrm dy|_{\bar g} +(\eta/l)|\mathcal H'|\right). \tag{154}\] Set \(A_t=(k^2-1)/8\) and \(B_y=k(k-1)/2\). Young’s inequality absorbs the first term of (154) into the two gradient terms in (145), at a cost at most \(C^2\mathcal H^2\chi_k\), where \[ \chi_k=\frac{k^2}{4}\left(\frac1{A_t}+\frac1{B_y}\right) =\frac{2k^2}{k^2-1}+\frac{k}{2(k-1)}\leq3 \quad(k\geq3). \tag{155}\] There is no use of the \(y\)-gradient term where \(y<0\), because \(b'=0\) there. Where \(y\geq0\), that term was retained in full in the preceding lemma. Hence no curvature term is used twice. As \(nk/2\geq6\), the remaining coefficient \[ d_k=\frac{nk}{2}-\chi_k\geq3 \tag{156}\] is positive in every dimension under consideration.

It is now enough that \[ (\eta/l)|\mathcal H'|\leq c'(1+\mathcal H^2), \qquad 0<c'\leq \min\left\{\frac{c_*}{kC_+},\frac{d_kC_-}{k}\right\}. \tag{157}\] Indeed this makes the second term in (154) at most \(c_*+d_kC^2\mathcal H^2\). Choose a smooth cutoff \(\chi_h\) which is zero on \([0,b_3/4]\) and one on \([b_3/2,b_3]\). For the desired plateau \(M<\infty\), define \[ \mathcal H(s_1)=\tan\bigl((\arctan M)\chi_h(s_1)\bigr). \tag{158}\] Then \[ \frac{|\mathcal H'|}{1+\mathcal H^2} \leq\frac\pi2\|\chi_h'\|_\infty, \tag{159}\] independently of \(M\). Since \(\eta/l\leq\ell^{1/2}\to0\), one sufficiently large \(N\) ensures (157) for every finite plateau. Off the band the tensor vanishes on the required exterior and \(R_\gamma\geq0\). Although \(|h|\) need not be smooth at its zero set, its composition with \(\mathcal H\) is smooth because \(\mathcal H\) vanishes on a neighborhood of zero. This proves the lemma. ◻

Proof. Keep \(s\), a smooth profile, and \(\varepsilon\) fixed. Take \(N\) sufficiently large for all preceding constructions. Sard’s theorem gives a regular value \(c_h\in(b_3/2,b_3)\) of \(|h|\). Let \(\mathcal D_N\) be the closure of the component of \(\{|h|<c_h\}\) containing the entire far end. By (108), this component cannot cross \(S\) into the filling. Its entire boundary is a compact smooth embedded two-sided hypersurface in the original interior. It is nonempty: a path in the original connected exterior from the far end to \(S\) must cross the chosen level. All boundary components are retained.

The mean curvature of this fixed compact boundary in \(\gamma\) is finite. We may now choose the plateau \(M\) in Lemma 39 so large that, with normal toward the end, \[ H_{\partial\mathcal D_N} +\operatorname{tr}_{\partial\mathcal D_N}K_{\mathrm{new}} =H_{\partial\mathcal D_N}-k\mathfrak a<0 \quad\hbox{on every component}. \tag{161}\] This late choice does not require repeating either scalar solve or increasing \(N\), by (159). Restricted to \(\mathcal D_N\), the new tensor has compact support, since \(h\to0\) and \(\mathcal H\) vanishes near zero.

We check the neutral input’s hypotheses explicitly. The exterior \(\mathcal D_N\) is connected and oriented, smooth up to its nonempty compact boundary, and has exactly the original coordinate end with compact complement. It is complete with boundary included: it is closed in the original complete exterior, and (143) bounds its metric below by a positive constant times \(g\). The metric is smooth and has all-order differentiated decay of order \(2-n\) in normalized end coordinates. Thus the metric and tensor satisfy the stronger falloffs in Theorem 4, with any exponent strictly between \((n-2)/2\) and \(n-2\). The neutral DEC follows from Lemma 39; its matter densities are integrable because the tensor is compactly supported and the scalar curvature is integrable at infinity. The ADM momentum is zero. The ADM energy exists and is given by (144). Finally the new boundary is strictly future outer trapped by (161).

Every full cut of \(\mathcal D_N\), regarded as a subset of the original exterior, is also a full cut there. This includes any portions coincident with \(\partial\mathcal D_N\), and counts every component. Therefore its full-cut infimum \(a_N\) satisfies \[ a_N\geq e^{ky_0}a_*>0. \tag{162}\] Indeed the metric comparison acts on all \(k\) tangent directions, and the class of available cuts has only been restricted. No minimizing boundary is assumed to be smooth or to exist.

It remains to verify the timelike energy hypothesis of the neutral input for these auxiliary data. Write \(e_N=\mathcal E(\gamma)\). If \(e_N>0\), their zero momentum makes that hypothesis immediate. If \(e_N\leq0\), apply the conformal end increase of Lemma 12, taking its positive function \(\phi\) constant on a core containing the boundary and the new tensor’s support. For \(U=1+\lambda\phi\), transform the data by \[ \gamma_\lambda=U^{4/(n-2)}\gamma, \qquad K_\lambda=U^{2/(n-2)}K_{\mathrm{new}}. \tag{163}\] The neutral DEC and trapping signs persist, the stronger asymptotic derivatives persist, momentum remains zero, and full-cut areas do not decrease. Its energy is \(e_N+2\lambda\). For every arbitrarily small \(e>0\) choose \(\lambda=(e-e_N)/2\geq0\). These data satisfy all hypotheses of Theorem 4, which would give \[ e\geq\frac12(a_N/\omega)^{(k-1)/k} \geq\frac12e^{(k-1)y_0}X_*>0. \tag{164}\] Letting \(e\downarrow0\) is a contradiction. Thus \(e_N>0\), and the same neutral input applied directly gives \[ e^{(k-1)\varepsilon}(\mathcal E_*+o_N(1)) -\frac{H_0(0)}k|Q| \geq\frac12 e^{(k-1)y_0}X_* =\frac{1-s^2}{2}X_*. \tag{165}\]

The parameter order is as follows. For the fixed prepared data, fix \(s\in(0,1)\), a profile tolerance, and \(\varepsilon>0\) first. Complete the two scalar solves and the regular cut for each sufficiently large \(N\); the tensor plateau is chosen last for that \(N\). Now let \(N\to\infty\) in (165), then let \(\varepsilon\downarrow0\), and then let the profile tolerance decrease to zero. Equations (119) and (116) give (160). Only fixed-parameter estimates were required in the second scalar solve. The electromagnetic input used in this proof was precisely (112) and its lower bound by \(I_g(w)\), which also holds for the indicated compact magnetic form. ◻

Proof. Apply the rest preparation and strict preparation in Proposition 17 and Lemma 15, retaining the two closed forms. The magnetic support stays in the compact original region. There the strictly positive margin absorbs the arbitrarily small rescaling errors. The gluing and bending annuli can be taken outside that support, so on those annuli and on the end the electric estimates are unchanged. This yields prepared data with charge \(Q\), energies \(\mathcal E_{*,j}\to m\), and full-cut infima \(a_{*,j}\geq(1-o_j(1))a\). For each fixed \(s\), apply Proposition 40 and then pass to this limit. The right side is increasing in the area variable, so the lower comparison of infima is sufficient. This proves (168). The elementary optimization is given in Proposition 19. The argument does not require the component electric fluxes to have one sign. ◻

Completion of Theorem 5. Proposition 40 establishes (43) for every prepared data set with positive energy and zero momentum, and every fixed \(s\in(0,1)\). The Reduction of the Numerical Theorem (Proposition 19) therefore proves Theorem 5, including positivity of the full enclosing area, strict future timelikeness of the original ADM vector, and the stated branch distinction. The endpoint limits in \(s\) are taken after the fixed-\(s\) inequality is established, so no profile estimate uniform at \(s=0\) or \(s=1\) is required. ◻

Proof. Large coordinate spheres have positive mean curvature, uniformly for the finite-dimensional metric paths considered below. On the region outside one such sphere, the outward unit normal to their foliation has positive divergence. Applying the divergence theorem to the portion of a filled competitor outside that sphere shows that clipping off that portion cannot increase perimeter, and strictly decreases it if the discarded portion has positive volume. The argument holds for almost every cutting radius by finite-perimeter slicing. Choose two fixed radii, both outside the compact core. Clipping at the smaller radius gives a common compact truncation, strictly inside the larger one, for minimization and for all sufficiently small parameters of each path. BV compactness and lower semicontinuity now give a minimizer; see (Ambrosio et al. 2000).

A smooth full cut bounds a filled competitor with exactly the same area, including its contact portions. Conversely, first push \(O\) and an arbitrary filled competitor slightly outward by a smooth ambient diffeomorphism supported in a collar of \(S\). The pushed competitor contains a neighborhood of the original \(O\), and its perimeter tends to the original perimeter. Strict approximation of finite-perimeter sets by smooth domains can therefore be carried out while retaining that neighborhood. It is supported inside a fixed compact truncation. Retain the exterior component containing the end, filling any bounded exterior components of the smoothed set. This only removes boundary components. Its entire smooth intrinsic boundary is a full cut. Passing through these two approximations proves equality of the infima. The same removal observation shows that a minimizing frontier cannot leave a removable bounded complementary open component: a nonempty such component has positive perimeter, whose removal would improve the minimum. The usual indecomposable-component decomposition gives the same conclusion for finite-perimeter representatives (Ambrosio et al. 2000).

Here is the local estimate needed at the obstacle. For a local competitor \(G\), replace it by \(G\cup O\). Submodularity gives \[ P(G\cup O)\le P(G)+P(O)-P(G\cap O). \tag{173}\] Extend the smooth outward normal of \(O\) to a smooth vector field of norm at most one near its boundary. Its divergence is bounded. The calibration inequality for this field gives \(P(O)-P(G\cap O)\le C|O\setminus G|\) for changes in a sufficiently small coordinate neighborhood. Since \(F\) minimizes among sets containing \(O\), it follows that \[ P(F;B)\le P(G;B)+C|F\mathbin{\triangle}G| \tag{174}\] whenever \(F\mathbin{\triangle}G\) is compactly contained in the small ball \(B\). This is an unconstrained perimeter almost-minimizing estimate. In coordinates the Riemannian area integrand is the ellipsoidal integrand associated with \(\mathcal A=(\det g)g^{-1}\), which is smooth and uniformly elliptic. The error in (174) is \(O(r^n)\) and therefore is bounded by \(C_\gamma r^{n-1+\gamma}\) for any fixed \(0<\gamma<1\) on sufficiently small balls. The almost-minimizer results in (Simmons 2022, Propositions 4.5–4.7 and Theorems 1.1, 9.5, 10.2–10.3) apply. They give density and perimeter bounds, local compactness, regularity off a set of dimension at most \(n-8\), and convergence as single \(C^1\) graphs near free regular limit points. On free patches, the uniform local \(C^{1,\gamma/4}\) graph bounds and the smooth minimal-graph equation upgrade this to smooth convergence on smaller patches by interior elliptic estimates. Tangent sets are perimeter-minimizing cones. The minimal-boundary formulation and dimension reduction are also described in (Simon 2014, chap. 7, Section 5).

At a contact point a tangent filled cone contains the tangent half-space of \(O\). The corresponding minimizing hypercone is therefore supported on one side of its boundary plane, and is that plane. One way to see the last assertion is to integrate the signed coordinate-height equation on its stationary spherical link; the height has one sign and its spherical Laplacian is a negative constant times itself. The singular set can be removed in this integration by its dimension and the perimeter bounds. Equivalently one can use the half-space cone theorem in (Simon 2014, chap. 7, Section 4, Theorem 4.5). The almost-minimizer regularity theorem now gives a \(C^{1,\beta}\) graph at contact for some \(\beta>0\); one may take \(\beta=\gamma/4\). Estimate (174), differentiated along smooth flows, bounds its weak mean curvature. Difference quotients in the uniformly elliptic prescribed-mean-curvature graph equation give \(W^{2,2}\) regularity. Written in nondivergence form, that equation has Hölder coefficients and bounded right-hand side, so local \(W^{2,p}\) estimates give all finite \(p\); see (Gilbarg and Trudinger 2001).

If two minimizing frontiers meet at a free regular point of one of them, their union and intersection also minimize by (173), with equality in the sum of the perimeters. They are ordered relative to the given regular sheet. Their tangent cones at contact are one-sided relative to its tangent plane, and hence are planes. Regularity and the strong comparison principle for the minimal graph equation imply local coincidence. Strict compactness and regular convergence then give the asserted uniform graph neighborhoods: otherwise a sequence of failing frontiers would converge strictly to a minimizer through the point, where multiplicity-one regular convergence gives precisely the missing neighborhood. The difference of two distinct ordered graphs solves a linear uniformly elliptic equation. Positive-solution Harnack and interior derivative estimates imply, on smaller patches, that its gradient is bounded by a uniform constant times its pointwise separation.

Finally let \(F_t\) minimize for \(g_t\). Compactness, lower semicontinuity and comparison with a fixed \(g_0\) minimizer show that every convergent subsequence at \(t=0\) converges strictly in perimeter to an element of \(\mathcal F_*\). Strict continuity for vector measures then gives convergence of the tangent-plane area measures (Ambrosio et al. 2000). The area Taylor expansion has a uniform remainder on the resulting perimeter-bounded family. Fixed minimizers give the upper bound in (172); the varying minimizers give the lower bound. This also proves compactness of the minimizing family and continuity of \(F\BeginAccSupp{method=hex,unicode,ActualText=21A6}\OriginalMapsto\EndAccSupp{}a'_F(h)\) in its strict topology. ◻

Proof. Consider the linear image of the test space with components \[ \left(\mathfrak C'/\rho_0,\ -\theta'_+,\ \{c a'_F(h)-2k\omega\mathcal B'\}_{F\in\mathcal F_*}\right). \tag{180}\] The constraint test space is the one-point compactification of the closed active-ray set, with one artificial infinity point added even if that set is bounded. At infinity assign \(2k\) times the coefficient of the strict direction. The other two test spaces are \(S\) and the compact minimizing family. The normalized functions are continuous by (177) and Proposition 42.

This image does not meet the open cone of functions strictly positive on the disjoint union of the three compact test spaces. Indeed such a direction has positive strict-direction coefficient because of the infinity test. Apply that positive conformal direction exponentially to the data after making the other variations additively. On compact subsets, strict first-order positivity on the active rays implies feasibility for all sufficiently small positive parameters: rays outside a fixed neighborhood of the active set have a positive zeroth-order gap. On the end the nonconformal linear errors are \(O(r^{-n-q})\) and are dominated by \(\rho_0\). After retaining the nonnegative background through the conformal formula, the quadratic errors are bounded by \(Ct^2\rho_0\). The strict first-order term therefore gives \(\mu_m-|J_m|\ge c_t\rho_0\), with \(c_t>0\). The forms remain closed; a newly created magnetic form has fixed compact support. Trapping is strict, the ADM vector remains future timelike, and completeness, decay, integrability and existence and differentiability of the fluxes are preserved. These are precisely the hypotheses of Corollary 41. The area infimum stays positive by uniform metric comparison on the compact core and the controlled tail. Equation (172) and strict positivity of the objective components would now violate the numerical inequality for small positive parameter.

Hahn–Banach separation of a linear subspace from an open convex cone gives a nonzero positive functional on the continuous functions on this compact disjoint union. The Riesz representation theorem supplies the three nonnegative measures. Its objective measure has positive mass. Otherwise testing the strict direction, uniformly positive on the two constraint test spaces, would force both remaining measures to vanish as well. Normalize the objective measure to have mass one. On finite active rays divide the constraint measure by \(\rho_0\) to obtain \(\Lambda\). This measure is finite over compact subsets, including the boundary. The possible infinity atom contributes nothing on compact tests or prototypes. Rearranging the annihilation identity gives (179). ◻

Proof. Fix a smooth lapse test \(s\) compactly supported in the open exterior, and choose a local Gaussian Lorentzian metric inducing \(g,K\) at time zero. Its spatial metric and first normal jet are fixed, while its second normal metric jets are free. Choose a two-form whose initial electric and spatial components are \(c_n E\) and zero. Its first normal derivatives can be chosen so that both Maxwell equations hold on the initial slice. In general dimension the tangential Maxwell constraints are \(\mathrm dD_m=0\) and \(\mathrm d\alpha=0\); the remaining equations prescribe the \(n(n-1)/2\) magnetic and \(n\) electric normal evolution components. The induced flux-form variations under normal motion with velocity \(sN\) are compactly supported and closed, as follows also from Cartan’s formula applied to these initial Maxwell jets.

Let \(\mathsf S\) denote the electromagnetic stress tensor with the normalization in Definition 2. It is divergence-free at the initial slice. For each \(\delta>0\), prescribe there the spatial components of Einstein minus \(\mathsf S\) to be \[ T^{(\delta)}_{ij} =\frac{(J_m)_i(J_m)_j}{\mu_m+\delta}. \tag{182}\] Spatial trace reversal of the second normal metric jets is invertible, so these are legitimate smooth jet prescriptions. The normal and mixed components remain \(\mu_m,J_m\) by Gauss–Codazzi. The resulting variation has \(h=2sK\).

We check that \(\mathfrak C'\to0\) uniformly on compact active test sets. The tensors in (182) converge spatially in \(C^1\). On \(\{\mu_m>0\}\) this is ordinary smooth convergence; near a zero of \(\mu_m\), DEC gives \(|J_m|\le\mu_m\) and smoothness gives \(\mathrm d\mu_m=\nabla J_m=0\) at that zero. The bounds \[ |T^{(\delta)}|\le\mu_m, \qquad |\nabla T^{(\delta)}| \le 2|\nabla J_m|+|\mathrm d\mu_m| \tag{183}\] make the convergence uniform across the zero set. At a positive-density active ray, the limiting spacetime tensor is \(\gamma\otimes\gamma/\mu_m\), where \(\gamma(N)=\mu_m\) and \(\gamma|_{T\Omega}=J_m\). It is causal and annihilates the null vector \(N+v\). Parallel-transport that null vector along spatial curves. Its contraction with the causal covector is nonnegative and vanishes at the base point; differentiation at the minimum therefore shows that all relevant spatial covariant derivatives, with last argument \(N+v\), vanish. Bianchi and the initial Maxwell equations then give the same limiting vanishing for the normal derivative contracted with \(N,N+v\). Changes of the first argument cost zero, and changes of the second cost zero because it remains null and \(\gamma\) is proportional to its metric dual. At a matter zero, both the tensor and its spatial derivatives vanish by (183), and the divergence equation again suffices. These observations prove the asserted uniform convergence of \(\mathfrak C'\).

Apply (179) to these tests and pass to the limit. They have no boundary or ADM variation, so \[ \int_{\mathcal F_*}\int_{\partial^*F} s\operatorname{tr}_{T\partial^*F}K\,\mathrm dA_g\,\mathrm d\pi(F)=0. \tag{184}\] At free regular points, Proposition 42 provides uniform noncrossing graph patches, and frontiers through a common point have the same tangent plane. Consequently the tangential trace is a single-valued continuous function on each such patch of the integrated surface measure. There is no cancellation between different tangent planes in (184). A countable cover and Fubini imply the first conclusion, initially almost everywhere and then everywhere on regular sheets by smoothness.

At contact, the \(W^{2,p}\) graph and the smooth obstacle have the same second jets almost everywhere on their coincidence set. The frontier therefore satisfies \(H+\operatorname{tr}_{\rm tan}K=0\) almost everywhere throughout the contact graph, including its free portion. Uniform ellipticity first improves its regularity and then bootstraps it to a smooth solution of the expansion equation. The strong comparison principle gives coincidence with \(S\) near contact. The set of coincidence is open and closed along the connected \(S\), so the whole \(S\) is present. Its area is already the minimum \(A\), leaving no area for another sheet. If there is no contact, the compact frontier is separated from \(S\), and the stated equations hold on all its regular points. ◻

Proof. For a compactly supported interior variation \(K'=p\), the coefficient after integration by parts is \[ 2\bigl[u(\tau g-K)-\operatorname{sym}\nabla X +(\operatorname{div}X)g\bigr]:p. \tag{187}\] Its trace gives \(\operatorname{div}X=-u\tau\), and its trace-free part then gives (185). For metric tests the scalar-curvature principal adjoint is \(\operatorname{Hess}u-(\Delta u)g\). Connection variations in the momentum term give smooth linear expressions in \(u,X,\nabla X\); fixed-flux field terms and the ray-transport term are algebraic in those moments. The area term has coefficient \(-c(g-\nu_F^\flat\otimes\nu_F^\flat)/2\) on this side of (179). Inverting trace reversal gives (186), with the displayed positive normal-normal measure.

Differentiating (185) and commuting covariant derivatives expresses every component of \(\nabla^2X\) as a smooth linear combination of \(\operatorname{Rm}*X\), \(K*\mathrm du\) and \((\nabla K)*u\). Taking traces gives a strongly elliptic Laplace system for \((u,X)\) with smooth lower-order coefficients. Its source measure has local growth \(O(r^{n-1})\), uniformly in the minimizer, by the local perimeter bound. A localized elliptic parametrix has order \(-2\) and kernel bounds \(C|x-y|^{2-n-j}\) for \(j=0,1\). Such a potential of a measure with this growth is locally \(C^\gamma\) for every \(0<\gamma<1\): split the potential difference at twice the point separation, use the growth bound nearby and the first kernel derivative on successive outer annuli to get \(O(r(1+|\log r|))\). Lower-order terms are included in the parametrix, and the homogeneous remainder is smooth. Thus \(u,X\) are Hölder continuous.

The traced differentiated equation for \(X\) now has the form of the divergence of a Hölder field plus a Hölder function; local elliptic estimates give \(X\in C^{1,\gamma}\), after decreasing \(\gamma\) if needed. The lower term in (186) is therefore continuous. Its positive measure term implies a local lower bound for the covariant Hessian in the distributional sense. Slice this inequality in smooth flow boxes whose longitudinal curves are unit-speed geodesics. Multiplying by the smooth volume Jacobian gives the one-dimensional distributional inequality \((u\circ\gamma)''\ge-C\) on almost every such curve. Continuity extends its integrated convexity inequality to every curve in a smaller box. This is local geodesic semiconvexity, which implies a local Lipschitz bound by bounding one-sided slopes on short geodesic segments with endpoints in a fixed larger ball. The differentiated equation for \(X\) consequently has bounded right-hand side componentwise, so \(\nabla^2X\) is locally bounded and \(X\in C^{1,1}\). The measure inequalities now reduce to pointwise inequalities by continuity. ◻

Proof. Fix any point of \(\operatorname{graph}(f_{\ell_0})\cap W\) and take signed normal-distance coordinates \((y,s)\) about that sheet on a smaller patch compactly contained in \(W\), writing the sheet as \(\Sigma=\{s=0\}\). Put \(m_0=\lambda(\{\ell_0\})>0\). The \(ss\) equation in (190) is a one-dimensional second-derivative equation after slicing by almost every normal line. The bounded nonmeasure terms include \(B\) and the coordinate connection terms. The atom at \(s=0\) has coefficient \(cm_0/2\): on the reference graph the unit conormal is \(ds\) and the sheet and volume Jacobians agree at \(s=0\). No other graph produces an atom there, by (192). Consequently \(\partial_su\) has bounded one-sided traces \(v^\pm(y)\) for almost every \(y\), and \[ v^+(y)-v^-(y)=\frac c2m_0. \tag{193}\] The existence and boundedness of these traces can also be read directly from local semiconvexity of \(u\) along the normal lines.

We compare the tangential equations from the two sides. Let \(A_s\) be the second fundamental form of the parallel sheet \(\{s=\mathrm{constant}\}\), with normal \(\partial_s\). With \(A_s(V,W)=g(\nabla_V\partial_s,W)\), \[ (\nabla^2u)|_{T\Sigma_s} =\nabla^2_{\Sigma_s}(u|_{\Sigma_s})+A_s\partial_su. \tag{194}\] Average this identity on each of \(0<s<\varepsilon\) and \(-\varepsilon<s<0\), pairing with any smooth compactly supported tangential tensor in the \(y\) variables. The intrinsic Hessian terms have the same limit: integrate twice in \(y\), then use uniform convergence of \(u(y,s)\) to \(u(y,0)\) and smooth convergence of the parallel metrics. The \(B\) terms also have the same limit, since \(u,X,\nabla X\) are continuous.

The averaged tangential components of the sheet measure tend to zero. Indeed the conormal of a nearby graph has tangential component bounded by \(C|s|\) relative to \(\Sigma\) at a point of height \(s\). This is the graph-difference estimate recalled above, with the smooth change from graph coordinates to normal coordinates included. Thus in a slab of width \(\varepsilon\) the tangential-tangential source is bounded in total variation by \(C\varepsilon^2\) times the total sheet measure in a fixed box. Dividing by \(\varepsilon\) still gives a quantity tending to zero. The atom on \(\Sigma\) itself has no tangential component.

Subtracting the two limits in (194) gives \(A_0(v^+-v^-)=0\) as a tensor distribution on \(\Sigma\). Equation (193) and \(c m_0>0\) imply \(A_0=0\). The sheet is smooth, so the conclusion holds at every point of the patch. ◻

Proof. Choose a base point \(p\) in the specified patch. Since its graph parameter is a nonatom in the support of the leaf measure, choose distinct realized graphs converging smoothly on compact subpatches to the reference graph, with representatives \(D_j\) in the full-measure family. Such representatives exist because every graph neighborhood has positive pushforward measure, whereas the reference fiber has measure zero. Compactness of \(\mathcal F_*\) permits passing to a subsequence with \(D_j\to D'\) for some minimizing filled set \(D'\). We do not assert that \(D'=D\) away from the selected component.

The limiting frontier of \(D'\) contains the reference graph near \(p\). Indeed points on the converging graphs are frontier points of \(D_j\), and uniform density and local \(L^1\) convergence put their limits on the frontier of \(D'\). Set \[C=\Sigma\cap\operatorname{frontier}(D').\] This set is nonempty and relatively closed in \(\Sigma\). It is also relatively open: at each of its points the reference frontier is free and regular, and the common-sheet property of Proposition 42 makes the frontier of \(D'\) agree with it in a neighborhood. Connectedness gives \(C=\Sigma\). In particular every point of \(\Sigma\) is a regular point of the limiting frontier \(D'\).

Choose a nested exhaustion of \(\Sigma\) by connected precompact smooth domains \(\Omega_m\) containing \(p\), with \(\overline{\Omega_m}\subset\Omega_{m+1}\); if \(\Sigma\) is compact, use \(\Sigma\) itself. Use the fixed normal of \(\Sigma\) and its normal exponential map on a tubular neighborhood of each compact stage. Regular multiplicity-one convergence to \(D'\) gives, for sufficiently large \(j\) at that stage, a unique normal height \(h_j\) over \(\Omega_m\). A finite covering of the slightly larger compact stage by regular graph patches supplies this assertion uniformly on \(\overline{\Omega_m}\). Uniqueness on each normal fiber makes the heights agree on overlaps and on successive exhaustion stages. Thus there is no choice of branch or monodromy in this construction.

The heights have a strict constant sign on each connected stage. If \(h_j\) vanished at a point, the common-sheet property would make its zero set relatively open as well as closed there; connectedness would force coincidence at \(p\), contrary to the distinct initial graphs. Passing to a subsequence fixes the sign at \(p\), and hence on every exhaustion stage. Replace \(h_j\) by \(-h_j\) in the linear difference equations if that sign is negative, and normalize by \(h_j(p)>0\). The equations \(H=0\) and \(\operatorname{tr}_{\mathrm{tan}}K=0\) are unaffected by choosing the fixed graph normal, even if a representative filled set induces the opposite orientation on its local graph. Subtract the three graph equations (189). The integral of the linearization along the segment between the two graphs gives linear elliptic equations for \(h_j\), whose coefficients converge smoothly on compact subsets to those of \(L_0,L_+,L_-\). Harnack’s inequality and interior elliptic estimates applied to the first equation bound \(h_j/h_j(p)\) above and below, with all derivatives, on each exhaustion stage. A diagonal subsequence converges smoothly to \(\varphi\), with \(\varphi(p)=1\). Harnack’s inequality gives positivity. Passing to the limit in all three equations proves (195). The same argument applies when the approach is from the other side, since all three linearized equations are homogeneous. ◻

Proof. We first identify the sign of the singular term in the stationary curvature calculation. For smooth positive \(u\) and smooth \(X\), set \[ \mathbf g=-u^2 dz^2+ g_{ij}(dx^i+X^i dz)(dx^j+X^j dz),\qquad \mathbf n=u^{-1}(\partial_z-X). \tag{199}\] The second fundamental form of a stationary slice is \(b=-\operatorname{sym}\nabla X/u\), with the convention for \(K\) used in the theorem. Define \(\mathsf B(P,Q)=P:Q-(\operatorname{tr}P)(\operatorname{tr}Q)\). Integrating the momentum term in the gravitational constraint pairing at fixed lapse and shift gives the algebraic expression \[\begin{align*} u\bigl(R_g-\mathsf B(K,K)+2\mathsf B(K,b)\bigr) &=u\bigl(R_g+\mathsf B(b,b) -\mathsf B(K-b,K-b)\bigr). \tag{200}\end{align*}\] At \(b=K\) its first metric variation agrees with that of the stationary scalar-curvature action, up to divergences. The spatial metric coefficient of that variation is \(-u\mathbf G_{ij}\), where \(\mathbf G\) is the Einstein tensor of (199). This identity can equally be obtained by substituting \(b=-\operatorname{sym}\nabla X/u\) and integrating the derivatives of a compactly supported metric variation; it uses no field equation.

Let \(\mathsf S\) be the Maxwell stress of a two-form whose slice electric field is \(c_nE\) and whose slice magnetic part is zero. Only these slice values are needed here. Holding the electric flux form fixed under \(g'=h\) gives \[ \delta\bigl(2u c_n^2|E|^2\,dV_g\bigr) =u c_n^2(2E_iE_j-|E|^2g_{ij})h^{ij}\,dV_g =-u\mathsf S_{ij}h^{ij}\,dV_g. \tag{201}\] Adding the variation of the volume density in the full matter pairing does not alter the adjoint coefficient, by \(u\mu_m+J_m(X)=0\). Transport of a unit-ball ray contributes \(-(J_m)_{(i}X_{j)}\), where parentheses include the factor \(1/2\). It follows from the actual metric-adjoint expression in (190) that a Hessian source \(M\) contributes \[ u(\mathbf G-\mathsf S)_{ij} =-(J_m)_{(i}X_{j)}+ \bigl((\operatorname{tr}_g M)g-M\bigr)_{ij}. \tag{202}\] For a positive semidefinite \(M\), the last tensor is positive semidefinite: for a unit vector \(v\), complete \(v\) to an orthonormal basis and sum \(M\) on the remaining basis vectors. In particular the sign is preserved by trace reversal in precisely this direction.

The lower Hessian term can be read off explicitly. Write \(\tau_b=\operatorname{tr}_g b\) and \((b^2)_{ij}=b_i{}^l b_{lj}\). The spatial stationary Einstein identity is \[\begin{align*} u\mathbf G_{ij} &=(\Delta_g u)g_{ij}-(\nabla_g^2u)_{ij} +(\mathcal C_b)_{ij}, \tag{203}\\ \mathcal C_b &=u\left[\operatorname{Ric}_g+\tau_b b-2b^2 -\tfrac12(R_g+\tau_b^2+|b|^2)g\right] -\mathcal L_Xb+X(\tau_b)g. \end{align*}\] At \(b=K\), put \(T=u\mathsf S-\operatorname{sym}(J_m\otimes X^\flat) -\mathcal C_K\). Equation (202) then gives \[ B(u,X,\nabla X)=\frac{\operatorname{tr}_gT}{n-1}g-T. \tag{204}\] Thus \(B\) has no \(du\) term. All its coefficients are smooth fixed-data coefficients multiplying \(u,X,\nabla X\). For the mollified pair, the same identity with \(b=b_\epsilon\) shows that \(C^1\) convergence of \(b_\epsilon\) controls every non-Hessian spatial term.

We justify the use of this calculation at the regularity of (190). Fix relatively compact coordinate patches \(U'\Subset U\Subset\{u>0\}\), and choose \(a>0\) with \(u\geq a\) on \(U\). All smoothing below is performed in \(U\), then restricted to \(U'\). Let \(\rho_\epsilon\) be a nonnegative standard mollifier, and convolve the coordinate components of \(u\) and \(\beta=X^\flat\): \(u_\epsilon=u*\rho_\epsilon\), \(\beta_\epsilon=\beta*\rho_\epsilon\), \(X_\epsilon=g^{-1}\beta_\epsilon\). The spatial metric is not changed. In components the first equation of (190) reads \(\partial_{(i}\beta_{j)}=\Gamma_{ij}^{l}\beta_l-uK_{ij}\). Hence the residual \[ R_\epsilon:=\operatorname{sym}\nabla\beta_\epsilon+u_\epsilon K = (\Gamma\beta)*\rho_\epsilon-\Gamma\beta_\epsilon -(uK)*\rho_\epsilon+u_\epsilon K \tag{205}\] tends to zero in \(C^1(U')\). More explicitly, for a smooth coefficient \(A\) and a Lipschitz function \(f\), \[ \|(Af)*\rho_\epsilon-A(f*\rho_\epsilon)\|_{C^1(U')} \leq C\epsilon\bigl( \|A\|_{C^2(U)}\|f\|_{L^\infty(U)} +\|A\|_{C^1(U)}\|Df\|_{L^\infty(U)}\bigr). \tag{206}\] For its derivative, differentiate the product to obtain the sum of the commutator of \(DA\) with \(f\) and the commutator of \(A\) with \(Df\). Both are bounded by the first moment of \(\rho_\epsilon\) times the displayed norms. No second derivative of \(f\) is used. Since \(u_\epsilon\geq a/2\) and \(Du_\epsilon\) is uniformly bounded, \[ b_\epsilon:=-\operatorname{sym}\nabla X_\epsilon/u_\epsilon =K-R_\epsilon/u_\epsilon\longrightarrow K \quad\hbox{in }C^1(U'). \tag{207}\]

In the Hessian equation, first express the measure-valued components relative to coordinate Lebesgue measure, dividing by the smooth positive volume density. Positivity is unchanged. Convolve this already un-trace-reversed equation. It gives \[ \nabla_g^2u_\epsilon =B_\epsilon+\mathcal M_\epsilon+ (\Gamma\,du)*\rho_\epsilon-\Gamma\,du_\epsilon, \qquad B_\epsilon\longrightarrow B\quad\hbox{uniformly on }U'. \tag{208}\] The connection commutator is \(O(\epsilon)\) in \(C^0\) because \(du\) is bounded. For every \(x\), \(\mathcal M_\epsilon(x)\) is a positive semidefinite matrix. We trace this matrix using \(g(x)\) only after convolution. Consequently \[ P_\epsilon(x):= (\operatorname{tr}_{g(x)}\mathcal M_\epsilon(x))g(x) -\mathcal M_\epsilon(x)\geq0 \tag{209}\] exactly, even if its norm diverges as \(\epsilon\downarrow0\). There is no product commutator involving a variable metric times an unbounded singular Hessian.

Apply the smooth stationary calculation to \(u_\epsilon,X_\epsilon\) and \(b_\epsilon\). Formula (207) controls the induced constraint densities and all the non-Hessian spatial adjoint terms uniformly. Indeed (203) uses only \(b_\epsilon,\nabla b_\epsilon,u_\epsilon,X_\epsilon, \nabla X_\epsilon\) outside the Hessian term. Equation (204) shows that uniform convergence of \(du_\epsilon\) is unnecessary. Writing \(\mathbf T_\epsilon=\mathbf G_\epsilon-\mathsf S_\epsilon\), we obtain uniformly on \(U'\) \[\begin{align*} \mathbf T_\epsilon(\mathbf n_\epsilon,\mathbf n_\epsilon) &=\mu_m+o(1),& \mathbf T_\epsilon(\mathbf n_\epsilon,\cdot)|_{TU'} &=J_m+o(1),\tag{210}\\ u_\epsilon(\mathbf T_\epsilon)_{ij} &=-(J_m)_{(i}X_{j)}+(P_\epsilon)_{ij}+o(1).&& \end{align*}\] To see the null sign of the bounded terms, when \(\mu_m>0\), (181) forces \(|X|=u\) and \(J_m=-\mu_m X^\flat/u\). Their limiting block tensor is \(\mu_m\ell\otimes\ell\), with \(\ell(\mathbf n)=1\) and \(\ell(V)=-g(X,V)/u\). When \(\mu_m=0\), all its blocks vanish. This description, or the direct formula \(u\mu_m(1\mp g(X,\nu)/u)^2\), proves nonnegativity on \(\mathbf n\pm\nu\). Maxwell stress is also nonnegative on null vectors. The exact positivity of \(P_\epsilon\) therefore gives \[ u_\epsilon\operatorname{Ric}_{\mathbf g_\epsilon} (\mathbf n_\epsilon\pm\nu, \mathbf n_\epsilon\pm\nu)\geq-o(1). \tag{211}\] This is a one-sided estimate; it requires no uniform bound on the full spacetime curvature.

For completeness, apply the null shape-operator equation to the outgoing null geodesics orthogonal to the fixed sheet in the smooth metric \(\mathbf g_\epsilon\). Use nearby stationary slices as the parameter surfaces. At the initial slice the null velocity with \(dz/d\tau=1\) is \(u_\epsilon(\mathbf n_\epsilon+\nu)\). Subtracting the stationary translation \(\partial_z\) leaves the spatial velocity \(u_\epsilon\nu-X_\epsilon\), whose normal component is \(u_\epsilon-g(X_\epsilon,\nu)\). The Riccati equation gives the expansion derivative \[ -u_\epsilon\bigl( |A_\Sigma+(b_\epsilon)_{\mathrm{tan}}|^2+ \operatorname{Ric}_{\mathbf g_\epsilon} (\mathbf n_\epsilon+\nu,\mathbf n_\epsilon+\nu)\bigr), \tag{212}\] up to a bounded renormalization factor times the initial expansion. Tangential transport contributes a bounded tangential velocity times the tangential derivative of that expansion. Both error terms tend to zero uniformly: \(H+\operatorname{tr}_\Sigma K=0\) and \(b_\epsilon\to K\) in \(C^1\). The spatial expansion linearization has coefficients converging uniformly to those of \(L_+\). Its principal part is always the fixed \(-\Delta_\Sigma\); convergence of its action on \(u_\epsilon-g(X_\epsilon,\nu)\) therefore holds in distributions, using uniform convergence of these functions and uniform bounds on their first derivatives. Combining (211)–(212) proves the plus inequality in (198). Replace the future normal by \(-\mathbf n_\epsilon\), equivalently reverse stationary time and \(X_\epsilon,b_\epsilon\), to obtain the minus inequality. Since \(U'\Subset\{u>0\}\) was arbitrary, this proves the stated local result. ◻

Proof. The singular dimension bound is \(k-7\), and local perimeter bounds give \(\mathop{\mathrm{Area}}(T\cap B_\rho)\leq C\rho^k\); see (Simon 2014). If the singular set is nonempty, cover a compact part of it by balls \(B_{r_i}\) with \(r_i\) arbitrarily small and \(\sum_i r_i^{k-2}\) arbitrarily small. Choose cutoffs equal to one on \(B_{r_i}\), supported on \(B_{2r_i}\), with gradient bounded by \(C/r_i\). The maximum of these cutoffs has Dirichlet energy at most \(C\sum_i r_i^{k-2}\) and support of area tending to zero. Truncation and smooth approximation produce cutoffs \(\chi_j\), vanishing near the singular set and tending to one almost everywhere, with \[ \int|\nabla_T\chi_j|^2\,dA\longrightarrow0. \tag{214}\] The same construction applies on compact cone annuli and their spherical links, where the singular link dimension is at most \(k-8\).

On the regular part, test the subsolution inequality by \(\eta^2\chi_j^2w\), using smooth approximations or positive truncations if needed. Integration by parts and Young’s inequality bound the Dirichlet integral of \(w\) on smaller patches by \[ \int\eta^2\chi_j^2|\nabla_Tw|^2 \leq C\|w\|_\infty^2 \int\bigl(|\nabla_T\eta|^2+ \eta^2|\nabla_T\chi_j|^2\bigr) +C C_0\|w\|_\infty\int\eta^2. \tag{215}\] Thus \(w\) has locally square-integrable weak gradient across the singular set. In a bounded compact test, the extra terms containing \(\nabla\chi_j\) tend to zero by Cauchy–Schwarz and (214). This proves the weak extension. The same proof on any union of regular components is valid, since its boundary within the support lies in the singular set.

The Michael–Simon surface Sobolev inequality (Michael and Simon 1973), applied in coordinate or isometrically embedded compact patches, gives for compactly supported \(v\) \[ \left(\int|v|^{2k/(k-2)}\,dA\right)^{(k-2)/k} \leq C\int\bigl(|\nabla_Tv|^2+v^2\bigr)\,dA. \tag{216}\] The ambient embedding contributes a uniformly bounded mean-curvature term. The inequality extends to the above weak functions by the same capacity cutoffs. After scaling a ball to radius one, apply the energy estimate to successive positive powers of \(w+C_0r^2\) and nested radial cutoffs. The Sobolev exponent multiplies each power by \(k/(k-2)\); the resulting geometric iteration gives the \(L^2\)-to-\(L^\infty\) estimate (213). All constants depend only on the ambient patch, dimension, and perimeter bound. Their scaled versions remain bounded as \(r\downarrow0\). ◻

Proof. Only \(k\geq7\) can occur. Let \(L=C\cap\mathbb S^k\) be the link, let \(s=\log r\), and put \(V=|A_L|^2\). The inequality becomes \[ z_{ss}+(k-2)z_s+\Delta_Lz\geq Vz. \tag{218}\] We first justify testing across the singular link. On each compact axial slab, the Caccioppoli test \(\eta^2\chi_j^2z\) gives local \(W^{1,2}\) bounds and a bound for \(\int V\eta^2\chi_j^2 z^2\); the axial first-order term is bounded by the same energy inequality. Passing through the capacity cutoffs of Lemma 49 proves the weak inequality across the singular set. Testing with a nonnegative function equal to one on a smaller slab also bounds \(\int Vz\) there. Positivity allows Fatou’s lemma for this term. The terms involving \(\nabla\chi_j\) tend to zero by the energy bound and (214). These observations justify integration of (218) over the whole link against any nonnegative compactly supported axial test.

In particular the bounded nonnegative function \(q(s)=\int_Lz(s,\theta)\,dA_L(\theta)\) satisfies \[ q''+(k-2)q'\geq0 \tag{219}\] on the whole real line. Convolve in \(s\). Each smooth bounded convolution satisfies the same inequality, so \(e^{(k-2)s}q'(s)\) is nondecreasing for that convolution. If \(q'(s_0)<0\), then for \(s<s_0\) \(q'(s)\leq q'(s_0)e^{(k-2)(s_0-s)}\). Integrating backward contradicts boundedness. Thus every convolution is nondecreasing, and the original \(q\) has a nondecreasing representative. If \(z\) is nonzero, its forward limit \(q_\infty\) is strictly positive.

Choose \(w\in C_c^\infty((1,2))\), \(w\geq0\), with integral one, and put \(w_T(s)=T^{-1}w(s/T)\). The functions \[ \overline z_T(\theta)=\int_{\mathbb R}w_T(s)z(s,\theta)\,ds \tag{220}\] are bounded uniformly, and their integrals tend to \(q_\infty\). Take a weak-star convergent subsequence in \(L^\infty(L)\) with limit \(\overline z\geq0\). For a compactly supported nonnegative regular link test \(\psi\), integration by parts in \(s\) bounds the axial terms by \(\|z\|_\infty\|\psi\|_{L^1(L)}(\|w_T''\|_1+(k-2)\|w_T'\|_1)\), which tends to zero. Therefore \[ \Delta_L\overline z\geq V\overline z, \qquad \int_L\overline z\,dA_L=q_\infty>0. \tag{221}\] Bounded subsolution regularity and the capacity argument give the needed local weak derivatives. Test this inequality by \(\chi_j^2\overline z\) and use Young’s inequality to obtain \[ \frac12\int_L\chi_j^2|\nabla_L\overline z|^2 +\int_LV\chi_j^2\overline z^2 \leq2\|\overline z\|_\infty^2 \int_L|\nabla_L\chi_j|^2\longrightarrow0. \tag{222}\] Thus \(\overline z\) is constant on every connected regular component of \(L\), and a positive constant can occur only on a totally geodesic component.

We show that a nonflat minimizing boundary cone has no such component. A connected totally geodesic regular component lies in an equator \(Q\cong\mathbb S^{k-1}\). It is relatively open there; its relative boundary lies in the singular set, since at a regular boundary point the same smooth sheet continues. A closed set of Hausdorff dimension less than \((k-1)-1\) does not disconnect \(Q\): in coordinate balls one connects points by polygonal paths whose vertices are chosen outside the measure-zero exceptional sets of segments meeting that set, and then concatenates such paths. Here the singular set has dimension at most \(k-8\). It follows that this component contains \(Q\) minus the singular set, and its closure contains the entire equator.

Any other regular component lies strictly in one open hemisphere determined by \(Q\). It cannot meet \(Q\) at a regular point without being part of the same regular sheet. On that component the signed coordinate height \(h\) is strictly of one sign and satisfies \(\Delta_Lh=-(k-1)h\). It and its gradient are bounded. Integrate against capacity cutoffs, extended by zero on other components. The boundary terms tend to zero by Cauchy–Schwarz; hence \(\int h=0\), a contradiction. There are no other regular components. Regular points are dense in the support, and multiplicity one now says that \(C\) is the plane through \(Q\). The componentwise interpretation is also consistent with the decomposition of a multiplicity-one oriented minimizing boundary into minimizing regular components; see (Simon 2014, chap. 7, Section 5, Corollary 5.11 and Remark 5.12). This contradicts nonflatness. Therefore (221) is impossible and \(z=0\). ◻

Proof. The Ricci curvature is bounded on a compact neighborhood of the frontier; choose \(C\) with \(\operatorname{Ric}_g(\nu,\nu)\geq-C\). For \(\delta>0\) put \(w_\delta=(1-\varphi/\delta)_+\) on \(\Sigma\), and extend it by zero on other regular components. Convex truncation of the Jacobi equation gives \(\Delta w_\delta\geq-C\); on \(\{\varphi<\delta\}\) this follows from \(-\Delta\varphi/\delta=(|A|^2+\operatorname{Ric}(\nu,\nu)) \varphi/\delta\geq-C\), and the truncation contributes a nonnegative measure. Moreover \(0\leq w_\delta\leq1\) and \(w_\delta\to0\) almost everywhere as \(\delta\downarrow0\).

Choose first a fixed radius \(r>0\) so small that the \(Cr^2\) term in (213), including its constant, is less than \(1/4\). Cover the compact frontier by finitely many balls of radius \(r/2\). Dominated convergence in their radius-\(r\) enlargements then permits choosing \(\delta>0\) so that the integral term is less than \(1/4\) in every ball. Thus \(w_\delta\leq1/2\) on \(\Sigma\), and \(\varphi\geq\delta/2\). This proves the first assertion with the required order of choices, \(r\) before \(\delta\).

Set \(z=1/\varphi\) and extend it by zero to the other regular components. It is bounded and satisfies \[ \Delta_\Sigma z =\bigl(|A|^2+\operatorname{Ric}_g(\nu,\nu)\bigr)z +2\varphi^{-3}|\nabla_\Sigma\varphi|^2 \geq(|A|^2-C)z. \tag{224}\] Fix a singular point \(p\) and any sequence of scales \(r_j\downarrow0\). Pass to a subsequence along which the rescaled frontiers converge to a multiplicity-one minimizing tangent cone \(C_p\). The cone is nonflat: a multiplicity-one planar tangent cone would imply regularity of the frontier at \(p\). These are the standard compactness and excess-regularity conclusions for minimizing boundaries (Simon 2014).

On compact subsets of the regular cone the convergence is smooth. The rescaled \(z\) have subsequences converging locally uniformly to a bounded nonnegative function \(z_0\). Here is a useful precise way to obtain this convergence. On a connected regular patch, either \(z\) tends to zero at one base point, in which case Harnack for the positive Jacobi function \(\varphi\) forces \(z\to0\) on smaller patches, or \(z\) stays bounded below there along a subsequence. In the latter case the same Harnack inequality and interior estimates bound \(\varphi\) and its derivatives, giving smooth convergence of \(z\). The uniform upper bound for \(z\) was already established. Branches on which the zero extension is used cause no difficulty; choose the branch in each regular patch and diagonalize. Thus one obtains \(z_0\) on the whole regular cone, allowing zero components. Scaling (224) and passing to regular patches gives \(\Delta_{C_p}z_0\geq|A_{C_p}|^2z_0\). Lemma 50 implies \(z_0=0\).

This regular convergence implies vanishing of the normalized \(L^2\) integral of \(z\) on each fixed bounded rescaled ball. To check points near the singular cone, first discard a small neighborhood of that set and of the cone vertex. Their limiting area can be made arbitrarily small. Area convergence and the uniform bound for \(z\) control the discarded part, while smooth regular convergence controls its complement. The scaled version of (213), with source bounded by \(Cr_j^2\|z\|_\infty\), now gives uniform decay on smaller balls.

In particular, if \(x_j\in\Sigma\) tends to \(p\), choose scales \(r_j\) comparable to \(d_g(x_j,p)\). The preceding argument and the mean-value estimate show \(z(x_j)\to0\); any contrary subsequence would have a tangent-cone subsequence of the type just treated. A sequence approaching the compact singular set has a subsequence tending to a particular singular point. Applying this fixed-point argument to that subsequence proves the uniform sequential assertion in (223). ◻

Proof. On \(W=B_0\times I_0\), the coordinate \(tt\) equation in (190) is the following equality of measures: \[ \partial_t^2u =b(y,t)\,dy\,dt+ dy\int_{\mathcal G}w(\ell,y) \mathbf1_{I_0}(f_\ell(y)) \delta_{f_\ell(y)}(dt)\,d\lambda(\ell). \tag{226}\] Here \(y\in B_0\) and the left side and bounded-density term are restricted to \(W\) as measures. The indicator records the restriction of the graph measure to the working \(t\)-interval; it is not a differentiation of a characteristic function. The density \(b\) is bounded, and \(0<w_0\leq w(\ell,y)\leq w_1\) for the contributing graphs. The weight is \(c/2\) times the squared conormal on \(\partial_t\), times the graph area Jacobian, divided by the volume Jacobian. All contributions on \(W\) are included because their frontiers lie in \(\mathcal H_V\). Pairing with product tests and using the graph area formula proves the display; measure approximation then gives its linewise disintegration.

For almost every \(y\), \(t\BeginAccSupp{method=hex,unicode,ActualText=21A6}\OriginalMapsto\EndAccSupp{}u(y,t)\) is Lipschitz and its derivative \(v_y=\partial_tu(y,\cdot)\) is a one-dimensional BV function. This follows from the sliced equation, or from semiconvexity and Fubini. Write \(Z_y=\{t:u(y,t)=0\}\). At any point of \(Z_y\) where the two one-sided precise values of \(v_y\) agree, their common value is zero: a nonzero common value would contradict the nonnegative minimum of the primitive \(u(y,\cdot)\) at that point. The diffuse derivative of a one-dimensional BV function does not charge a fixed precise-value level. In the usual \(D^av+D^cv\) notation this locality statement is \[ |D^av+D^cv|\bigl(\{\widetilde v=0\}\bigr)=0; \tag{227}\] it follows from the BV chain rule, or directly from the one-dimensional coarea formula; see (Ambrosio et al. 2007, Proposition 8, equation (16)). Consequently the diffuse part of \(Dv_y\) gives no mass to \(Z_y\). The countably many jump points do not affect that assertion.

The bounded term \(b\) is zero almost everywhere on \(\{u=0\}\). Indeed \(X=0\) there by \(u\geq|X|\), and the first derivatives of \(u\) and \(X\) vanish almost everywhere on their respective zero sets. Thus the linear expression \(B(u,X,\nabla X)\) and the connection term involving \(du\) both vanish almost everywhere there. Fubini gives \(b(y,\cdot)=0\) almost everywhere on \(Z_y\) for almost every line.

Finally, by (192) the pushforward of \(w(\ell,y)d\lambda^{\mathrm{na}}(\ell)\) under \(\ell\BeginAccSupp{method=hex,unicode,ActualText=21A6}\OriginalMapsto\EndAccSupp{}f_\ell(y)\) is atomless for every \(y\). All line atoms of the area source in (226) therefore come from transverse atoms of \(\lambda\); there are no hidden atoms produced by the evaluation map. On almost every line, restrict the diffuse part of (226) to \(Z_y\) and use (227) and the preceding vanishing of \(b\). It follows that \[ \int_{\mathcal G}w(\ell,y) \mathbf1_{I_0}(f_\ell(y)) \mathbf1_{\{u(y,f_\ell(y))=0\}} \,d\lambda^{\mathrm{na}}(\ell)=0 \tag{228}\] for almost every \(y\). The integrand is defined to be zero whenever \(f_\ell(y)\notin I_0\), so \(u\) is evaluated only inside \(W\). For each leaf set \[m_W(\ell)=\int_{B_0}w(\ell,y) \mathbf1_{I_0}(f_\ell(y))\,dy.\] This quantity is strictly positive whenever its graph meets \(W\): that intersection projects to a nonempty open subset of \(B_0\), and \(w\geq w_0>0\). A uniform positive lower bound for \(m_W\) over all leaves is neither asserted nor needed. Integrate (228) in \(y\) and use Tonelli’s theorem. On \(\mathcal Z_W\) the result is \[\int_{\mathcal Z_W}m_W(\ell)\,d\lambda^{\mathrm{na}}(\ell)=0.\] Strict positivity of \(m_W\) there gives \(\lambda^{\mathrm{na}}(\mathcal Z_W)=0\), proving the claim. ◻

Proof. First consider a connected regular component \(\Sigma\) with a nonatomic support patch. Choose the common field \(\varphi\) of Lemma 47. The functions \(Q_\pm\) of (197) are bounded on the compact frontier. By Lemma 51, \(Q_\pm/\varphi\) tends to zero at every singular end. If one of these ratios is positive somewhere, it therefore attains a positive maximum at a regular interior point. This is also immediate when \(\Sigma\) is smooth and compact.

Write \(Q_\pm=\varphi v\). For an operator \(L=-\Delta+b\cdot\nabla+d\) with \(L\varphi=0\), direct product differentiation gives \[ L(\varphi v)=\varphi\left[ -\Delta v+(b-2\nabla\log\varphi)\cdot\nabla v\right]. \tag{230}\] At any point where \(Q_\pm>0\), one has \(u>0\), so Lemma 48 applies. The weak strong maximum principle for the locally Lipschitz \(v\) and the smooth coefficients in (230) implies that \(v\) is constant at its positive maximum on the connected component of \(\{Q_\pm>0\}\) containing that point. This component is also closed in \(\Sigma\): at a regular limit point, \(Q_\pm=v\varphi>0\) by continuity. Since \(\Sigma\) is connected, it is all of \(\Sigma\). Substitution in (198) then gives \[ A_\Sigma\pm K_{\mathrm{tan}}=0, \qquad |A_\Sigma|\leq |K|. \tag{231}\] The weak maximum principle used here is the ordinary local elliptic one, for example (Gilbarg and Trudinger 2001); no maximum principle on an unresolved singular boundary is used. If neither ratio is positive anywhere, \(Q_+=Q_-=0\) on \(\Sigma\), hence \(u=0\) there and \(X=0\) there by domination. Lemma 52 excludes multiplier mass on this alternative in every nonatomic patch. In atomic patches Lemma 46 already gives \(A=0\).

We make the exceptional set independent of the regular incidence point. The open inner boxes \(W\) constructed above cover all free regular points of all minimizing frontiers. Their union is an open subset of the second-countable ambient manifold, so choose a countable subcover \(W_i\), retaining the corresponding middle boxes \(V_i\) and graph maps \(q_i\) on the open families \(\mathcal H_{V_i}\). In particular the open sets \(\mathcal H_{V_i}\times W_i\) cover the regular incidence set in the family-times-manifold topology.

For each \(i\) let \(\lambda_i=(q_i)_*(\pi|_{\mathcal H_{V_i}})\), let \(\mathcal A_i\) be its countable set of atoms, and let \(\mathcal Z_i=\mathcal Z_{W_i}\) be the zero-leaf set from Lemma 52. These are measurable sets. For the last assertion, the condition that a graph meets \(W_i\) is open in the graph topology. The condition that \(u\) vanishes on its intersection with \(W_i\) can be checked at a countable dense set of base points: failure gives, by continuity, an open base set on which the graph stays inside \(W_i\) and \(u>0\). Thus it is a Borel condition.

The set \[\mathcal N_i= \bigl(C^\infty(B_{1,i})\setminus\operatorname{supp}\lambda_i\bigr) \cup(\mathcal Z_i\setminus\mathcal A_i)\] has \(\lambda_i\)-measure zero, since the second term has \(\lambda_i\)-measure \(\lambda_i^{\mathrm{na}}(\mathcal Z_i)=0\). Consequently \[\pi\bigl(\{F\in\mathcal H_{V_i}:q_i(F)\in\mathcal N_i\}\bigr)=0.\] Remove the countable union of these pullbacks, together with the already excluded null set in the good family. For a remaining frontier and any of its regular points \(p\), choose \(i\) with \(p\in W_i\). Its leaf is either an atom, giving \(A=0\) locally, or a nonatom in the support of \(\lambda_i\). In the latter case the common Jacobi argument applies on its entire connected regular component. If that component had \(u=0\) everywhere, the leaf would belong to \(\mathcal Z_i\setminus\mathcal A_i\), which was removed. Therefore the positive-ratio alternative gives \(|A|\leq |K|\). This proves the curvature bound simultaneously at every regular point of every remaining frontier. In particular, \[ |A|\leq\sup_{\mathcal K}|K|, \tag{232}\] where \(\mathcal K\) is a common compact set containing the minimizing frontiers.

Fix such an interior frontier and any proposed singular point. Blow it up at scales tending to zero. The tangent cone is a multiplicity-one minimizing boundary cone, with smooth convergence on its regular part. The rescaled bound (232) tends to zero, so every regular component of the cone is totally geodesic. The equator and height argument in Lemma 50 shows that the cone is a plane. Multiplicity-one planar tangent regularity contradicts the point being singular. The entire compact frontier is therefore smooth.

This frontier is a genuine enclosing cut. In particular the minimizing filled set has no removable bounded open component of its complement: filling one would preserve the obstacle condition and strictly reduce its full perimeter. Its components carry the outward normals of that enclosing set, and (189) gives zero future expansion on every component. The assumed exclusion of a smooth interior enclosing marginal frontier rules it out. Consequently \(\pi\)-almost every frontier is \(S\). The filled set with frontier \(S\) and the prescribed obstacle is unique modulo null sets, and \(\pi\) is a probability. This proves (229). ◻

Proof. Smoothness and boundary moments. Proposition 53 removes the interior surface measure from (186). Together with the differentiated KID equation, the resulting equations express all second derivatives of \(u,X\) as smooth linear expressions in their first jets. Ordinary elliptic regularity makes them smooth in the open exterior. The full first-jet system is closed; along the normals of a smooth collar it is a linear ODE with smooth coefficients. Starting on an interior collar face, solve that ODE to \(S\). Smooth dependence on the tangential parameters gives smooth extension of all fields and their jets. This argument also shows that a zero full first jet at one interior point would force the solution to vanish throughout the connected open exterior.

There can still, at this point, be constraint moment measures supported on \(S\). Integrate the smooth interior fields by parts in (179). Take \(K'=0\) and both form variations zero, and choose a metric test with value and first jet zero at \(S\). The trace of its normal second derivatives in the tangential slots is arbitrary, so its scalar-curvature variation at \(S\) is arbitrary. Its area and expansion variations, all integrated interior boundary terms, momentum variation and ray-transport terms at \(S\) vanish. Equation (179) therefore kills the scalar boundary moment. Positivity and domination of the vector moment by the scalar moment kill the vector boundary moment as well. Henceforth the moments in that identity are just the smooth functions \(u,X\) times volume.

The end constants and positive lapse. The closed first-jet system gives, for \(Y=(u,X)\) in end coordinates, \[ |D^2Y|\le C r^{-1-q}|DY|+C r^{-2-q}|Y|. \tag{241}\] Indeed curvature, \(\nabla K\), \(K^2\), the constraints and the electric stress multiply values, whereas \(K\) and metric first derivatives multiply first derivatives. These orders require only the stated \(O_2/O_1\) tails. Radial integration and Gronwall first give \(Y=O(r)\) and \(DY=O(1)\): write \(|Y(r)|\) as its initial value plus the radial integral of \(|DY|\) and use the integrability of \(r^{-1-q}\). Equation (241) now gives a limit of \(DY\) on each ray with error \(O(r^{-q})\). The limits agree on all rays, since integration of \(D^2Y\) along a path of length \(O(r)\) on a large sphere gives an \(O(r^{-q})\) difference. Denote the common linear part by \(\mathcal A\).

The scalar affine part cannot be nonzero: \(u\ge0\) on all large spheres. Once that part is zero, \(|X|\le u\) rules out every growing vector affine part as well. Thus \(\mathcal A=0\). Since \(n\ge4\) and \(q>(n-2)/2\), we have \(q>1\). Integration gives bounded \(Y\), a common constant limit \(Y_\infty\), and initially \(Y-Y_\infty=O(r^{1-q})\). Reinserting boundedness in (241) improves the Hessian to \(O(r^{-2-q})\), and two integrations give \(Y-Y_\infty=O_2(r^{-q})\). The stated weaker exponent \(q_0\) is therefore available.

Apply (179) to the end prototypes and integrate by parts on large truncations. The interior adjoint equations and the compact boundary terms cancel. Only the constant scalar multiplying the linear scalar ADM flux, and twice the constant vector multiplying the linear momentum flux, remain. All other boundary terms vanish by (241), the prototype orders and the background decay. The surviving identity is \[ 2k\omega\bigl(b^0\delta\mathcal E_{\mathrm{ADM}} +b^i\delta(\mathbf P_{\mathrm{ADM}})_i\bigr) =2k\omega\bigl(u_\infty\delta\mathcal E_{\mathrm{ADM}} +X_\infty^i\delta(\mathbf P_{\mathrm{ADM}})_i\bigr). \tag{242}\] The prototype fluxes span all entries, proving (234). If \(u\) vanished at an interior point, smooth nonnegativity would give \(\mathrm du=0\) there. Domination would give \(X=0\) and \(\nabla X=0\) there as well, since \(|X|\le u=O(d^2)\). The full-jet uniqueness noted above would contradict \(u_\infty=b^0>0\). Thus \(u>0\) in the interior.

The stationary Einstein equation. The stationary metric in (235) has second fundamental form \(b=-\operatorname{sym}\nabla X/u=K\) on each slice. We recall the metric-adjoint computation, now with no interior area source. Put \(\mathcal Q(A,B)=A:B-(\operatorname{tr}A)(\operatorname{tr}B)\). Integration of the momentum pairing gives \[ u\bigl(R_g-\mathcal Q(K,K)+2\mathcal Q(K,b)\bigr) =u\bigl(R_g+\mathcal Q(b,b)-\mathcal Q(K-b,K-b)\bigr). \tag{243}\] At \(b=K\) its metric first variation equals that of the stationary scalar action, at fixed lapse and shift, up to the integrated divergence. Its spatial coefficient is \(-u\operatorname{Ein}_{ij}\). Equation (188) supplies the electromagnetic coefficient. The variation of the constraint volume factor vanishes by complementarity, and isometric transport of the active unit-ball rays adds \(-(J_m)_{(i}X_{j)}\). Thus \[ u(\operatorname{Ein}-\mathsf S)_{ij} =-(J_m)_{(i}X_{j)}. \tag{244}\] Gauss–Codazzi gives the remaining normal and mixed components as \(\mu_m,J_m\). Where \(\mu_m=0\), DEC gives \(J_m=0\). Where \(\mu_m>0\), equality in \[\begin{equation*} -J_m(X)\le |J_m||X|\le\mu_m u \end{equation*}\] forces \(|X|=u\) and \(J_m=-\mu_m X^\flat/u\). Consequently all components combine to give (236), and the extra matter can occur only at \(W=0\).

Maxwell tests and a global potential. Test (179) with the metric and second tensor fixed and a compactly supported closed variation of \(\alpha\). The resulting identity, after dividing by the nonzero factor \(4c_n^2\), is \[ \int_\Omega uE^\flat\wedge\alpha'=0 \quad\hbox{for every compactly supported closed } (n-1)\hbox{-form }\alpha'. \tag{245}\] Exact tests imply \(\mathrm d(uE^\flat)=0\). All closed tests, not only exact ones, are available. Compact-support de Rham duality on the oriented open manifold \(\operatorname{int}\Omega\) therefore makes \(uE^\flat\) exact; see (Bott and Tu 1982). This step does not presume simple connectivity. Define \(p\) by \(\mathrm dp=(k-1)uE^\flat\). Since \(u\) is bounded at infinity and \(E=O_1(r^{-k})\), radial integration and integration along large spheres give one common limit of \(p\) at infinity and \(p-p_\infty=O_2(r^{2-n})\). Choose \(p_\infty=0\). Integration in local collars extends \(p\) smoothly to \(S\).

Independent closed compact two-form tests of \(D_m\) give \[ \int_\Omega D_m'\wedge i_X\alpha=0, \qquad\hbox{and hence}\qquad \mathrm d(i_X\alpha)=0. \tag{246}\] The degree of \(i_X\alpha\) is \(n-2\), so this pairing and its differential have the required degrees in every dimension. With orientation \(\mathop{\mathrm{vol}}_{\mathbf g}=u\mathrm dz\wedge\mathop{\mathrm{vol}}_g\), a direct orthonormal-frame computation gives \[ *_\mathbf g\mathbf F=-c_n \bigl(\alpha+\mathrm dz\wedge i_X\alpha\bigr). \tag{247}\] Equations (245), (246) and \(\mathrm d\alpha=0\) therefore give both Maxwell equations. Contraction of \(\mathbf F\) with \(N=(\partial_z-X)/u\) and pullback to a slice yields \(c_nE^\flat\); its spatial pullback is zero. This verifies all conventions in the claimed original-data realization.

Boundary laws. The outward integration normal of \(\Omega\) at \(S\) is \(-\nu\). The boundary coefficient of a free second-tensor variation in (179) is \[ -2\operatorname{sym}(X^\flat\otimes\nu^\flat) +2X_\nu g -\frac{\mathrm d\zeta}{\mathrm dA_g} (g-\nu^\flat\otimes\nu^\flat). \tag{248}\] Interpreting this first as an equality of measures, its tangential and mixed entries show \(X_{\rm tan}=0\) and \(\mathrm d\zeta=2X_\nu\mathrm dA_g\). Now use compact scalar tests with \((h,K')=(2s g,sK)\) and the forms fixed. Their constraint differential terms are \(-2ku\Delta s+2kK(X,\nabla s)\); the other terms cancel by the interior adjoint equation. Since \(\theta_+=0\), their expansion derivative is \(k\partial_\nu s\), while their area derivative is \(k\int_Ss\). Thus \[ -kc\int_Ss\,\mathrm dA_g =2k\int_S\left[u\partial_\nu s -(\partial_\nu u+K(X,\nu))s\right]\mathrm dA_g -k\int_S\partial_\nu s\,\mathrm d\zeta. \tag{249}\] The boundary value and normal derivative of \(s\) are independent. The derivative coefficient gives \(u=X_\nu\), and the value coefficient gives \(\partial_\nu u+K(X,\nu)=c/2\).

For the last boundary identity, take Lie variations of all the data along a vector field extending arbitrary \(s\nu\) on \(S\). Such tests are legitimate field variations on the fixed manifold: the jets may be defined by a smooth extension across \(S\); no flow preserving the boundary is asserted. The active interior constraints have zero first variation by spatial transport at a minimum of their nonnegative contractions. The induced rotation of the transported unit ball contributes zero for the same reason. The area and expansion derivatives at \(S\) are, respectively, \(\int_SHs\) and \(L_+s\). Hence \[\begin{equation*} c\int_SHs\,\mathrm dA_g=2\int_SuL_+s\,\mathrm dA_g, \end{equation*}\] which proves \(cH=2L_+^*u\). One-sided outer area minimality gives \(H\ge0\) by nonnegative outward tests. The strong minimum principle for this smooth elliptic equation and \(u\ge0\) on connected \(S\) gives the boundary dichotomy in the proposition.

The boundary laws give \(W=0\) on \(S\). If \(u>0\) there, differentiate \(W\) and use (185); tangential derivatives vanish and \[\begin{equation*} \partial_\nu W =2u\bigl(\partial_\nu u+uK(\nu,\nu)\bigr) =2\kappa u. \end{equation*}\] This is (239) and implies positivity in a deleted exterior collar. If \(u=0\) on \(S\), then \(X=0\) there and \(\partial_\nu u=\kappa\). All tangential derivatives of \(X\) vanish at \(S\); the normal-normal and mixed parts of the KID equation then give \(\nabla_\nu X=0\) there. Smooth division by the boundary defining function gives (240). In particular \(W=\kappa^2s^2+O(s^3)>0\) in a deleted collar. Finally (234) and \((b^0)^2-|b|^2=1\) imply \(W\to1\). ◻

Proof. Suppose first that \(u>0\) on \(S\). The stationary metric is smooth and nondegenerate up to the hypersurface \(\mathbb R\times S\). Equations (255) imply that \(\xi\) is a null normal to this hypersurface: it is null, is tangent to it, and is orthogonal to all its tangent vectors. Its null second fundamental form vanishes, since its flow preserves the metric induced on the transverse sections. The null expansion equation consequently gives \(\operatorname{Ric}_{\mathbf g}(\xi,\xi)=0\). This assertion only uses the one-sided smooth jets on the hypersurface. The nonaffine normalization of its null generators contributes a multiple of the expansion, which is zero.

The extra tensor in (253) has zero contraction with \((\xi,\xi)\) there. Maxwell null positivity therefore gives \[0=2\lvert i_\xi F\rvert_{\mathbf g}^2.\] A covector annihilating a null vector has nonnegative Lorentz norm, and its norm is zero precisely when it is proportional to that null vector’s metric dual. Thus \(i_\xi F\) annihilates the tangent space of the horizon. Since \(i_\xi F=c_nuE^\flat\), the tangential component of \(E\) on \(S\) is zero. Equations (251) and (255) give \(X\wedge E=0\) and \(dp|_{TS}=0\). If \(u=0\) on \(S\), these two assertions follow immediately from \(X=0\) and \(dp=(k-1)uE^\flat\). Connectedness of \(S\) gives a single boundary value, denoted by \(p_b\).

The divergence-free constraint and Cartan’s formula give \[0=d(i_X\alpha)=\mathcal L_X(i_E\mathop{\mathrm{vol}}_g) =i_{[X,E]+(\operatorname{div}X)E}\mathop{\mathrm{vol}}_g.\] In particular, \[ [X,E]+(\operatorname{div}X)E=0, \qquad \mathcal L_E(X\wedge E)=0. \tag{260}\] Alignment therefore propagates along every nonzero electric trajectory once it is known at one point of that trajectory.

Here is the measure argument ensuring sufficiently many boundary hits. Let \(\Pi\) be a relatively compact regular level patch \(p=p_0\ne0\) in the interior. Follow \(V=\operatorname{sgn}(p_0)E\) from this patch until its first boundary hit, with maximal time \(T(y)\in(0,\infty]\). Along each trajectory \(|p|\) increases strictly, because \[dp(E)=(k-1)u|E|^2>0\] where \(E\ne0\). All these trajectories remain in a fixed compact subset of the original exterior: outside a sufficiently large coordinate sphere, \(|p|<|p_0|\). The trajectory map \((t,y)\BeginAccSupp{method=hex,unicode,ActualText=21A6}\OriginalMapsto\EndAccSupp{}\Phi_t(y)\) is injective for \(0<t<T(y)\), by uniqueness of the flow and strict monotonicity of \(p\).

Let \(d\mathfrak f\) be the positive transverse flux measure \(i_V\mathop{\mathrm{vol}}_g\) on \(\Pi\), oriented appropriately. Since \(\operatorname{div}V=0\), the volume element in this flow tube is \(dt\,d\mathfrak f\). Consequently \[ \int_\Pi T(y)\,d\mathfrak f(y) \leq \operatorname{Vol}_g(K_\Pi)<\infty, \tag{261}\] where \(K_\Pi\) is the fixed compact region just described. Hence \(T(y)<\infty\) for almost every \(y\) in flux measure. A finite-time trajectory cannot terminate in the interior of that compact region. This follows by smoothly extending the vector field locally and using ordinary differential equation existence. Nor can it reach a zero of \(E\) at a finite time, by uniqueness. It therefore hits \(S\) with \(E\ne0\).

Equation (260) transports the boundary alignment back to almost every point of \(\Pi\). The flux density is smooth and strictly positive on this regular patch, so continuity gives alignment on the whole patch. Varying the patch proves it at every point with \(p\ne0\) and \(E\ne0\). If \(p=0\) and \(E\ne0\) at an interior point, the derivative of \(p\) along \(E\) is nonzero there, so such a point is a limit of the preceding points. At zeros of \(E\) alignment is automatic. This proves (259). ◻

Proof. In spacetime dimension \(k+2\), the electromagnetic trace reversal is \[\operatorname{Ric}_{ab} =2F_{ac}F_b{}^c-\frac1k(F_{cd}F^{cd})\mathbf g_{ab}\] apart from the matter term in (253), which is trace-free wherever it is nonzero. In an orthonormal frame whose timelike vector is \(\mathbf n\), the electric magnitude is \(c_n|E|\). The mixed Ricci eigenvalue on the plane spanned by \(\mathbf n\) and \(E\) is therefore \[-2c_n^2\left(1-\frac1k\right)|E|^2 =-(k-1)^2|E|^2.\] Lemma 55 places \(\xi=u\mathbf n+X\) in this plane wherever \(E\ne0\). The additional matter tensor annihilates \(\xi\), since it is supported where \(\mathbf g(\xi,\xi)=0\). The same formula holds at \(E=0\). This proves (264), including \[ \operatorname{Ric}(\xi,\xi)=(k-1)^2|E|^2W. \tag{267}\]

For a Killing field, \(D\) is skew and \(\boldsymbol\nabla_a(d\xi^\flat)^{ab} =-2\operatorname{Ric}^{b}{}_{a}\xi^a\). Applying the Killing wave identity to \(W=-\mathbf g(\xi,\xi)\) gives (265): a stationary scalar has wave operator \(u^{-1}\operatorname{div}(uC\,d\,\cdot)\) because the spatial inverse block of \(\mathbf g\) is \(C\) and \(\mathop{\mathrm{vol}}_{\mathbf g}=u\,dz\wedge\mathop{\mathrm{vol}}_g\). The spatial components of the right side of the Killing divergence identity vanish by (264). Antisymmetry removes the remaining Christoffel term, and hence \(\nabla_i\widetilde Q^{ij}=0\). Finally, \[\xi^\flat=-W(dz-A_s),\qquad d\xi^\flat=-dW\wedge(dz-A_s)+WF_s.\] Raising both spatial indices eliminates the first summand and gives (266). ◻

Proof. The interior zero set \(Z_0=\{W=0\}\cap\Omega^\circ\) is compact, since \(W\to1\) at infinity and is positive on a deleted boundary collar. On a fixed relatively compact neighborhood of \(Z_0\), both \(u\) and \(|X|\) have positive lower bounds. Put \(a=|X|\), \(e=X/a\), and \(T=e^\perp\) there. The eigenvalues of \(C\) are one on \(T\) and \(W/u^2\) on \(e\). Smooth nonnegativity of \(W\) implies the local estimate \[ |dW|^2\leq C_0W. \tag{268}\] For completeness, this follows by restricting to short coordinate line segments, using a bound for the Hessian, and minimizing the resulting quadratic Taylor upper bound at a point where the function is nonnegative. A finite cover gives a uniform constant on the chosen neighborhood.

Use a slice-orthonormal frame \((e,e_A)\) and complete it by the future normal \(\mathbf n\). Since \(\xi=u\mathbf n+ae\), \[D_{i\mathbf n}=-\frac{W_i/2+aD_{ie}}u.\] An explicit expansion of the Lorentz norm gives \[\begin{align*} -2D_{ab}D^{ab} ={}&\frac{|dW|^2}{u^2} +\frac{4a}{u^2}\sum_A W_A D_{Ae} -\frac{4W}{u^2}\sum_A D_{Ae}^2 -4\sum_{A<B}D_{AB}^2. \tag{269}\end{align*}\] The original fields and their derivatives are bounded on this fixed neighborhood. Equations (268), (267), and (265) therefore imply \[ u^{-1}\operatorname{div}(uC\,dW) \leq C_1\bigl(W+|d_TW|\bigr). \tag{270}\]

Choose a smooth compactly supported \(\eta\) in this neighborhood. For \(\delta>0\), test (270) with \(\eta^2/(W+\delta)\). There is no singular integration here: \(C,W\) are smooth and the denominator is positive. Integration by parts gives, with \(L W=u^{-1}\operatorname{div}(uC\,dW)\), \[\begin{align*} I_\delta &:=\int u\eta^2\frac{|dW|_C^2}{(W+\delta)^2} =\int\frac{u\eta^2LW}{W+\delta} +2\int\frac{u\eta\,C(d\eta,dW)}{W+\delta},\\ |dW|_C^2&=|d_TW|^2+\frac W{u^2}\bigl(dW(e)\bigr)^2. \end{align*}\] Since \(0\leq C\leq g^{-1}\), Young’s inequality absorbs both the cutoff term and the \(|d_TW|/(W+\delta)\) term. Explicitly, each is at most \(I_\delta/4\) plus a constant times \(\int u(\eta^2+|d\eta|^2)\). The remaining \(W/(W+\delta)\) term is bounded. Consequently \[ I_\delta\leq C_\eta\quad(0<\delta<1),\qquad \int_{\mathcal U}\eta^2|d_T\log W|^2<\infty. \tag{271}\] Take finitely many such cutoffs equal to one near \(Z_0\).

We record the exact flux used at the zero-set cutoff. Contract (266) with \(A_s\), and lower the remaining index using \(g\). Since \(C X=(W/u^2)X\), direct substitution of \(F_s=d(W^{-1}X^\flat)\) gives \[ J^i=Q^{ij}(A_s)_j,\qquad J^\flat_i=\frac1u\left( X^j(dX^\flat)_{ij}-|X|^2(d_T\log W)_i\right). \tag{272}\] In particular \(J\) is exactly orthogonal to \(X\), and near \(Z_0\) \[ |J|\leq C\bigl(1+|d_T\log W|\bigr). \tag{273}\]

Choose a smooth cutoff \(\chi:[0,\infty)\to[0,1]\), zero on \([0,1]\) and one on \([2,\infty)\). A zero-set cutoff \(z_\delta\) can be chosen to equal \(\chi(W/\delta)\) near \(Z_0\) and one outside the fixed neighborhood. Indeed place the spatial localization transition on a compact set disjoint from \(Z_0\); \(W\) has a positive lower bound there, so for all sufficiently small \(\delta\) no transition derivative remains. On the support of \(dz_\delta\), transversality in (272) yields \[ |J(dz_\delta)| \leq C\frac W\delta \left(|d_T\log W|+|d_T\log W|^2\right) \leq C\left(1+|d_T\log W|^2\right). \tag{274}\] This is supported where \(\delta<W<2\delta\). Its integral tends to zero by (271) and dominated convergence. In particular, an open portion of \(Z_0\) causes no omitted contribution: all test fields used below vanish on a neighborhood of it.

Next choose a distance cutoff \(\beta_\varepsilon\) at \(S\), zero for \(s\leq\varepsilon\) and one for \(s\geq2\varepsilon\), with derivative bounded by \(C/\varepsilon\). If \(u>0\) on \(S\), the boundary laws give \[W\asymp s,\qquad e=\nu_s+O(s),\qquad d_TW=O(s),\] where \(\nu_s=\nabla s\). Thus \(J=O(1)\) and \(J\cdot\nu_s=O(s)\) by its exact transversality. The resulting boundary error is \(O(\varepsilon)\). If \(u=0\) on \(S\), write \[W=s^2w,\qquad w=a^2-s^2|Y|^2>0\] using (256). Then \(A_s=Y^\flat/w\), \(F_s\), and \(C\) are smooth up to \(S\). Equation (266) gives \(Q=O(s^3)\) and \(J=O(s^3)\), so the boundary error is \(O(\varepsilon^3)\).

It remains to make the test field decay at infinity. In the end coordinates let \(A_\infty\) be the constant limit of \(A_s\). Choose a smooth function \[\psi(x)=\rho(r)(A_\infty)_ix^i,\] where \(\rho\) is zero near the compact core and one far out. Its support is chosen disjoint from the boundary and zero-set cutoff regions. Set \(B=A_s-d\psi\) on \(\mathcal U\). Then \(dB=F_s\) and \[ B=O_1(r^{-q_0}),\qquad Q=O(r^{-1-q_0}). \tag{275}\] There is no bulk term from the transition of \(\psi\), since \(Q^{ij}\nabla_i\nabla_j\psi=0\) by antisymmetry. If \(\theta_R\) is one for \(r\leq R\), zero for \(r\geq2R\), and \(|d\theta_R|\leq C/R\), its error is bounded by \[ \left|\int Q^{ij}B_j\partial_i\theta_R\right| \leq C R^{n-2-2q_0}\longrightarrow0. \tag{276}\]

Test \(\nabla_i\widetilde Q^{ij}=0\) with \(z_\delta\beta_\varepsilon\theta_R B_j\), extended by zero where necessary. This is a smooth compactly supported test in the interior. Antisymmetry gives \[\begin{align*} \int z_\delta\beta_\varepsilon\theta_R\,\mathcal T_{\mathrm{tw}} &=-\int Q^{ij}B_j \partial_i(z_\delta\beta_\varepsilon\theta_R), \tag{277}\\ \mathcal T_{\mathrm{tw}} &:=\frac12 Q^{ij}(F_s)_{ij} =\frac{uW}{2}C^{ia}C^{jb}(F_s)_{ij}(F_s)_{ab}\geq0. \tag{278}\end{align*}\] The three error estimates are independent of the other cutoffs, which lie between zero and one. Take \(R\to\infty\), \(\varepsilon\downarrow0\), and \(\delta\downarrow0\), or use a diagonal sequence. Fatou’s lemma in (277) gives \(\int_{\mathcal U}\mathcal T_{\mathrm{tw}}=0\); its integrability need not be assumed beforehand. Since \(C\) is positive definite on \(\mathcal U\), \(F_s=0\) there. ◻

Proof. The Maxwell equation in (251) gives \[ \operatorname{div}(u^{-1}\nabla p)=0. \tag{280}\] Its coefficients are smooth and elliptic on every compact subset of the interior. A positive maximum above both boundary and end values, or a negative minimum below them, would be attained in such a subset. The strong maximum principle therefore applies without requiring a uniform boundary bound for \(u^{-1}\). If \(p_b=0\), it would give \(p=0\) and then \(E=0\), contrary to \(Q\ne0\). Otherwise it gives the stated strict interval for \(p\).

There is a useful flux which is smooth even at interior zeros of \(W\): \[ \mathcal D^i=u(d\xi^\flat)^{iz} =\frac1u\left(\nabla^iW+X^j(dX^\flat)^i{}_j\right). \tag{281}\] The Killing divergence identity and (264) imply \[ \operatorname{div}\mathcal D=2u^{-1}|dp|^2, \qquad \mathcal D\cdot X=u^{-1}dW(X). \tag{282}\] On \(\mathcal U\), Lemma 57 and (281) also give \[ \mathcal D=\frac uW C\,dW. \tag{283}\]

Choose \(d_0\) as in (279), and put \[P=W-1-p^2-d_0p,\qquad j=\mathcal D-(2p+d_0)u^{-1}\nabla p.\] Equations (280) and (282) give \(\operatorname{div}j=0\). On \(\mathcal U\), alignment implies \(C\,dp=(W/u^2)\nabla p\); hence \[ j=\frac uW C\,dP, \qquad dP(j)\geq u^{-1}|dP|^2. \tag{284}\] Indeed the axial eigenvalue of \((u/W)C\) is \(1/u\), and its transverse eigenvalues are \(u/W\geq1/u\). This also covers points where \(X=0\). At an interior zero of \(W\), \(dW=0\) and \(X\ne0\). Alignment makes \(dp\) parallel to \(X\), and (282) gives \(\mathcal D\cdot dp=0\). Thus (284) holds there as well, with equality in its second assertion.

The current \(j\) is bounded up to \(S\). This is immediate when \(u>0\) there. In the other case \(dW=O(s)\) and \(X^j(dX^\flat)_{ij}=O(s^3)\), so the quotient in (281) extends smoothly by \(u=sa\). The other current is \(u^{-1}\nabla p=(k-1)E\), already smooth. Since \(P|_S=0\), a distance cutoff gives a boundary error tending to zero. At infinity, \[P=O_2(r^{-q_0}),\qquad j=O(r^{-1-q_0}),\] so the cutoff error in integrating \(P\operatorname{div}j\) is \(O(R^{n-2-2q_0})\). Testing with the product of the boundary and end cutoffs used above, and using nonnegativity in (284), gives \[\int_{\Omega^\circ}u^{-1}|dP|^2=0.\] Thus \(P\) is constant, and its limit at infinity is zero. This proves the first equality in (279).

Its root at \(p_b\) is simple. Otherwise \(W=(p-p_b)^2\). If \(u>0\) on \(S\), this contradicts \(dW=2\kappa X^\flat\ne0\) there. If \(u=0\) on \(S\), then \(dp=(k-1)uE^\flat\) gives \(p-p_b=O(s^2)\), whereas (256) gives \(W=\kappa^2s^2+O(s^3)\), again a contradiction. Writing \(p=t p_b\) with \(0<t<1\), the polynomial becomes \[W=(1-t)(1-p_b^2t).\] Every intermediate value is attained by continuity between the boundary and the end. Nonnegativity implies \(p_b^2\leq1\), and simplicity excludes equality. The displayed formula then makes \(W\) strictly positive at every interior point. ◻

Proof. In the positive-\(u\) case, \(W\) is an original boundary defining function. Equation (286) follows from the boundary identity \(dW=2\kappa X^\flat\) and smooth division by this defining function. More explicitly, write \[X^\flat=b(W,y)dW+W\beta_A(W,y)dy^A, \qquad b(0,y)=\frac1{2\kappa}.\] Upon setting \(W=\lambda^2\), the metric components are \[\begin{align*} (h_s)_{\lambda\lambda} &=4\lambda^2g_{WW}+4b^2,\\ (h_s)_{\lambda A} &=2\lambda(g_{WA}+b\beta_A),\\ (h_s)_{AB}&=g_{AB}+\lambda^2\beta_A\beta_B, \end{align*}\] where every coefficient on the right is evaluated at \((\lambda^2,y)\). Normal and tangential diagonal coefficients are even and the mixed coefficients are odd, which is exactly tensor invariance under reflection. At \(\lambda=0\) the normal coefficient is \(\kappa^{-2}\), the mixed coefficients vanish, and the tangential metric is \(g|_{TS}\).

In the zero-\(u\) case, put \(v=(a^2-s^2|Y|^2)^{1/2}\) in a small original collar. Then \[\lambda=sv,\qquad h_s=g+s^2v^{-2}Y^\flat\otimes Y^\flat,\qquad A_s=v^{-2}Y^\flat.\] These expressions are smooth and nondegenerate, and \(v|_S=\kappa\). They prove the remaining assertions. ◻

Proof. Where \(W>0\), equation (253) has no matter term. Lemma 57 makes \(A_s\) locally exact, so a local coordinate \(T\) has \(dT=dz-A_s\) and \(\mathbf g=-\lambda^2dT^2+h_s\). Alignment gives \(A_s\wedge dp=0\), proving (287). In this splitting the normalized static electric covector is \(dp/((k-1)\lambda)\).

The warped-product Ricci components are \[\begin{split} \operatorname{Ric}_{\mathbf g}(\partial_T,\partial_T) &=\lambda\Delta_{h_s}\lambda,\\ \operatorname{Ric}_{\mathbf g,ij} &=\operatorname{Ric}_{h_s,ij} -\lambda^{-1}\operatorname{Hess}_{h_s,ij}\lambda. \end{split}\] For \(F_{Ti}=c_n p_i/(k-1)\), trace reversal gives respectively \(|dp|_{h_s}^2\) and the right side of (289). The spatial divergence of the raised \(iT\) Maxwell component gives the second equation of (288). These calculations establish all constants in (288)– (289). Finally apply the conformal Ricci and Laplacian formulas with conformal logarithm \((k-1)^{-1}\log\lambda\). The Hessian terms cancel and yield (290)– (291). ◻

Proof. Lemma 58 already gives \(W>0\) throughout the interior. Set \[a_0=p_b+d_0/2=\frac{p_b^2-1}{2p_b}\ne0, \qquad V_0=\frac12\log \frac{1-p/p_b}{1-p_bp}.\] The logarithm is real and \(V_0\to0\) at infinity. Direct differentiation, using (279), gives \[ dV_0=a_0\frac{dp}{W},\qquad d\log\lambda\otimes d\log\lambda-W^{-1}dp\otimes dp =dV_0\otimes dV_0. \tag{295}\] The second identity follows also from \((p+d_0/2)^2-W=d_0^2/4-1=a_0^2\). Equations (290) and (291) now give \[\operatorname{Ric}_{\check h} =\frac{k}{k-1}dV_0\otimes dV_0, \qquad \Delta_{\check h}V_0=0.\] The inverse conformal formulas, with \(\lambda_0=e^{V_0}\) and \(h_0=\lambda_0^{-2/(k-1)}\check h\), prove (293). The polynomial factorization makes these definitions exactly (292). It also gives \(0<\lambda_0<1\), since for \(p=t p_b\), \(0<t<1\), \[\lambda_0^2=\frac{1-t}{1-p_b^2t}<1.\]

Near the boundary the simple root in Lemma 58 makes \(p\) a smooth function of \(W=\lambda^2\). Thus the positive factor \(1-p_bp\) is smooth in the attached coordinates of Lemma 59. Its boundary value is \(1-p_b^2>0\), and the normal rescaling gives (294). Smoothness of \(h_0\) and the equation \(\operatorname{Hess}_{h_0}\lambda_0 =\lambda_0\operatorname{Ric}_{h_0}\) imply that the Hessian vanishes at the boundary. Its tangential components are the nonzero normal derivative times the boundary second fundamental form, so the boundary is totally geodesic.

Completeness of \(h_s\) including its attachment follows from \(h_s\geq g\). Indeed a finite-length Cauchy escape has a limit in the complete original metric space. An interior limit is harmless because \(W>0\) there, and a boundary limit is contained in the regular attached collar. The same reasoning proves metric completeness, or equivalently the absence of any finite-length escape. Since \[1-p_b^2\leq1-p_bp\leq1,\] \(h_0\) and \(h_s\) are uniformly comparable. Completeness passes to \(h_0\). The attachment changes only the boundary collar’s smooth defining function, so the topological end and compact-complement properties are retained. ◻

Proof. The boundary value of \(p\) is constant by Lemma 55. Integrate \(\operatorname{div}(pE)=dp(E)=(k-1)u|E|^2\) on the original exterior truncated at radius \(R\). The boundary normal in this divergence theorem is \(-\nu\) on \(S\). The outer term tends to zero by the decay of \(p\) and \(E\), and the inner flux is \(-p_b\omega Q=0\). Since \(u\) is bounded and \(|E|^2\) is integrable, this gives \[(k-1)\int_\Omega u|E|^2=0.\] Positivity of \(u\) in the interior implies \(E=0\), and then \(p=0\) by its normalization at infinity.

On every component of \(\mathcal U\), Lemmas 57 and 60 now give \[ \operatorname{Hess}_{h_s}\lambda =\lambda\operatorname{Ric}_{h_s},\qquad \Delta_{h_s}\lambda=0, \qquad \operatorname{Scal}_{h_s}=0. \tag{296}\] Let \(\mathcal C\) be the positive component containing the whole sufficiently distant end. Any other positive component has compact closure in the original exterior and continuous boundary value \(\lambda=0\) on its whole topological boundary, including any part on \(S\). Its positive maximum would be attained in its interior, contradicting the strong maximum principle. Thus no other positive component exists, and all deleted boundary collars lie in \(\mathcal C\). The same compact-superlevel argument gives \(0<\lambda<1\) on \(\mathcal C\).

We next exclude a finite orbit-metric distance to an interior zero. Near any such zero, \(u\) and \(|X|\) are bounded below, and Lemma 57 makes the left side of (272) zero. Therefore \[ (d_T\log W)_i=|X|^{-2}X^j(dX^\flat)_{ij}. \tag{297}\] Its right side is bounded. Together with (268), this gives on \(\mathcal C\) near the zero \[ |d\log W|_{h_s}^2 =|d_T\log W|_g^2 +\frac{(dW(e))^2}{u^2W}\leq C. \tag{298}\] Consequently a finite \(h_s\)-length curve cannot approach the zero: integration of (298) would keep \(\log W\) bounded on its tail.

Attach \(S\) to the base of \(\mathcal C\) as in Lemma 59. By (296), the Hessian of \(\lambda\) vanishes at \(S\), and the boundary is totally geodesic. In Gaussian base distance \(\sigma\) from \(S\), smoothness gives the Taylor expansions \[\begin{align*} h_s&=d\sigma^2+ \bigl(\gamma_0+\sigma^2\gamma_2 +\sigma^3R\bigr)_{AB}dy^A dy^B, \tag{299}\\ \lambda&=\kappa\sigma+a_3(y)\sigma^3 +\sigma^4R_\lambda. \tag{300}\end{align*}\] The remainders are smooth for \(\sigma\geq0\). The absent quadratic term in the lapse is exactly \(\operatorname{Hess}_{h_s}\lambda(\partial_\sigma, \partial_\sigma)|_S=0\).

Take a circle coordinate \(T\) of period \(2\pi/\kappa\), and put \[ h_{\mathrm{circ}}=h_s+\lambda^2dT^2 \quad\hbox{on }\mathcal C\times (\mathbb R/(2\pi/\kappa)\mathbb Z). \tag{301}\] Collapse this circle at \(S\) by attaching a disk in the normal direction. The period is the same over the entire connected boundary because \(\kappa\) is constant. To check regularity, set \(\theta=\kappa T\), use disk coordinates \((x_1,x_2)=(\sigma\cos\theta,\sigma\sin\theta)\), and write \(\alpha_D=x_1dx_2-x_2dx_1\). Besides the Euclidean disk metric, the polar contribution is \[\begin{align*} \left[\left(\frac{\lambda}{\kappa\sigma}\right)^2-1\right] \sigma^2d\theta^2 &=\left[\left(\frac{\lambda}{\kappa\sigma}\right)^2-1\right] \frac{\alpha_D^2}{\sigma^2}\\ &=\frac{2a_3(y)}\kappa\alpha_D^2 +\sigma B(\sigma,y)\alpha_D^2 \end{align*}\] with smooth \(B\). The first term has smooth quadratic coefficients in \((x_1,x_2)\); the remaining coefficients have differentiated order at least three. The same holds for the remainder in (299). The extended metric is therefore \(C^{2,\alpha}\) for every \(0<\alpha<1\). It is nondegenerate. The warped-product Ricci formulas and (296) give Ricci curvature zero off the attached copy of \(S\), and continuity extends this identity across it. Harmonic-coordinate regularity for Einstein metrics then makes this extension smooth (DeTurck and Kazdan 1981).

The resulting manifold, denoted by \(\mathcal M\), is complete and has no boundary. To check completeness directly, a finite-length curve projects to a finite \(h_s\)-length curve, hence to a \(g\)-Cauchy curve because \(h_s\geq g\). Completeness of the original metric space supplies a limit in \(\Omega\). An interior zero of \(W\) is excluded by (298). At an interior point of \(\mathcal C\), the circle has uniformly positive size, so its coordinate is Cauchy as well. At a point of \(S\), the regular disk attachment makes all limiting circle phases converge to that point of the attached zero section. Thus no finite-length escape remains, which proves completeness.

Suppose an interior frontier of \(\mathcal C\) still existed. A sequence in \(\mathcal C\) approaching such a frontier, lifted at a fixed circle phase, escapes every compact subset of \(\mathcal M\). Choose a far coordinate sphere and its product with the circle, \[\Sigma_R\cong S^{n-1}\times S^1.\] It is a compact connected cross-section. Its outside is the asymptotic product end, while its inside contains the escaping sequence and is noncompact. The inside is connected: excursions of a path through the outside can be replaced by paths in a thin inside collar of the connected cross-section. A compact tubular neighborhood of \(\Sigma_R\) therefore separates two noncompact regions. In particular \(\mathcal M\) has at least two ends.

For clarity, the line needed for the splitting theorem can be constructed directly. Choose escaping sequences in the two regions and minimizing segments between them. Each segment meets the fixed compact separating cross-section, and its distances from that cross-section to both endpoints tend to infinity. Centering there and taking a subsequence gives a complete minimizing line. The smooth, complete, connected manifold \(\mathcal M\) has \(\operatorname{Ric}=0\), so the Cheeger–Gromoll splitting theorem applies (Cheeger and Gromoll 1971). It gives \(\mathcal M\cong\mathbb R\times N\) with \(N\) complete and connected. Since the product with a connected noncompact factor has only one end, \(N\) must be compact. Such a product has at most linear volume growth.

On the other hand, (258) shows that \(h_s\) approaches a positive constant Euclidean metric and \(\lambda\to1\). The circle has fixed period. Hence the product end in (301) gives \(\operatorname{Vol}_{h_{\mathrm{circ}}}(B_R)\geq cR^n\) for large \(R\): paths to a fixed far sphere have bounded cost, and the metric on the remaining end is uniformly comparable to its Euclidean product model. This contradicts linear growth. There is no interior frontier. Since \(\Omega^\circ\) is connected, \(\mathcal C=\Omega^\circ\) and \(W>0\) throughout.

Now \(h_s\) itself is complete up to its regular boundary by the same projection and collar argument, and (296) holds globally. The boundary assertions follow from Lemma 59 and the vanishing Hessian already proved. ◻

Proof. All interior, boundary, and completeness assertions follow from Propositions 61 and 62, together with Lemmas 55 and 57. On \(W>0\), (253) gives \(\mu_m=0\), and the matter dominant energy condition gives \(J_m=0\). Continuity gives the same conclusion up to \(S\).

In the original end coordinates, \(h_s\) tends to \(\delta+b^\flat\otimes b^\flat\), a positive constant matrix. Equations (258) and (262) give differentiated error \(O_2(r^{-q_0})\). In the charged case the additional conformal factor differs from one by \(O_2(r^{2-n})\), so the same bound holds for \(h_0\) and \(\lambda_0\). A fixed linear coordinate change normalizes the limiting matrix and preserves these decay orders.

Finally take Gaussian normal coordinates for \(h_0\) at the boundary. Total geodesicity makes the first normal derivative of its tangential metric zero. The vanishing boundary Hessian makes the second normal derivative of the lapse zero. Even metric reflection and odd lapse reflection consequently match their derivatives through order two and are \(C^{2,\alpha}\). This is the regularity required for the vacuum doubling argument in the next section. ◻