Proof. The end metric is uniformly comparable to \(b\) sufficiently far out. Radial infinity escapes compact subsets: otherwise a sequence tending to radial infinity would accumulate in a compact coordinate neighborhood, while pairwise disjoint background balls of a fixed small radius about a subsequence would have uniformly positive metric volumes and lie in a fixed compact neighborhood. This is impossible. A large coordinate sphere separates the product end from an inner side. If the latter were noncompact, a smooth compact exhaustion would yield an end separated from the given product end. One may choose this exhaustion with finitely many complement components at each stage, since its smooth compact frontier has finitely many components. Nested extraction then contradicts the one-end assumption. The inner side is therefore compact. Paths between its points can be pushed back across the product end, so the truncations can be taken connected.

Every component of \(S\) is a closed orientable surface and bounds a handlebody. Orient each handlebody and glue by a map reversing the induced boundary orientations. The collars then extend the orientation of \(\Omega\) across each seam. Extend the metric smoothly across \(S\). The added region is compact; completeness at infinity is unchanged. For a set containing \(O\) up to a null set, its characteristic function is constant almost everywhere in the cap interiors. Its perimeter therefore uses no metric values there. ◻

Proof. For an admissible \(D\), complement its manifold interior in \(X\). The result contains \(O\) and is precompact because \(D\) contains the whole sufficiently distant end. Local boundary coordinates show that its perimeter is bounded by the entire intrinsic area of \(\partial D\), including any portions on \(S\). This also follows by smoothing corners and passing to the perimeter lower limit. For the converse, take the closed exterior of the smooth enclosure. It is connected, has the required end, and has precisely the stated intrinsic boundary. Any admissible cut of this new exterior, complemented in \(X\), encloses the original enclosure; outward minimization gives the final claim. ◻

Proof. The potentials in (1) are the ambient coordinate functions of the future unit hyperboloid. Hyperbolic isometries act on their span by Lorentz transformations. The flux integrand is tensorial in \(b\), linear in \(V\), and natural under these isometries. It remains to justify replacing the transformed integration surfaces by the original large coordinate spheres. The static identity \(\nabla_b^2V=Vb\) gives \[\mathop{\mathrm{div}}_b\mathbb U(V,e)=V\,D R_b(e).\] The difference between \(D R_b(e)\) and \(R_g+6\) is bounded by a constant times \[|e|_b^2+|\nabla_be|_b^2+|e|_b|\nabla_b^2e|_b.\] Its \(V_0\)-weighted integral over \(r>D\) is \(O(D^{3-2\tau})\), since \(\mathop{}\!\mathrm dV_b\) is comparable to \(r\mathop{}\!\mathrm dr\mathop{}\!\mathrm dA_\sigma\). Also \(|V_i|\le V_0\). Thus the divergence is absolutely integrable on the end for each potential. The divergence theorem on regions between the two families of spheres makes their flux difference tend to zero. This is the covariance mechanism of Chruściel and Herzlich (2003, Proposition 2.2, Theorem 2.3, and Section 3) in the normalization (3).

A fixed hyperbolic isometry changes distance from the chosen origin by a bounded amount. The exponential weights and \(V_0\) are therefore comparable before and after the change, and uniform background balls are carried to uniform background balls. The rest-chart assertion is the elementary Lorentz normal form of a future-timelike covector. ◻

Proof. The trace assertion follows immediately from the definition of \(\pi\). To prove the first identity without any geometric interpretation, set \(a=\mathop{}\!\mathrm d\log s\). Since \(p^\flat=s^2\mathop{}\!\mathrm df\), \[\nabla_i p_j=N\pi_{ij}+a_i p_j-a_j p_i.\] Differentiating \(s^2=N^2-|p|^2\) and substituting this formula gives \[N^2a_i-(a\cdot p)p_i=N\nabla_iN-N\pi_{ij}p^j.\] Division by \(N^2\) and Equation (16) give Equation (19), as an equality of vectors.

For the curvature identity consider, solely for this computation, the Lorentzian metric \[\widetilde g=-s^2\mathop{}\!\mathrm dt^2+\bar g\] on \(\mathbb R\times U\). The graph \(t=f\) has induced metric \(g\). If \(X^\mathrm{tan}=X+\mathop{}\!\mathrm df(X)\partial_t\), its future unit normal is \(n=(\partial_t+p^\mathrm{tan})/N\). Thus \(\partial_t=Nn-p^\mathrm{tan}\). The tangential part of the Killing equation for \(\partial_t\) is \(N\widetilde g(\widetilde\nabla_{X^\mathrm{tan}}n,Y^\mathrm{tan}) =\operatorname{sym}\nabla p^\flat(X,Y)\). Its second fundamental form, with the sign convention specified after (8), is therefore \(\pi\).

Write \(\widetilde G\) for the Einstein tensor and \(n_0=s^{-1}\partial_t\) for the normal to a static slice. The contracted Gauss and Codazzi identities with timelike unit normal and second form \(\widetilde g(\widetilde\nabla n,\cdot)\) read \[2\widetilde G(n,n)=R_g+T^2-|\pi|^2, \qquad \widetilde G(n,X^\mathrm{tan}) =(\mathop{\mathrm{div}}_g\pi-\mathop{}\!\mathrm dT)(X).\] The static slices have zero second form. Applying these same contractions to a static slice gives \(\widetilde G(n_0,n_0)=R_{\bar g}/2\) and \(\widetilde G(n_0,X)=0\) for spatial \(X\). The static time component of \(n\) is \(N/s\), whereas that of \(\partial_t/N\) is \(s/N\). Consequently \[\frac12R_{\bar g} =\widetilde G(n,\partial_t/N) =\widetilde G(n,n)-\widetilde G(n,v^\mathrm{tan}),\] which proves Equation (20) with the claimed sign of the momentum term.

For completeness, the completion of the square uses symmetry to give \[\frac1N\mathop{\mathrm{div}}_g(Cp) =(\mathop{\mathrm{div}}_g\lambda-\mathop{}\!\mathrm d\ell)\cdot v +\lambda:\pi-\ell T.\] Also \(2\ell T-\ell^2=T^2-\kappa^2\) and \(|\lambda|^2-2\lambda:\pi=|\mathcal K|^2-|\pi|^2\). Substituting these two equalities in the right side of Equation (23) recovers Equation (20) term by term. ◻

Proof. Since \(w=x/s\), Equation (19) gives \[A\mathop{}\!\mathrm dw=N\bar\nabla x-Nx\bar\nabla\log s =N\bar\nabla x-x\nabla N+x\pi p.\] Adding \(B\) proves the first formula for \(Q\). Its divergence from the defining equations is \(\Phi-\Psi\). To compute the divergence of its other expression, Equation (16) gives \[\begin{align*} \mathop{\mathrm{div}}_g(N\bar\nabla x) &=N\Delta_{\bar g}x +N\bar g^{-1}(\mathop{}\!\mathrm d\log s,\mathop{}\!\mathrm dx)\\ &=N\Delta_{\bar g}x+(\nabla N-\pi p)\cdot\mathop{}\!\mathrm dx. \end{align*}\] The derivative-of-\(x\) terms in \(\mathop{\mathrm{div}}_g(xCp+(1-x)B)\) are \((Cp-B)\cdot\mathop{}\!\mathrm dx=(\pi p-\nabla N)\cdot\mathop{}\!\mathrm dx\). They cancel exactly, leaving \[\mathop{\mathrm{div}}_gQ=N\Delta_{\bar g}x+x\mathop{\mathrm{div}}_g(Cp)+(1-x)\Phi.\] This proves Equation (26). In dimension three the conformal change \(h=x\bar g\) has scalar curvature \[xR_h=R_{\bar g}-2x^{-1}\Delta_{\bar g}x +\frac32x^{-2}|\mathop{}\!\mathrm dx|_{\bar g}^2.\] Substitution of Equations (23) and (26) cancels the two occurrences of \(2N^{-1}\mathop{\mathrm{div}}_g(Cp)\) and proves Equation (27). The two stated systems now follow by substituting their values of \(T,\mathcal K,\Phi,\Psi\). In the AH case, the nondifferential cosmological terms after multiplication by \(x\) are \(6x^2-12x+6=6(x-1)^2\). Finally, \(|v|<1\) and the constraint definitions give the asserted sign of \(D\). ◻

Proof. On \(\mathcal S\), \(p=N\hat z n\). For tangent vectors \(X,Y\), \(\nabla^2f(X,Y)=f_n\operatorname{II}(X,Y)\); consequently \(\pi(X,Y)=\hat z\operatorname{II}(X,Y)\) and \(\pi(n,n)=T-\hat z H\). Taking the normal component in Equation (19) therefore yields \[d^2\partial_n\log s =\partial_n\log N-\hat z T+yH.\] Since \(\partial_nN=B_n+N(\mathcal K(n,n)+\ell)\hat z\) and \(\mathcal K(n,n)+\ell-T=-k_T\), this proves Equation (36).

In base normal coordinates, the tangential entries of \(\bar g\) and their first normal derivatives agree with those of \(g\) at the face: the extra products contain at least one tangential derivative of \(f\). The mixed entries vanish along the entire face, and the normal length factor is \(W\). Hence \(\bar n=dn\) and \(H_{\bar g}=dH\). The mean-curvature formula for \(h=x\bar g\) gives \[\sqrt{x}\,H_h=d(H+\partial_n\log x).\] Multiplication by \(s=Nd\), followed by Equation (36), gives \[s\sqrt{x}\,H_h =N(H-k_T\hat z)+B_n+Nd^2w_n/w.\] The final term equals \((A\mathop{}\!\mathrm dw)_n/x\), proving Equation (37). In the maximal case \(k_T=-k\), \(\hat z=-z\), and \(\partial_nN=-Nzk-b_S\); this proves Equation (38). The asserted strict sign in the compact case follows because \(H-k_T\hat z\le0\) and \(B_n<0\).

At the gluing face the tangential graph metrics equal their base metrics, which agree, and \(x_i=s_iw=N_ow=x_o\). The remaining inequality reduces by Equation (37) to \[N_i(H_i-k_T\hat z)\ge s_i\sqrt{H_i^2-k_T^2}.\] Both sides are nonnegative, and after division by \(N_i\) its squared difference is \[(H_i-k_T\hat z)^2-(1-\hat z^2)(H_i^2-k_T^2) =(k_T-H_i\hat z)^2\ge0.\] This proves the mean-curvature ordering. ◻

Proof. Use a \(g\)-orthonormal frame at the point with first vector \(e\) and transverse indices \(a,b\in\{2,3\}\). Put \(U_{ab}=\nabla^2_{ab}f/m\) and \(C_T=T/z\), where \(z=\sqrt y\). Differentiation of the constitutive law gives \[ \begin{aligned} Nm&=\frac z{1-y},\qquad q'=\frac{\mathop{}\!\mathrm dq}{\mathop{}\!\mathrm d(l+u)}=-\frac{4yq}{(1+y)^2},\\ \mathcal P:\nabla^2f&=mC_T-\beta\nabla f\cdot\mathop{}\!\mathrm du, \qquad \mathop{\mathrm{tr}}U=C_T-ql_1-\beta u_1. \end{aligned} \tag{45}\] Here and below \(q'\) differentiates the scalar constitutive function, not a spatial coordinate. The derivative of the height flux with respect to \(\mathop{}\!\mathrm df\) is \(s^2\mathcal P\); its derivative with respect to \(N\) is \(s^2(\beta/N)\nabla f\). This also proves that the nondivergence equation is smooth and elliptic through zero gradient.

Differentiating the height equation in direction \(e\), commuting the third derivatives of a scalar, and differentiating its norm gives \[\begin{align*} \mathcal P:\nabla^2l={}&\mathop{\mathrm{Ric}}(e,e)-\beta u_{11} -q'(l_1+u_1)^2-\beta\mathop{}\!\mathrm dl\cdot\mathop{}\!\mathrm du +|U|^2+(1-q)|\mathop{}\!\mathrm d_\perp l|^2-ql_1^2 \\ &+\frac{\partial_1T}{z} +C_T\bigl[(1-q)l_1-qu_1\bigr]. \tag{46}\end{align*}\] Here is an explicit account of the norm and coefficient terms in this calculation. In \(m^{-1}\mathcal P:\nabla^2m\), the second derivatives of the norm contribute \(q|\mathop{}\!\mathrm d_\perp l|^2+|U|^2\). The contraction \(m^{-1}(\nabla_e\mathcal P):\nabla^2f\) is \(q'(l_1+u_1)l_1+2(q-1)|\mathop{}\!\mathrm d_\perp l|^2\). Differentiating \(-\beta m u_1\) contributes \(-\beta u_{11}-\beta\mathop{}\!\mathrm dl\cdot\mathop{}\!\mathrm du -q'(l_1+u_1)u_1\). Passing from \(m\) to \(l\) subtracts \(\mathcal P(\mathop{}\!\mathrm dl,\mathop{}\!\mathrm dl)=ql_1^2+|\mathop{}\!\mathrm d_\perp l|^2\). Finally, \(m^{-1}\partial_1(mT/z)=T_1/z+C_T[(1-q)l_1-qu_1]\). These are precisely all terms of Equation (46).

Set \(I=(1+y)^{-1}\) and \(J=y/(1+y)\). Then \(G_l=J\), \(G_u=-I\), and all four second partial derivatives of \(G\) as a function of \((l,u)\) equal \(-q'/2\). Since \(-I(q-1)=J\beta\), the Hessian terms in \(u\) in \(\mathcal P:\nabla^2G\) reduce exactly to \(-I\Delta u\). The lapse equation, expanded using the symmetry of \(\mathcal K+\ell g\), is \[ \Delta u=\Phi/N+ [\mathop{\mathrm{div}}_g\mathcal K-\mathop{}\!\mathrm d\kappa+\mathop{}\!\mathrm dT]\cdot v +(\mathcal K+\ell g):\pi-|\mathop{}\!\mathrm du|^2. \tag{47}\] The coefficient of \(T_1\) is \(J/z-Iz=0\). Thus no derivative of \(T\) remains. More precisely, with \(a_y=-q'/2\), the resulting identity is \[\begin{align*} \mathcal P:\nabla^2G={}&J\mathop{\mathrm{Ric}}(e,e)-I\Phi/N -I(\mathop{\mathrm{div}}_g\mathcal K-\mathop{}\!\mathrm d\kappa)\cdot v -I(\mathcal K+\ell g):\pi \\ &+\mathcal Q+JC_T[(1-q)l_1-qu_1], \tag{48}\\ \mathcal Q={}&J|U|^2+a_y|\mathop{}\!\mathrm dl+\mathop{}\!\mathrm du|^2 -J\beta\mathop{}\!\mathrm dl\cdot\mathop{}\!\mathrm du +J(1-q)|\mathop{}\!\mathrm d_\perp l|^2-Jql_1^2+I|\mathop{}\!\mathrm du|^2. \tag{49}\end{align*}\] In obtaining the coefficient \(a_y\) of the normal square, one uses \(2J+q=1\): the two contributions are \(-Jq'\) and \(-qq'/2\).

At the specified maximum put \(D_0=j'm\). The vanishing first derivative is equivalent to \[ J(l_i+u_i)=u_i-D_0\delta_{i1}-\eta_i. \tag{50}\] First take \(\mathop{}\!\mathrm d\eta=0\) and separate \(|U|^2=|U_{\rm tf}|^2+(\mathop{\mathrm{tr}}U)^2/2\). The portion of \(\mathcal Q+j'\mathcal P:\nabla^2f\) independent of \(C_T\) is exactly \[\begin{align*} J|U_{\rm tf}|^2 &+\frac{(3-2y)u_1^2-6(1-y)D_0u_1 +3(1-y)^2D_0^2}{2y(1+y)} \\ &+\frac{2-y}{y(1+y)}|\mathop{}\!\mathrm d_\perp u|^2. \tag{51}\end{align*}\] This follows by inserting \(l_1=(Iu_1-D_0)/J\), \(l_a=Iu_a/J\), and \(\mathop{\mathrm{tr}}U=C_T+[-u_1+(1-y)D_0]/y\) in Equation (49). The terms containing \(C_T\), including the last term of Equation (48), are \[\begin{align*} \frac J2 C_T^2+ C_T[J(1-2q)l_1-J(\beta+q)u_1+D_0] =\frac J2C_T^2+qC_T(2D_0-u_1). \tag{52}\end{align*}\] This last simplification is the gain needed for a bounded trace target: \(q|D_0|\le(1-y)|D_0|\), rather than an unweighted \(|D_0|\).

For \(y\in[1/2,1)\) the positive coefficients in Equation (51) are bounded below. Young’s inequality therefore bounds that expression below by \(c(|U_{\rm tf}|^2+|\mathop{}\!\mathrm du|^2)-C(1-y)^2D_0^2\). The trace terms in Equation (52) have the same lower bound after reducing \(c\) and adding a constant, because \(|C_T|\le\sqrt2 C_0\). For general \(\mathop{}\!\mathrm d\eta\), use \(D_0+\eta_1\) in the normal substitutions. The unchanged term \(-\beta D_0u_1\) adds \(\beta\eta_1u_1\) relative to that substitution. The transverse substitutions add terms bounded by \(C(|\mathop{}\!\mathrm d\eta|^2+|\mathop{}\!\mathrm d\eta||\mathop{}\!\mathrm du|)\), and the additional trace term is \(-(1-2q)C_T\eta_1\). Another application of Young’s inequality thus adds only \(C|\mathop{}\!\mathrm d\eta|^2\) to the lower bound.

The components of the graph second form are \[ \pi_{11}=T-z\mathop{\mathrm{tr}}U,\qquad \pi_{ab}=zU_{ab},\qquad \pi_{1a}=\frac{z\beta}{2}(u_a+l_a). \tag{53}\] Equation (50) also gives \[|\mathop{\mathrm{tr}}U|\le C\bigl(1+|\mathop{}\!\mathrm du|+(1-y)|D_0|+|\mathop{}\!\mathrm d\eta|\bigr).\] Thus the term \((\mathcal K+\ell g):\pi\) is absorbed in the same positive quadratic expression, with an error \(C[1+(1-y)^2D_0^2+|\mathop{}\!\mathrm d\eta|^2]\). The curvature and momentum terms are bounded, and \(-I\Phi/N\ge-C-\Sigma/N\). At a local maximum the positive definite tensor \(\mathcal P\) has nonpositive contraction with \(\nabla^2(G+j(f)+\eta)\). Its remaining height term is \(j''\mathcal P(\mathop{}\!\mathrm df,\mathop{}\!\mathrm df)=j''qm^2\), and \(|\mathcal P:\nabla^2\eta|\le C|\nabla^2\eta|\). This proves Equation (41).

Multiply it by \(N^2\) and use \(N^2qm^2=J/(1-y)\), \(J\ge1/3\), and \(N^2(1-y)^2(j'm)^2=y(j')^2\). This proves Equation (42) without a lapse floor. Lastly, negate Equation (36) and use \(d^2f_n=\hat z/N\) to obtain Equation (43). In the AH case, \(\mathop{\mathrm{tr}}\pi=0\) also gives \(d^2\partial_n\log s=u_n+yH\). Substitution of \(u_n=-zk-b_S/N\) and \(H\le k\) proves Equation (44). ◻

Proof. At nonzero gradient write \(\gamma=\mathop{}\!\mathrm dG\). The identity \(\gamma=J\mathop{}\!\mathrm dl-I\mathop{}\!\mathrm du\) gives \(\mathop{}\!\mathrm du=y\mathop{}\!\mathrm dl-(1+y)\gamma\). Inserting it into Equation (49) gives the exact expression \[\begin{align*} \mathcal Q={}&J\left[|U|^2+(1-y)l_1^2 +(2-y)|\mathop{}\!\mathrm d_\perp l|^2\right]\\ &-\frac{2y(2-y)}{1+y}\mathop{}\!\mathrm dl\cdot\gamma +\frac{1+4y-y^2}{1+y}|\gamma|^2. \tag{56}\end{align*}\] On the bounded field range, \(1-y\) has a positive lower bound and \[\frac J{m^2}=\frac{N^2(1-y)^2}{1+y},\qquad |\nabla^2f|^2=m^2(l_1^2+2|\mathop{}\!\mathrm d_\perp l|^2+|U|^2).\] The first line of Equation (56) therefore controls \(|\nabla^2f|^2\). Its mixed term is at most \(Cy|\mathop{}\!\mathrm dl||\gamma|\) and is absorbed with an error \(C|\gamma|^2\). Moreover, \(|\mathop{}\!\mathrm du|^2\le C(y^2|\mathop{}\!\mathrm dl|^2+|\gamma|^2)\), so after reducing the positive coefficient the same lower bound controls \(E_i\).

In Equation (48), the trace term has \(|JC_T|\le C z\). Its absolute value is bounded by \(C\sqrt{E_i}\), since \(z\) is comparable to \(m\) on this field range. Equation (53), or direct differentiation of \(p(N,\mathop{}\!\mathrm df)\), gives \(|\pi|\le C\sqrt{E_i}\). The remaining terms in the master identity are bounded below by a constant minus \(C\sqrt{E_i}\). Young’s inequality proves Equation (54).

There is no singularity at zero slope. The expansion \(G=-\log N+\tfrac12N^2|\mathop{}\!\mathrm df|^2+O(|\mathop{}\!\mathrm df|^4)\) and the lapse equation give, directly at a point with \(\mathop{}\!\mathrm df=0\), \[\mathcal P:\nabla^2G =N^2|\nabla^2f|^2+|\mathop{}\!\mathrm du|^2-\Phi/N -N(\mathcal K+\ell g):\nabla^2f.\] This proves the same inequality there. The matrices \(\mathcal P\) and \(\rho\mathcal P\) are smooth functions of \((N,\mathop{}\!\mathrm df)\), despite their expressions in terms of \(e\); for example \(q-1=-2y/(1+y)\) cancels the denominator in \(e\otimes e\). Their derivatives consequently satisfy \(|\mathop{\mathrm{div}}_g(\rho\mathcal P)|\le C\sqrt{E_i}\).

For \(Y=e^{k_0G}\) the product rule yields \[\begin{align*} \mathop{\mathrm{div}}_g(\rho\mathcal P\mathop{}\!\mathrm dY) =k_0Y\bigl[\rho\mathcal P:\nabla^2G +\mathop{\mathrm{div}}_g(\rho\mathcal P)\cdot\mathop{}\!\mathrm dG +k_0\rho\mathcal P(\mathop{}\!\mathrm dG,\mathop{}\!\mathrm dG)\bigr]. \end{align*}\] The mixed coefficient term is bounded by \(C\sqrt{E_i}|\mathop{}\!\mathrm dG|\). Absorb half the positive \(E_i\) term with Young’s inequality, and then choose \(k_0\) so the final term absorbs the resulting multiple of \(|\mathop{}\!\mathrm dG|^2\). The upper and lower bounds on \(s\) bound \(Y\) on this range and prove Equation (55). Outside, direct differentiation gives \[\Delta N^{-k_0}=k_0N^{-k_0} \left[(k_0+1)|\mathop{}\!\mathrm d\log N|^2-\Phi/N\right],\] which proves the stated outer inequality.

On the face, \(\rho q=d\) and Equation (36) gives \[ (\rho\mathcal P\mathop{}\!\mathrm dY)_n =-\frac{k_0YB_n}{s} +\frac{k_0Y}{d}(k_T\hat z-yH), \qquad (\partial_nY)_o=-\frac{k_0YB_n}{s}. \tag{57}\] The second term in the inner expression is bounded under the stated finite bounds; the first terms agree. This proves the flux assertion. ◻

Proof. At nonzero gradient the vector multiplying \(\mathcal T_f\) can be written as \[\frac{y\nabla f}{d m^2}= \frac{s^3}{N}\nabla f, \qquad \frac{zT}{d}e=Ts\nabla f.\] For a transverse component, multiplication of the proposed identity by \(d\) gives \(-u_a+yl_a\) on both sides. For its first component, the two terms preceding \(Ts\nabla f\) on the right side, multiplied by \(d\), are \[-u_1-y\mathop{\mathrm{tr}}U=q(yl_1-u_1)-zT,\] by Equation (45). The last term adds \(zT\), giving exactly the first component of the left side. This proves the first identity, including at zero gradient by smoothness. Commuting the covariant derivatives of \(f\) gives \(\nabla^j\nabla^2_{ij}f=\nabla_i\Delta f+\mathop{\mathrm{Ric}}_{ik}\nabla^kf\), which is the second identity.

Multiply the first identity by \(k_0Y\). In Equation (59), the coefficients of \(\nabla N\) and of \(\mathcal T_f\) then vanish at the reference state: \[\begin{align*} \mathcal J_i={}& \left(c_*-\frac{k_0Y}{s}\right)\nabla N +\mathcal T_f\left(k_0Y\frac{s^3}{N}\nabla f-\mathfrak b_*\right)\\ &+c_*(B-\nabla N)+k_0YTs\nabla f. \end{align*}\] The last line is bounded, since \(B-\nabla N=-(\mathcal K+\ell g)p\). The two coefficient differences in the first line are Lipschitz in the bounded field variables, so they are bounded by \(C|\mathcal V-\mathcal V_*|\). On the outer side, \(\mathcal J_o=(c_*-k_0N_o^{-k_0-1})\nabla N_o\), whose coefficient also vanishes at \(N_o=s_*\). This proves Equation (60).

The reference vector \(\mathfrak b_*\) and its covariant derivative are uniformly bounded in the fixed chart. Equation (58) gives \[|\mathop{\mathrm{div}}_g(\mathcal T_f\mathfrak b_*)| \le C(1+|\nabla^2f|).\] Adding \(c_*B\) changes the divergence by the bounded scalar \(c_*\Phi\). Thus Equation (55) and Young’s inequality give \(\mathop{\mathrm{div}}\mathcal J\ge c'E-C\) on each open side. If reference coefficients vary smoothly with the metric in an interior chart, their derivatives add only terms bounded by \(C(1+\sqrt E)\); these are absorbed in the same way.

Across the face, the energy flux difference is bounded by Equation (57), and the added current has matching normal flux. The vector \(\mathfrak b_*\) is normal to the face. Its Newton contraction there uses only \[\mathcal T_f(n,n)=-\mathop{\mathrm{tr}}_{\mathcal C}\nabla^2f=-f_nH_i,\] which is bounded by the constant height trace and the slope bound. Consequently the modified normal flux difference is bounded. Integration by parts on the two sides, whose face area densities agree, gives the surface error in Equation (61). ◻

Proof. In a two-sided signed-distance collar of \(\partial E_0\), let \(q\) be positive on the exterior and choose \(\chi=q\zeta(q)\), where \(\zeta\) is a nonnegative smooth cutoff equal to one near zero. Extend \(\chi\) by zero beyond the collar. Then \(\chi=0\) and \(\partial_\nu\chi=1\) at the initial surface, while \(\chi\ge0\) outside \(E_0\). Set \(\widetilde h=e^{2a\chi}\widehat h\), with \(a>0\). The conformal mean-curvature formula gives \(H_{\widetilde h}=2a\) on \(\partial E_0\). Consequently its area is unchanged and its squared-mean-curvature integral is \(4a^2|\partial E_0|_{\widehat h}\).

Every outward competitor has its boundary outside \(E_0\), where the metric has only increased, so strict outward minimality is preserved. Finally, \[ \mathop{\mathrm{Scal}}_{\widetilde h} =e^{-2a\chi}\bigl(\mathop{\mathrm{Scal}}_{\widehat h} -4a\Delta_{\widehat h}\chi -2a^2|\mathop{}\!\mathrm d\chi|_{\widehat h}^2\bigr). \tag{64}\] Choose \(a\) so that \(4a\|\Delta\chi\|_\infty+2a^2\|\mathop{}\!\mathrm d\chi\|_\infty^2\le1\) and \(4a^2|\partial E_0|\le1\). Further decreasing \(a\) gives any prescribed \(C^2\) or tensor-comparison tolerance. There is no need for a uniform collar width or a uniform positive lower bound for the initial area. This modification will be used only for the flow calculation; mass inequalities are applied to the original comparison metric. ◻

Proof. Fix a time \(t\) with the stated regularity and with \(\mathop{\mathrm{Vol}}\{u=t\}=0\). Excluding the latter exceptional times removes at most a countable set. Suppose a component \(B\) of \(E_t\) did not meet \(E_0\). For almost every \(0<s<t\), the set \(E_s\cap B\) is compactly contained in \(B\) and separated from \(E_0\). To check this, continuity gives \(u\le s<t\) on its closure, and a neighborhood of \(\overline E_0\) belongs to \(E_t\). A connected component of that neighborhood cannot meet \(B\) without placing part of \(E_0\) in \(B\). The boundaries here are perimeter representatives; the same containment holds modulo null sets.

Removing \(E_s\cap B\) is consequently an allowed variation in (67). If \(p(s)=\mathop{\mathrm{Per}}_{\widetilde h}(E_s;B)\), minimality and coarea give \[ p(s)\le\int_{E_s\cap B}|\nabla u|\,\mathop{}\!\mathrm dV =\int_0^s p(\ell)\,\mathop{}\!\mathrm d\ell. \tag{69}\] There is no initial contribution because \(B\cap E_0=\varnothing\). Grönwall’s inequality implies \(p=0\) almost everywhere. A finite-perimeter indicator with zero variation on the connected open set \(B\) is constant there. Compact containment rules out the value one, so \(E_s\cap B\) is null for almost every \(s<t\). Letting such \(s\) increase to \(t\) contradicts that \(B\) is a nonempty open component of \(E_t\). This proves the first assertion. It also shows that each such component contains an entire component of \(E_0\), and hence meets \(O\).

Outward minimality excludes a bounded open component of the complement: filling it removes its nonempty boundary and strictly decreases perimeter. Because \(X\) has just one product end and \(E_t\) is precompact, only one component of the exterior can be unbounded. Its open exterior is therefore connected.

Let \(N_t^1,\ldots,N_t^m\) be the boundary components. Choose a large compact truncation \(Y\subset X\setminus\operatorname{int}O\) containing all of them. For each \(j\), an arc can start on \(\partial O\), travel in the adjacent component of \(E_t\), cross \(N_t^j\) once transversely, and then travel in the connected exterior to the outer boundary of \(Y\), without meeting any other \(N_t^k\). To construct its inner part, join a point just inside \(N_t^j\) to \(O\) in the relevant connected open component and stop at the first cap encountered; thus the arc stays outside the cap interiors. Small perturbations give the asserted transversality.

These arcs define relative one-cycles in \((Y,\partial Y)\) whose mod-two intersection pairings with the closed surfaces \(N_t^k\) are the separate coordinate vectors. Hence the classes \([N_t^k]\) are linearly independent in \(H_2(Y;\mathbb F_2)\). Extending the truncation along its product end changes no homology, so \(m\le b_2\). Each closed orientable surface has Euler characteristic at most two, which proves the second estimate. The arcs and the intersection pairings also make explicit why disconnected caps and arbitrary compact interior topology cause no difficulty. ◻

Proof. Let \(\mathcal A_t\) denote the weak second fundamental form of \(N_t\). For almost every pair \(0<r<t\), the cited growth formula, after dropping its nonnegative tangential gradient term, says \[ J(r)\ge J(t)+\int_r^t\int_{N_s} \bigl(2|\mathcal A_s|^2+2\mathop{\mathrm{Ric}}(\nu,\nu)-H^2\bigr) \,\mathop{}\!\mathrm dA\,\mathop{}\!\mathrm ds. \tag{71}\] After our adjustment, Smooth Start Lemma 2.4 identifies the weak flow with the smooth flow on an initial interval. Hence \(J(r)\to J(0)\le1\) as \(r\downarrow0\). Letting admissible Lebesgue endpoints tend to zero supplies the initial upper trace in the integrated inequality below. At positive times only the almost-everywhere inequality is used.

For completeness, Gauss and Gauss–Bonnet apply to these levels with their weak curvature. A local \(C^{1,\gamma}\) graph with bounded weak mean curvature belongs to \(W^{2,2}\) by difference quotients in its uniformly elliptic mean-curvature equation. Smoothing graph representations gives smooth embeddings converging in \(C^1\cap W^{2,2}\); the graph formula for the second fundamental form then gives convergence of the quadratic curvature terms in \(L^1\). Thus the smooth Gauss identity and Gauss–Bonnet pass to the limit. This is also the content of Lemmas 5.4–5.5 of Huisken and Ilmanen (2001).

The Gauss equation and \(|\mathcal A_s|^2\ge H^2/2\) give \[\begin{align*} \int_{N_s}\bigl(2|\mathcal A_s|^2+2\mathop{\mathrm{Ric}}(\nu,\nu)-H^2\bigr) &=\int_{N_s}\bigl(|\mathcal A_s|^2+ \mathop{\mathrm{Scal}}_{\widetilde h}-\mathop{\mathrm{Scal}}_{N_s}\bigr)\\ &\ge\tfrac12J(s)-C_0A(s)-8\pi b_2. \tag{72}\end{align*}\] Consequently, in distributions in time, \(J'+J/2\le C_0A+8\pi b_2\), with initial upper value at most one. One can see this directly from (71) at Lebesgue endpoints: \(J\) plus the integral of the lower bound in [eq:flow-gauss-bound] has a nonincreasing representative. Any singular part of its derivative is nonpositive, so the integrating factor is valid. It yields \[ J(t)\le e^{-t/2} +\int_0^t e^{(s-t)/2}\bigl(C_0A(s)+8\pi b_2\bigr)\,\mathop{}\!\mathrm ds. \tag{73}\] Using \(A(s)=e^{s-t}A(t)\) bounds the two integrals by \(\frac23C_0A(t)\) and \(16\pi b_2\), respectively. ◻

Proof. First take \(N\) smooth and transverse to the spheres used in the construction. Let \(D\) be its closed end-side domain. For part (i), choose a transverse sphere \(r=r_1\) with \(r_p/2\le r_1\le r_p\), and replace \(D\) by \[ D'=D\cup\{r\ge r_1\}. \tag{77}\] This domain is connected because both sets contain the entire remote end. Its boundary inside \(r_1\) is part of \(N\); the remaining boundary is a portion of the sphere \(r=r_1\). Every point of this newly exposed sphere portion is covered by radial projection of \(N\cap\{r\ge r_1\}\): the ray starts outside \(D\) and eventually lies in its end, so must cross \(N\).

In the hyperbolic reference metric the radial map from radius \(r\) to \(r_1\) has two-dimensional Jacobian at most \(r_1^2/r^2\) on every two-plane. The original metric has uniform relative error \(o(1)\) on \(r\ge r_p/2\). Its corresponding Jacobian is therefore at most \[ (1+o_{r_p}(1))\frac{r_1^2}{r^2} \le(1+o_{r_p}(1))\rho. \tag{78}\] The area formula, allowing multiplicity, bounds the new sphere area by the weighted area of the discarded portion of \(N\).

The boundary of \(D'\) has transverse corners along finitely many smooth curves. In local coordinates given by defining functions for the two faces, round each quadrant edge in a shrinking tubular neighborhood, toward the side enlarging \(D'\). The added area tends to zero: the modification has vanishing width along a fixed compact edge, with bounded local slopes. The rounded end-side domain stays connected and gives an admissible cut, disjoint from \(S\). Since the definition of \(A_*\) uses its full intrinsic boundary, it follows that \(A_*\le(1+o_{r_p}(1))\int_N\rho\,\mathop{}\!\mathrm dA_g\). This proves (75).

For part (ii), keep the part of \(D\) in the original compact inner region and attach the whole original \(\ensuremath{\mathrm{AH}}\) region outside \(\mathcal C\). The resulting end-side domain is connected. Indeed, every point of its inner part can be joined inside the old \(D\) to infinity, and the initial portion of such a path reaches \(\mathcal C\). Its newly exposed boundary in \(\mathcal C\) is covered by projection of the old outer boundary along the parallel rays. This projection has two-dimensional Jacobian at most one: \(\gamma_t\ge\gamma_0\) by strict convexity, its normal derivative is zero, and the positive coefficient \(U^2\) creates no mixed terms. Rounding the resulting transverse corners as above gives admissible original cuts with area at most \(\int_N\mathop{}\!\mathrm dA_{g_{\mathrm{base}}}\) plus a vanishing error.

Finally, a \(C^1\) embedded enclosing surface can be approximated by smooth embeddings in \(C^1\), within a collar disjoint from \(S\), by an ambient isotopy. This preserves the end side and its connectedness. Choose the approximations transverse to the cutting sphere. Areas and weighted areas converge. In the joined case, first approximate on the two smooth sides and make the approximations transverse to \(\mathcal C\); round across a shrinking seam neighborhood. The base metric is bounded and smooth on either side at this fixed construction, and the surface has zero area in the seam. The area of its portion in that neighborhood tends to zero, so these local roundings contribute a vanishing area error. This procedure also applies when the smooth collar coordinates chosen for the comparison metric differ from the original side coordinates: the identifications are smooth on each closed side and preserve the joining surface. For almost every flow level the zero-area intersection follows by applying the level-measure decomposition to the fixed finite-area seam. Passing to the limit proves both assertions. ◻

Proof. Make the change from Lemma 15. It preserves the initial area and all topological properties. Take its size sufficiently small that the comparison (79) for the resulting \(\widetilde h\) holds with an error \(\bar\eta\le\varepsilon/32\), by initially requiring, for example, \(\eta\le\eta_0=\varepsilon/64\). Its scalar lower bound is \(-C_0\), where \(C_0=C_R+1\). Also take \(q_0=\delta_0=\varepsilon/32\).

Suppose, to the contrary, that \(A(0)<(1-\varepsilon)A_*\). Exponential area growth gives an interval \(I=[t_-,t_+]\) on which the endpoint areas are \((1-\varepsilon)A_*\) and \((1-\varepsilon/2)A_*\). Its length is the fixed positive number \[ \Delta_\varepsilon=\log\frac{1-\varepsilon/2}{1-\varepsilon}. \tag{83}\] Although \(t_-\) may tend to infinity when \(A(0)\) tends to zero, Lemma 17 bounds \(J(t)\) throughout \(I\) by the same constant \(C_H\).

Write \(\mathcal B=\{x<1-\delta\}\). On the good part of a level, (79) and \(\bar g\ge g_{\mathrm{base}}\) give \[ \int_{N_t\setminus\mathcal B}\rho\,\mathop{}\!\mathrm dA_{g_{\mathrm{base}}} \le\frac{A(t)}{(1-\bar\eta)(1-\delta)}. \tag{84}\] Combine this with (80). Since \((1-\bar\eta)(1-\delta)\ge1-\varepsilon/16\), for almost every \(t\in I\) we obtain \[ \int_{N_t\cap\mathcal B}\rho\,\mathop{}\!\mathrm dA_{g_{\mathrm{base}}} \ge (1-q)A_* -\frac{(1-\varepsilon/2)A_*}{(1-\bar\eta)(1-\delta)} \ge \frac{\varepsilon A_*}{4}=:c_A. \tag{85}\] The same metric comparison on the bad part therefore implies \[ \int_{N_t\cap\mathcal B}\rho x^{-1}\,\mathop{}\!\mathrm dA_{\widetilde h} \ge(1-\bar\eta)c_A\ge c_A/2. \tag{86}\]

Here is the precise weighted coarea step. Put \(F=\rho x^{-1}\) and \(H=|\nabla u|_{\widetilde h}\). For almost every level \(H>0\) at almost every point in its area measure. Hölder with exponents \(3/2\) and \(3\) gives \[\begin{align*} \int_{N_t\cap\mathcal B}F\,\mathop{}\!\mathrm dA &=\int_{N_t\cap\mathcal B} \left(\frac{F^{3/2}}H\right)^{2/3}(H^2)^{1/3}\,\mathop{}\!\mathrm dA\\ &\le \left(\int_{N_t\cap\mathcal B}\frac{F^{3/2}}H\,\mathop{}\!\mathrm dA\right)^{2/3} J(t)^{1/3}. \tag{87}\end{align*}\] Infinite integrals cause no difficulty; otherwise this identity can first be applied where \(H\) is bounded away from zero and passed to the limit. Equations (70) and (86) yield \[ \int_{N_t\cap\mathcal B} \frac{\rho^{3/2}x^{-3/2}}{|\nabla u|}\,\mathop{}\!\mathrm dA_{\widetilde h} \ge \frac{(c_A/2)^{3/2}}{\sqrt{C_H}}=:c_V. \tag{88}\] Integrating over \(I\) and applying Lipschitz coarea gives \[\begin{align*} \Delta_\varepsilon c_V &\le\int_{\mathcal B\cap\{t_-<u<t_+\}\cap\{|\nabla u|>0\}} \rho^{3/2}x^{-3/2}\,\mathop{}\!\mathrm dV_{\widetilde h}\\ &\le(1+\bar\eta)^{3/2} \int_{\mathcal B}\rho^{3/2}W\,\mathop{}\!\mathrm dV_{g_{\mathrm{base}}}. \tag{89}\end{align*}\] The last inequality uses \(\mathop{}\!\mathrm dV_{\widetilde h}\le (1+\bar\eta)^{3/2}x^{3/2}W\,\mathop{}\!\mathrm dV_{g_{\mathrm{base}}}\). Thus it is enough to take \(v_0=\Delta_\varepsilon c_V/2^{3/2}\). Notice that no contribution from a plateau of the level function has been assumed: coarea integrates only where its gradient is nonzero, and the full bad-set volume is an upper bound. This proves the proposition. ◻

Proof. Only \(|Z|\le |K|\) is used. First consider the density equation \[ \mathop{\mathrm{div}}_g(\nabla H_1-ZH_1)=3H_1 \tag{108}\] with the prescribed outer flux and inner outward flux \(t b_S\). Its adjoint is \(\Delta_g+Z\cdot\nabla-3\), with ordinary homogeneous Neumann boundary condition. The maximum and boundary point principles give zero adjoint kernel. Scalar elliptic Fredholm theory, deforming the lower terms to those of \(\Delta_g-3\), gives index zero and hence unique solvability of the primal problem. Green’s identity with the adjoint solution of a nonnegative forcing, which is nonpositive, proves \(H_1\ge0\): both prescribed outward fluxes are nonnegative, and the outer one is strictly positive. Local Harnack then gives strict positivity.

We give the weighted estimate used here, including its uniformity in the outer radius. For \(-\Delta_g+3\), scalar and divergence forcing of size \(O(r^{-j})\), \(0<j<3\), with natural flux data in \(L^{p_1}(S)\) and of size \(O(R^{-j})\) on \(S_R\), yield an \(O(r^{-j})\) solution, with a constant independent of \(R\). The natural datum includes the flux of the divergence forcing. To see this, construct a positive strict supersolution comparable to \(r^{-j}\), constant on a fixed inner region. Change its logarithmic slope slowly from \(0\) to \(-j\) in a long interval of \(\log r\) lying sufficiently far out. The limiting radial indicial slopes are \(-3\) and \(1\), so this retains \((-\Delta_g+3)\varphi\ge c\varphi\). Add a fixed sufficiently large multiple of \(R^{-j-a}r^a\), similarly continued as a constant inside, where \(0<a<1\), to make its outer normal derivative nonnegative. The resulting barrier stays comparable to \(r^{-j}\) on \(r\le R\).

For divergence forcing the barrier argument is justified by normalization and compactness. If the asserted bound fails, divide a violating solution by the maximum of its absolute value relative to the barrier, and normalize locally by the barrier’s size at a maximizing point. The normalized forcing and boundary data tend to zero. Local Laplace \(W^{1,q}\) estimates are available with \(3<q<3p_1/2\): the boundary \(L^{p_1}\) datum belongs to the required negative trace space by the two-dimensional boundary Sobolev embedding. These estimates give local uniform compactness. The outer sphere charts have uniformly controlled geometry. They may be formed by flowing \(\nabla\eta/|d\eta|^2\), \(\eta=\operatorname{arsinh}r\); the normal direction is orthogonal and the reflected principal metric is uniformly Lipschitz. Thus no high derivatives of geodesic normal coordinates are needed. At points escaping to infinity, subsequences of the normalized geometries converge to hyperbolic geometry, the relative radius to an exponential horospherical coordinate, and a nearby outer boundary to a horosphere. The strict supersolution and its Neumann sign persist in these limits. A nonzero maximum of the limiting solution divided by that supersolution contradicts the maximum or boundary point principle. The same argument in a fixed chart handles bounded maximizing points. This proves the claimed estimate.

Apply it to \(H_1-V_0\), retaining the drift as divergence forcing. The background errors are \(O(r^{1-\tau})\), and \(Z=O(r^{-\tau})\). It follows that \[ |H_1-V_0|\le Cr^{-\kappa}. \tag{109}\] Here and below \(V_0\) is extended smoothly across the compact part. There is one additional compactness point in this application: a failure with bounded maximizing points could produce a nonzero limit \(H_\circ\ge0\), with bounded measurable limit drift, zero natural flux, and \(H_\circ=O(r^{-\kappa})\), solving (108). The sign follows from positivity of \(H_1\) before subtracting \(V_0\) and dividing by an unbounded normalizing constant. Regard \(ZH_\circ\) as divergence forcing for \(-\Delta_g+3\) and apply the preceding estimate repeatedly. This improves the decay to \(O(r^{-j'})\) for some \(2<j'<3\). To justify each improvement on the full exterior, solve the baseline problem on truncations with zero natural outer flux and pass to a limit; its difference from \(H_\circ\) decays in the previous weight and vanishes by the maximum principle. The improved decay is integrable. Testing the density equation with radial cutoffs, equal to one near the finite boundary, and moving the Laplacian onto the cutoffs gives \(3\int_\Omega H_\circ=0\). All shell terms vanish since \(j'>2\). This is impossible. On bounded limiting truncations the same contradiction follows by integrating the zero-flux equation directly. This proves (109) with the asserted uniformity. For \(b_S=0\), Harnack, also by homogeneous-flux reflection at \(S\), and connectedness now give \(H_1\ge cr\).

The dual maximum principle compares \(N\) to the homogeneous solution with the same drift and boundary data, giving \(N\le H_1\le Cr\). For the lower bound divide \(N\) by the homogeneous solution with the same drift and \(b_S=0\). If this ratio is below \(c\epsilon/R_s\), with \(c\) sufficiently small, then \(N\le\epsilon\) wherever the spatial cutoff permits \(\Sigma\ne0\). Hence \(\Sigma=0\) on this sublevel. The ratio there solves an elliptic drift equation with no zeroth term. Its outward derivative is nonnegative at the inner face and strictly positive at the outer face when the ratio is below the threshold. The minimum and boundary point principles therefore exclude such a minimum. This proves the positive floor, and also excludes negative lapses when constitutive arguments have been extended.

Finally, on \(2R_s\le r\le R\), subtract \(V_0\) and treat \(ZN\) as divergence forcing of size \(O(r^{1-\tau})\). The inner Dirichlet values have size \(O(R_s)\), and the natural outer error has the corresponding decaying size. The weighted argument above, now with the inner Dirichlet face, gives \[|N-V_0|\le C R_s^{1+\kappa}r^{-\kappa}.\] The homogeneous contribution of the inner data is bounded by the same barriers. In the normalized argument the zero Dirichlet trace also excludes a limit maximum at that face. This proves the final estimate in (107). On fixed patches write the lapse equation as a metric Laplace equation with bounded divergence forcing \(ZN\), bounded scalar forcing \(3N+\Sigma\), and the prescribed \(L^{p_1}\) boundary data. The local \(W^{1,q}\) and Hölder estimates from Proposition 8 prove the last assertion. ◻

Proof. The maximum principle for \(\mathop{\mathrm{div}}_g p=0\) gives the height bound. We spell out how the differential estimate in Lemma 12 supplies all the finite bounds used here. At nonzero gradient write \[z=|v|,\quad y=z^2,\quad m_f=|df|,\quad q=\frac{1-y}{1+y},\quad \beta=1+q,\quad \mathcal P=g^{-1}+(q-1)e\otimes e,\quad e=\nabla f/m_f .\] The height equation is \(\mathcal P:\nabla^2 f+\beta\,d\log N\cdot df=0\). At an interior maximum of \(G+j(f)+d_0\), where \(G=-\log s\), \(j'<0\), \(j''>0\), and \(y\ge1/2\), that lemma gives \[ j''q m_f^2\le C\left[1+\frac{\Sigma}{N} +(1-y)^2(j'm_f)^2+|dd_0|^2+|\nabla^2d_0|\right]. \tag{113}\] Its hypotheses hold because the lapse equation has precisely the drift \(K(v)\), the tensor \(K\) and its divergence are bounded on the relevant patches, and \(\Sigma\ge0\). The cancellation of the Hessian of \(\log N\) in that lemma requires no prior second-derivative estimate for \(N\). Multiplying (113) by \(N^2\) controls \((1-y)^{-1}\), using \(Nm_f=z/(1-y)\) and the upper lapse bound; a positive lower lapse is not needed in this step.

At either constant-height face let \(n\) point towards increasing end direction. When \(df\ne0\), \(e=-n\), and the boundary part of the same lemma gives \[G_n=-\frac{yH_g+\partial_n\log N}{1-y}.\] At \(S\), \(\partial_\nu\log N=-zk-tb_S/N\), so \(G_\nu\ge zk/(1+z)\), a uniform lower bound. Since \(j'f_\nu=-j'm_f>0\), a maximum at this face bounds \(m_f\). At \(S_R\), \(H_g\) is bounded below by a positive constant, \(K=o(1)\), and \(N\partial_{\nu_R}\log N=K(p,\nu_R)+\partial_{\nu_R}V_0\). Taking \(|j'|\) sufficiently small makes the outward derivative of \(G+j(f)\) negative at large \(y\), bounded above by \(-c/(1-y)\). Thus an outer maximum is excluded, or its slope is bounded by a localization derivative. A sufficiently small decreasing exponential \(j\) has both required derivative signs.

On a compact truncation these arguments give a finite global slope bound. To make it independent of \(R\) locally, set \(d_0=k_0\log\chi\), \(k_0\ge2\), where \(\chi\) is a smooth cutoff on a slightly larger compact region, constant normally near \(S\) if that face is present. Equation (113) bounds \(W\chi\) at an interior maximum; the preceding face argument does the same at \(S\). When \(y<1/2\) this bound is immediate. Applying the lapse floor to the weighted maximum gives (110).

Since \(|p|\le N\), the flux through \(S\) is bounded independently of all internal large parameters. Test \(\mathop{\mathrm{div}}_g p=0\) with a clipped height whose derivative is one on \(I\) and zero elsewhere. Its boundary oscillation is at most \(|I|\), giving the first estimate in (111). The identity \(p\cdot df=W^2-1\), tested with the entire height, gives the second one.

We next prove the initial uniform end bound \(f\le C/r\); it cannot be obtained from a compact-region bound depending arbitrarily on \(M\). By Lemma 21, \(N\asymp r\) when \(r\ge CR_s\), for a fixed large \(C\). Fix a radial segment of \(\eta\)-length at most one starting at radius \(r_*\) there, and set \(F=r_*f\). In a bounded number of uniformly controlled metric charts covering its neighborhood, consider the subgraph of \(F\) in the product with a Euclidean line. The product vector \[\mathcal Z=(-p/r_*,1)\] is bounded and divergence free. Its pairing with the upward graph normal is \[\frac{1+p\cdot df}{\sqrt{1+r_*^2|df|^2}} =\frac{W^2}{\sqrt{1+r_*^2|df|^2}}\ge c>0,\] using \(N/r_*\asymp1\) and the constitutive relation. The divergence theorem therefore makes the subgraph locally quasiminimizing for perimeter, with a uniform constant. Filling or emptying a product ball bounds the graph perimeter there by a constant times the trace area of either phase. The four-dimensional isoperimetric inequality then gives, for the positive volume \(V(h)\) of either phase in a ball, \(V'(h)\ge cV(h)^{3/4}\). Integration yields uniform two-sided volume density, and relative isoperimetry yields a fixed positive graph-area lower bound in each sufficiently small ball centered on the graph.

If the height span of \(F\) on the radial segment is \(D\), choose graph points with separated heights and disjoint fixed-size product balls. The preceding estimate bounds the graph area in the surrounding tube and relevant height band below by \(cD-C\). Conversely, the constitutive relation gives \(\sqrt{1+r_*^2|df|^2}\le C(1+p\cdot df)\). The base tube has bounded volume and the relevant interval of \(f\) has length at most \(C(D+1)/r_*\). Equation (111) bounds the graph area above by \(C+C(D+1)/r_*\). For large fixed starting radius this bounds \(D\) uniformly. If the segment meets \(S_R\), oddly reflect \(f\), evenly reflect \(N\), and reflect the metric in the orthogonal charts described in the lapse proof. The zero trace of \(f\) makes the divergence equation valid weakly after reflection. The reflected height band adds only an interval of comparable length to the energy estimate. Thus the same estimate holds there. Summing the bounds \(C/r_*\) over consecutive outward unit segments, up to the zero height at \(S_R\), proves \(f\le C/r\).

On far-out unit balls rescale \((f,N,p,s)\) by \((r_*,r_*^{-1},r_*^{-1},r_*^{-1})\), respectively. The equations there have \(\Sigma=0\) and retain their form. The localized gradient estimate now gives uniform bounds for the rescaled height gradient and uniform ellipticity. The outer-face argument applies with its rescaled positive flux datum. The rescaled lapse has uniform \(W^{1,q}\) bounds. Differentiating the divergence height equation, or using difference quotients, gives scalar uniformly elliptic divergence equations for its first derivatives, with divergence forcing in \(L^q\), \(q>3\). The inhomogeneous Hölder estimate therefore bounds \(df\) in a Hölder norm, also at a constant-height boundary by odd reflection. The lapse divergence equation then gives \(C^{1,\gamma}\) control, followed by the nondivergence height equation and successive Schauder estimates. These steps use only the stated \(C^{2,\alpha}\) metric and \(C^{1,\alpha}\) tensor bounds. They also prove compact-set gradient Hölder estimates away from infinity without differentiating \(b_S\).

On each far-out ball, the energy estimate for a height interval of length \(C/r_*\), together with this Hölder control, implies \(z=o(1)\) uniformly as \(r_*\to\infty\). The derivative version of (107) implies that \(\nabla\log N\) tends to the hyperbolic radial direction as \(r/R_s\to\infty\). The limiting height operator is \(\Delta_b+2\,d\eta\cdot d\); its decaying radial indicial root is \(-4\). For any sufficiently small fixed \(\xi>0\), a positive multiple of \(r^{-4+\xi}\) is therefore a strict supersolution sufficiently far beyond \(R_s\). Comparison with the already proved \(C/r\) bound at a fixed multiple of \(R_s\) gives the first estimate in (112). Treating the actual height equation as a linear equation on these smaller height ranges gives the corresponding derivative bounds by Schauder theory. Inserting them in \(p=s^2\nabla f\) and \(\pi=N^{-1}\operatorname{sym}\nabla p^\flat\) gives the remaining assertions. ◻

Proof. The lapse floor confines \(N<6\epsilon\) to a compact set depending on \(M\), independently of \(\epsilon\). Equation (110) on a slightly larger compact set therefore permits a choice of \(C_2(M,L)\) such that \(sT_2\ge1\) throughout the required low-lapse region.

For every Lipschitz test \(\zeta\) vanishing at \(S_R\), positivity of \(w\), its equation, and its zero inner flux give \[ \int P\,\frac{\zeta^2}{w}\,\mathop{}\!\mathrm dV_g \le \int A(d\zeta,d\zeta)\,\mathop{}\!\mathrm dV_g . \tag{117}\] Indeed test with \(\zeta^2/w\) and complete the square in \(2\zeta\,dw\cdot A\,d\zeta/w-\zeta^2 A(dw,dw)/w^2\). All the other terms on the right side of the \(w\)-equation are nonnegative. On the support of \(P_1\), \(f/M\ge1/2\) and \(w\le4T\). With \(\zeta=f/M\), the algebraic inequality \(A(df,df)=W-W^{-1}\le W^2-1=p\cdot df\) and (111) prove \(\int P_1\le CT/M\).

For \(P_2\) take a cutoff of \(N/\epsilon\), equal to one below \(4\) and zero above \(6\). It vanishes near \(S_R\) for large \(R\). Since \(A\le N^2g^{-1}\) and \(w\le4T_2\) on this source’s support, \[ \int P_2\le C T_2\int_{\{N<6\epsilon\}}|\nabla N|^2\,\mathop{}\!\mathrm dV_g. \tag{118}\] Test the lapse equation with \((6\epsilon-N)^+\). The outward current flux at \(S\) is \(t b_S\ge0\), so its boundary contribution has the favorable sign. On the sublevel set \(|Kp|\le CN\le C\epsilon\). Cauchy’s inequality then gives \[ \int_{\{N<6\epsilon\}}|\nabla N|^2 \le C(M)\epsilon^2 +C\epsilon\int_{\{N<6\epsilon\}}\Sigma . \tag{119}\]

We must show that the final integral tends uniformly to zero; boundedness of \(\Sigma\) alone would not suffice. For any potentially violating sequence, use the uniform local lapse Hölder bounds to take \(N_j\to N_0\ge0\) uniformly on the compact region containing these sublevels. After a subsequence, \[Kp_j\rightharpoonup Z_0N_0,\qquad \Sigma_j\rightharpoonup^*\Sigma_0,\qquad |Z_0|\le C,\quad 0\le\Sigma_0\le L .\] The weak limit equation is \(\Delta_gN_0=\mathop{\mathrm{div}}_g(Z_0N_0)+3N_0+\Sigma_0\). We claim \(\Sigma_0=0\) almost everywhere on \(\{N_0=0\}\). Choose an interior density point of that zero set which is also a Lebesgue point of \(\Sigma_0\). In balls of radius \(h\downarrow0\), rescale \(N_0\) by \(h^{-2}\). The rescaled fields are nonnegative and vanish at the center. Their divergence drift has size \(O(h)\) and their scalar forcing is bounded. The inhomogeneous Harnack estimate therefore gives a uniform bound on smaller balls. Because the rescaled zero sets have asymptotically full measure, these functions tend to zero in local \(L^1\). Testing the rescaled equation against compactly supported smooth functions shows that the limiting constant scalar forcing is zero, proving the claim. Boundary points need not be considered, since \(S\) has zero volume.

For completeness, the changing sublevel sets cause no loss: for every \(a>0\), uniform convergence gives \(\{N_j<6\epsilon_j\}\subset\{N_0\le a\}\) eventually. Nonnegativity and weak convergence, using continuous cutoffs of \(N_0\) that majorize this fixed sublevel, imply \[\limsup_j\int_{\{N_j<6\epsilon_j\}}\Sigma_j \le \int_{\{N_0\le2a\}}\Sigma_0.\] Letting \(a\downarrow0\) proves the asserted uniform convergence. Equations (118) and (119), with \(T_2=C_2/\epsilon\), prove the second cost in (115). Choose \(\epsilon=\epsilon(M)\) sufficiently small that this cost, as well as \(T/M=M^{-1/5}\), tends to zero as \(M\to\infty\).

Now fix \(M,L,\epsilon\) and increase the finite strengths. The previously proved continuity of \(f,N\) supplies uniform neighborhoods of the targets \(S\) and \(N\le2\epsilon\) inside, respectively, the weaker triggering regions \(f\ge3M/4\) and \(N\le3\epsilon\). The cost bounds imply that the volume in these neighborhoods where \(w\le2T\), or \(w\le2T_2\), tends to zero as \(d_*\to\infty\). For either threshold \(D=T,T_2\), the bounded nonnegative function \((2D-w)^+\) is a weak subsolution of the homogeneous divergence operator \(\mathop{\mathrm{div}}_g(A\,d\cdot)\). Its coefficients are uniformly elliptic on the relevant compact region for these fixed parameters, independently of the strength. The scalar local supremum estimate bounds its supremum on a smaller neighborhood by its \(L^2\) mean on the larger one. At \(S\), even reflection using the zero natural flux gives the same estimate. Choosing \(d_*\) sufficiently large makes that supremum smaller than \(D\). This proves (116), with room in the thresholds if desired. Finally, if \(x<1-\delta\), the second target excludes \(N\le2\epsilon\), so the volume penalty is at full strength when also \(r\le R_s\). ◻

Proof. Only the upper bound for \(w\) remains to be established. The right side of its equation is bounded on compact sets: \(\Sigma x\le L\), since a nonzero \(\Sigma\) requires \(x<1\), and \(P\) has fixed finite strength. By (107) and (112), a multiple of \(r^{-1}\) is a supersolution for \(-\mathop{\mathrm{div}}_g(A\,dw)=3N\) sufficiently far beyond \(R_s\). Comparison therefore bounds the end values of \(w\) by \[C\left(1+\sup_{\{r\le C_0R_s\}}w\right)r^{-1}.\] If its compact supremum were unbounded, divide by that supremum. Uniform ellipticity, local divergence estimates, and convergence of \(N,df\) then give a nonzero nonnegative limit satisfying the homogeneous \(w\)-equation, with zero inner flux and decaying to zero at infinity. The maximum principle excludes such a function; Harnack and reflection at \(S\) handle a maximum at that face. Thus the compact upper bounds hold. Local Hölder estimates for \(w\) and the already proved estimates for \(N,df\) give the stated precompactness. The lapse floor and the strictly positive forcing \(3N\) in the supersolution equation imply positivity of both limiting fields. Scalar elliptic bootstrapping gives classical interior regularity.

On the end, (114) and these upper bounds permit comparison for \(x-1\) with a multiple of \(r^{-3+\xi}\). For a finite truncation add \(C(r/R)^{a_0}\), \(0<a_0<1\), to dominate its outer boundary values. The first term dominates the decaying forcing \(U_0\), while the second vanishes on fixed sets as \(R\to\infty\). The scalar zeroth coefficient is positive after increasing the fixed end radius. The maximum principle proves the zeroth-order bound in (120), and interior Schauder estimates prove its derivative and Hölder versions. These estimates are uniform over all limit solutions at the fixed parameters, without bounds on derivatives of \(b_S\).

Since \(\bar g-g=O_{\mathrm{fixed},2,\gamma}(r^{-6+2\xi})\), the choice \(\xi<3-\tau\) ensures that \(h-b\) has the original admissible decay exponent, with a possibly smaller Hölder exponent. Put \(\varphi=x-1\). The change in each defining mass flux is, up to an error tending to zero, the flux of the one-form \[-2\{V_\mu\,d\varphi-\varphi\,dV_\mu\}.\] All nonlinear conformal errors and graph terms disappear by the displayed decay. This flux has a finite limit: its divergence is \(-2V_\mu(\Delta_b-3)\varphi\), and (114) gives absolute weighted integrability. Indeed \(U_0=O(r^{-\lambda})\), \(\lambda>3\); the difference between \(\Delta_{\bar g}\) and \(\Delta_b\) on \(\varphi\) is \(O(r^{-\tau-3+\xi})+O(r^{-9+3\xi})\). Multiplication by \(V_0\) and integration against the hyperbolic volume, of radial size \(r\,\mathop{}\!\mathrm dr\), is integrable. The scalar identity (101), the original weighted matter and \(K^2\) integrability, and (112)–(120) similarly prove integrability of \(V_0(\mathop{\mathrm{Scal}}_h+6)\).

On each limit end the lapse equation also gives \[(\Delta_g-3)(N-V_0) =O(r^{1-\tau})+\mathop{\mathrm{div}}_g(Kp).\] Together with (107) and the end derivative estimates this yields \(|\nabla(N-V_0)|=O_{\mathrm{fixed}}(r^{-\kappa})\). Insert the second expression for \(Q\) in (100). Its sphere flux differs from that of \(V_0\,d\varphi-\varphi\,dV_0\) by a quantity tending to zero. The \(N-V_0\) terms have integrated size \(O(r^{-1-\kappa+\xi})\); the \((\pi-K)p\) terms have integrated sizes bounded by \(O(r^{-3+2\xi})+O(r^{-\tau+\xi})\). Metric replacement errors vanish by the same decay. It follows that \[\lim_{r\to\infty}\int_{\{r=\mathrm{const}\}}Q\cdot\nu_r \,\mathop{}\!\mathrm dA_g=-8\pi\{p_0(h)-p_0(g)\}.\] At the inner boundary the outward normal of the exterior is \(-\nu\), and \(Q\cdot(-\nu)=t b_S\). Integrating \(\mathop{\mathrm{div}}_g Q=-(x-1)\Sigma-P\) proves (121). This integration remains valid for limits with only continuous fields at \(S\): use the first expression \(Q=A\,dw+B\), the weak equations and their prescribed natural fluxes on compact regions. The second expression is needed only at infinity, where the fields are classical. Finally, \((x-1)\Sigma\le0\), the \(L^{p_1}\) bound makes \(\int_S b_S=o(1)\), and Lemma 23 gives \(\int P=o(1)\). ◻

Proof. On \(r>r_1\) solve \[(\Delta_{\bar g}-3-U_0)Y=0,\qquad Y|_{r=r_1}=0,\qquad Y/V_0\longrightarrow1 .\] Existence follows by solving with outer Dirichlet datum \(V_0\) on larger truncations, applying positive barriers, and passing to a limit. Choose \(\xi<b_0<\min(1,\tau)\). Positive and negative barriers of size \(Cr(r_1/r)^{b_0}\) for \(Y-V_0\) work by the end estimates. They give \(0\le Y\le Cr\), and the corresponding first derivative control follows from elliptic estimates. On an annulus from \(r_1\) to a large fixed multiple of \(r_1\), compare from below with a small positive multiple of \(r_1((r/r_1)^2-1)\). This is a subsolution with zero inner value, and its outer value can be made smaller than that of \(Y\). The boundary point principle and local boundary estimates therefore give, in the end direction, \[cr_1\le\partial_nY\le Cr_1 \quad\hbox{on }\{r=r_1\},\] where \(n\) is the \(\bar g\)-unit normal.

For \(\varphi=x-1\), the operator just used satisfies \((\Delta_{\bar g}-3-U_0)\varphi=U_0\). Green’s identity, the mass-flux calculation, and \(b_0>\xi\) give \[ \int_{\{r=r_1\}}(x-1)\partial_nY\,\mathop{}\!\mathrm dA_{\bar g} =\int_{\{r>r_1\}}YU_0\,\mathop{}\!\mathrm dV_{\bar g} +8\pi\{p_0(h)-p_0(g)\} \le Cr_1^3 . \tag{123}\] The source integral is bounded by \(Cr_1^3\), since \(\lambda>3\), \(Y\le Cr\), and \(W\) is uniformly bounded on this end region. The mass increment is bounded above by (121). Add \(\int\partial_nY\le Cr_1^3\) to both sides of (123), and then use \(x>0\) and \(\partial_nY\ge cr_1\). This yields \(\int_{\{r=r_1\}}x\,\mathop{}\!\mathrm dA_{\bar g}\le Cr_1^2\). Two-dimensional conformal scaling makes this integral exactly the area in (122). ◻

Proof. Consider all limits at \(t=1\) arising from arbitrary smooth choices \(b_S\in\mathcal B_M\) on arbitrary expanding truncations. Their restrictions to each fixed compact region form a compact family in the uniform field topology. Precompactness follows from Lemma 24; closedness follows by a diagonal choice of actual truncation solutions from the defining sequences of a convergent sequence of limits. This definition uses no existence assertion, and the family may initially be empty.

The fixed-parameter end estimates are uniform over this family. Choose a large fixed radial sphere, beyond \(r_1\), so that all spheres from slightly before it to infinity are strictly outward mean convex in every limit metric. Adjoin smooth compact orientable handlebodies to all components of \(S\), possible because each is a closed orientable surface. Choose the gluing maps to reverse the induced boundary orientations, so the orientation of \(\Omega\) extends over the caps. Let \(O\) denote their union and \(X\) the resulting one-ended smooth manifold without boundary; let \(K_c\) be the compact region cut off by the chosen large sphere. Extend each continuous positive metric through \(S\) to the cap side, continuously with respect to its uniform field values. For example reflect in a fixed base collar and interpolate to a fixed cap metric. The mixed metric entries vanish at \(S\), because \(f\) has constant trace. Perimeters of sets containing \(O\) depend only on the metric outside \(O\) and its boundary values.

For each of these metrics minimize perimeter among finite-perimeter subsets of \(K_c\) containing \(O\) up to null sets. Compactness and lower semicontinuity give a minimizer. These facts hold for a continuous positive metric by uniform smooth approximation of the area integrands. The sphere at \(r_1\) is an admissible comparison, so every minimizer has perimeter at most \(CR_s^2\).

Define contact points on \(S\) as points in the support of a minimizer’s perimeter measure. The union of all such contacts equals the contact set of one minimizer. To prove this, minimize volume in an auxiliary fixed smooth metric among the perimeter minimizers. Perimeter submodularity shows that the intersection and union of any two minimizers are still minimizers. The volume-minimizing one, denoted \(E_-\), is therefore contained almost everywhere in every other minimizer. If a contact of a second minimizer were absent from the perimeter support of \(E_-\), a small ball there would have zero \(E_-\)-perimeter. Since \(O\) occupies a positive part of that ball, the ball would be filled by \(E_-\), and hence by the second minimizer, a contradiction. The contact union is consequently the closed contact set of \(E_-\).

These contact unions are upper semicontinuous in the uniform field topology. Indeed, along uniformly convergent metrics, any sequence of minimizers has an \(L^1\) subsequence converging to a minimizer for the limit metric. At a contact point, the complement has uniform positive volume density: filling a chart ball is an allowed competitor, so its perimeter in the ball is bounded by a constant times its trace area on the sphere. Isoperimetry, applied to the complement inside the ball, gives \(V^{2/3}\le CV'\), and thus \(V(r)\ge cr^3\). Constants are uniform on the compact family of positive continuous metrics. The cap supplies positive density for the set on the other side. These two density bounds persist when both the metric and the contact point converge, so the limit point is in the perimeter support of a limit minimizer.

The same density bounds place all contacts in the measure-theoretic boundary. Up to a set of two-dimensional Hausdorff measure zero they therefore belong to the reduced boundary, by the finite-perimeter structure theorem. At almost every point of their intersection with the smooth surface \(S\), the tangents are those of \(S\). Since \(df\) is tangentially zero, the area density there is \(x\,\mathop{}\!\mathrm dA_g\). Writing \(C_h\) for the contact union, we obtain \[ \int_{C_h}x\,\mathop{}\!\mathrm dA_g\le CR_s^2. \tag{125}\]

Set \(\Theta=[N(H_g-zk)]_+\) on \(S\). Since \(H_g\le k\), its positive part is bounded by \(Cs\). For example, when \(k\ge0\), \(N(H_g-zk)\le Nk(1-z)\le ks\); when \(k<0\) the left side is nonpositive. The upper lapse bound also gives \(\Theta\le C\), with \(C\) independent of the pointwise cap \(B_*\). Fix \(B_*\) strictly above this latter bound. By (116), \(s\le x/T\) at \(S\), and hence \[ \int_{C_h}\Theta^{p_1}\,\mathop{}\!\mathrm dA_g \le C\int_{C_h}s\,\mathop{}\!\mathrm dA_g \le C R_s^2/T=CM^{-1/5}. \tag{126}\] The admissible \(p_1\)-power budget in (105) is \(M^{-1/10}\). Thus for sufficiently large \(M\) there is strict slack. For each field in the compact family, approximate its continuous nonnegative threshold on the closed set \(C_h\) from above by a smooth function, with a small positive margin, and taper it to zero in a sufficiently small neighborhood of that set. Regularity of surface measure and (126) keep its \(L^{p_1}\) norm within the budget; the fixed gap below \(B_*\) keeps its pointwise values admissible. This constructs a datum strictly majorizing the threshold on \(C_h\). When \(C_h\) is empty the condition is vacuous.

Upper semicontinuity of contacts and continuity of the threshold show that the same datum works in an open neighborhood of the given field in the compact limit family. Take a finite such cover, with associated smooth data \(b_1,\dots,b_m\), and enlarge the neighborhoods to open subsets of the ambient uniform field space so that their intersections with the limit family retain this property. A continuous partition of unity, supplemented by the complement of the compact limit family carrying datum zero, yields \(\mathfrak b\). Explicitly, distance to the complements of these finitely many open neighborhoods, together with distance to the compact limit family, gives nonnegative continuous weights with nonzero sum after a harmless subordinate refinement. Normalize and take the corresponding convex combination. At a field in the limit family the zero datum has weight zero, and every active \(b_i\) is a strict majorant at all its contacts. Convexity preserves both constraints in (105). If the family is empty, use the constant zero rule. Only the original compact subdomain’s fields enter this rule; no trial fields on the caps are required. ◻

Proof. We describe the compact map precisely, because a frozen source need not preserve positivity of a lapse output away from fixed points. Fix a small Hölder exponent \(\gamma>0\), to be decreased below the exponents in the a priori estimates, and use triples of \(C^{1,\gamma}\) functions on the fixed truncation. Choose \(a>0\) so that \(2a\) is below the fixed-point floor in Lemma 21, and a smooth function \(\mathcal E:\mathbb R\to[a,\infty)\) equal to the identity on \([2a,\infty)\).

Given trial fields and \(t\), first solve the linear nondivergence height equation, freezing \(\mathcal P,\beta,d\log\mathcal E(N)\) from the trial arguments, with Dirichlet values \(tM\) and zero. At zero trial gradient the smooth continuations are \(\mathcal P=g^{-1}\) and \(\beta=2\). The first-order coefficient acts on the new height. Next compute \(v\) from this new height and the extended trial lapse, and set \(Z=K(v)\). Solve the linear density equation for the raw new lapse: \[\mathop{\mathrm{div}}_g(\nabla N_{\mathrm{new}}-ZN_{\mathrm{new}}) =3N_{\mathrm{new}}+\Sigma_{\mathrm{trial}} .\] Use the prescribed outer current flux and inner current flux \(-t\mathfrak b\) evaluated on the trial fields. The source here is frozen from the trial fields, including the raw trial lapse in its cutoff at \(N\le\epsilon\). The scalar multiplying the drift and the zeroth term is the raw output lapse, not \(\mathcal E(N_{\mathrm{new}})\). The adjoint argument in Lemma 21 proves unique linear solvability.

Finally solve a linear \(w\)-equation whose coefficient \(A\) is formed from the new height and \(\mathcal E(N_{\mathrm{new}})\). Its right side is \[3\mathcal E(N_{\mathrm{new}}) +\Sigma_{\mathrm{trial}}x_{\mathrm{trial}}^+ +P_{\mathrm{trial}}, \qquad x_{\mathrm{trial}}^+ =s(\mathcal E(N_{\mathrm{trial}}),df_{\mathrm{trial}}) (w_{\mathrm{trial}})^+ .\] It has the zero inner natural flux and the prescribed positive outer Dirichlet datum. The three solves are sequential, and define a continuous compact map \(\mathcal T_t\). Both frozen sources include their prescribed factor \(t\). Indeed bounded sets of input fields give uniform ellipticity and Hölder coefficients. The height output is bounded in \(C^{2,\gamma}\), so the drift in the lapse step is \(C^{1,\gamma}\); its inverse estimates are uniform on bounded input sequences by Schauder theory, coefficient compactness in a lower exponent, and the zero-kernel adjoint argument. The lapse output and then the \(w\) output have second-order Hölder bounds. The rule \(\mathfrak b\) has values in a fixed finite convex hull of smooth functions, hence uniform bounds of every spatial derivative. The compact embedding of these \(C^{2,\gamma}\) bounds into \(C^{1,\gamma}\) proves compactness. The same estimates and uniqueness prove continuity without differentiating the selection rule.

At a fixed point, the cutoff in the actual lapse variable and the bounded drift give the positive floor in Lemma 21. Thus every extension \(\mathcal E\) becomes the identity. The last equation and positive outer datum give \(w>0\), so the positive part also disappears. Every fixed point therefore satisfies the original equations and boundary data.

All such fixed points for \(0\le t\le1\) have bounded \(C^{1,\gamma}\) norms for the fixed truncation and fixed parameters. Here is the regularity chain underlying this assertion. The lapse bound, height gradient bound, and bounded sources give lapse \(W^{1,q}\) estimates, \(q>3\), and Hölder estimates for the height gradient, as in Lemma 22. The lapse divergence equation then gives \(C^{1,\gamma'}\) estimates for some \(\gamma'>0\); its scalar-forcing portion can equivalently be estimated by the Laplace \(W^{2,q}\) theorem. The selected boundary data have bounded smooth norms. The nondivergence height equation gives \(C^{2,\gamma'}\) estimates, after decreasing \(\gamma'\) if necessary. The \(w\)-equation has bounded forcing, since \(\Sigma x\le L\) and \(P\) has finite strength. Mixed-boundary coercivity first gives an \(H^1\) bound, then scalar local boundedness and Hölder estimates give a uniform bound and continuity. Successive Schauder estimates now give second-order Hölder bounds for all fields. Take the Banach exponent \(\gamma<\gamma'\). The same bootstrap makes every fixed point smooth.

At \(t=0\), the map is constant. The first output is \(f_0=0\), so the drift in the second step is zero independently of the input. Its output is the unique positive solution \(N_0\) of \(\Delta_gN_0=3N_0\), with zero inner flux and the prescribed outer flux. The third output \(w_0\) uses \(\mathcal E(N_0)=N_0\), by the homogeneous lapse floor, and is the unique solution of \[-\mathop{\mathrm{div}}_g(N_0^2\nabla w_0)=3N_0,\qquad \partial_\nu w_0|_S=0,\qquad w_0|_{S_R}=1/V_0 .\] In general \(w_0\ne1/N_0\), because the outer lapse condition is Neumann. Choose a ball in the Banach space containing this constant output and all homotopy fixed points. On its boundary \(\mathop{\mathrm{Id}}-\mathcal T_t\) never vanishes, and its Leray–Schauder degree is \(+1\) at \(t=0\). Homotopy invariance gives a fixed point at \(t=1\).

Now let \(R\to\infty\). The fixed-parameter compact estimates give a subsequential limit on every compact set, and a diagonal subsequence gives a global limit. It belongs to the family used to define \(\mathfrak b\). Uniform convergence of \((N,f,df,w)\) on its defining compact domain implies convergence of the selected datum, in every spatial norm because its range is a finite smooth convex hull. Thus the limit realizes the rule, and the elliptic bootstrap is valid up to \(S\) as well as in the interior. ◻

Proof. Choose \(j_0\) with \(3/2<j_0<\tau\), where \(\tau\) is the original end decay exponent. Extend \(r\) to a smooth positive function on the compact part. For large \(B\), let \(\chi_B\) be a smooth radial cutoff equal to one for \(r\le B\) and zero for \(r\ge2B\), with a fixed profile in \(r/B\). Set \[g_B=b+\chi_B(g-b)\] on the end and \(g_B=g\) on the compact part. For sufficiently large \(B\) this is positive definite, and its uniform metric charts and \(C^{2,\alpha}\) bounds can be chosen independently of \(B\). Fix a smooth \(k>0\) equal to \(r^{-4}\) far out. We seek a small \(u\) with \(\partial_\nu u=0\) on \(S\) satisfying \[ -8\Delta_{g_B}u+(R_{g_B}+6)(1+u) +6\big((1+u)^5-(1+u)\big) =\chi_B(R_g+6)+\delta k. \tag{135}\] The constant-term discrepancy, excluding \(\delta k\), is supported in \(B<r<2B\) and is \(O(r^{-\tau})\) in local \(C^{0,\alpha}\) norms. Consequently it tends to zero in the scalar weighted norm of weight \(r^{-j_0}\). Also \(R_{g_B}+6\ge-o(1)\) uniformly, because it equals the nonnegative \(R_g+6\) on the inner part, vanishes beyond \(2B\), and is \(O(B^{-\tau})\) on the intervening annulus.

For completeness, the linearized operator divided by eight is \[\mathscr L_B=-\Delta_{g_B}+3+(R_{g_B}+6)/8.\] It has an inverse, with homogeneous Neumann condition, bounded uniformly from weighted \(C^{0,\alpha}\) to weighted \(C^{2,\alpha}\) at weight \(j_0\) (a smaller Hölder exponent may be used). To prove the required supremum bound, take a positive function \(\beta\) constant on a large compact set, comparable to \(r^{-j_0}\) at infinity, and let its logarithmic slope decrease slowly from zero to \(-j_0\) as a function of \(\log r\). For the limiting radial operator, \[(-\partial_\eta^2-2\partial_\eta+3)e^{-j_0\eta} =(3+2j_0-j_0^2)e^{-j_0\eta}>0.\] Every intermediate slope has the same strict sign. Slow variation of the slope, placement of the transition sufficiently far out, and the uniformly small negative part of \((R_{g_B}+6)/8\) therefore give \(\mathscr L_B\beta\ge c\beta\) with \(c>0\) independent of large \(B\). Here \(\partial_\nu\beta=0\) on \(S\). Solve on compact truncations with zero outer Dirichlet data. The maximum principle, applied to positive and negative multiples of \(\beta\), bounds the solution by the weighted norm of its forcing. Uniform local interior and Neumann Schauder estimates give the stated second-order bound. Compactness gives a solution on \(\Omega\), and the maximum principle gives uniqueness. This proves the inverse assertion without an assumption about solvability on the noncompact manifold.

The nonlinear remainder in (135) is \(O(u^2)\) and is locally Lipschitz with Lipschitz constant tending to zero on small weighted balls. The uniform inverse and contraction mapping therefore give \(u=u_{B,\delta}\) tending to zero in weighted \(C^{2,\alpha}_{j_0}\) as \(B\to\infty\) and \(\delta\downarrow0\). Local elliptic bootstrapping makes \(u\) smooth. For small parameters \(1+u>0\), and the conformal scalar-curvature formula gives, for \(\widetilde g=(1+u)^4g_B\), \[ R_{\widetilde g}+6 =(1+u)^{-5}\big(\chi_B(R_g+6)+\delta k\big)>0. \tag{136}\] The boundary conformal formula and \(\partial_\nu u=0\) give \(H_{\widetilde g}=(1+u)^{-2}H_g\le0\). The uniform metric convergence in (133) follows from the weighted bound and the cutoff construction.

We next justify the end expansion for each fixed \(B,\delta\). Beyond \(2B\), \(g_B=b\) and \[ (-\Delta_b+3)u=F(u)+\delta r^{-4}/8, \qquad F(u)=O(u^2),\quad F'(u)=O(u). \tag{137}\] All fixed orders of angular derivatives of \(u\) are \(O(r^{-j_0})\). Indeed rotations commute with \(\Delta_b\). Their difference quotients solve a linear equation with small potential \(O(u)\), with smooth data on a fixed inner sphere and decaying data at infinity. The same \(r^{-j_0}\) barriers and local weighted estimates bound these quotients. Induction, with the lower angular derivatives as forcing, proves the claim at every fixed order. Radial derivatives at the initial weighted order follow from local elliptic estimates. Since \[\Delta_b=\partial_\eta^2+2\coth\eta\,\partial_\eta +r^{-2}\Delta_\sigma,\] Equation (137) reduces, with all these angular derivatives, to \[-u_{\eta\eta}-2u_\eta+3u=O(e^{-(3+\xi)\eta})\] for any fixed sufficiently small \(0<\xi<\min\{2j_0-3,j_0-1,1\}\). Variation of constants for the two solutions \(e^\eta,e^{-3\eta}\) excludes the growing mode by the known decay and gives \(u=c(\omega)e^{-3\eta}+O(e^{-(3+\xi)\eta})\). The same argument after angular differentiation makes \(c\) smooth; the equation gives all stated radial derivatives. Changing from \(\eta\) to \(r\) gives (134).

We finally check mass convergence, since convergence only in a weight less than three would not suffice. Write \(h=b+e\) for either the original metric or one of the approximants, and let \(\mathbb U_{V_0}(e)\) be the one-form in the mass definition. The linearized scalar curvature is \[(DR)_b(e)=\mathop{\mathrm{div}}_b\mathop{\mathrm{div}}_b e-\Delta_b\mathop{\mathrm{tr}}_b e+2\mathop{\mathrm{tr}}_b e.\] Differentiation, using \(\nabla_b^2V_0=V_0b\), yields the exact linear identity \(\mathop{\mathrm{div}}_b\mathbb U_{V_0}(e)=V_0(DR)_b(e)\). The difference between \(R_h+6\) and \((DR)_b(e)\) is bounded by \(C(|e|\,|\nabla_b^2e|+|\nabla_be|^2+|e|^2)\). The uniform weighted second-order bound and \(dV_b\asymp r\,dr\,dA_\sigma\) thus give \[ 16\pi p_0(h)-\int_{r=D}\mathbb U_{V_0}(h-b)(\nu_b)\,\mathop{}\!\mathrm dA_b =\int_{r>D}V_0(R_h+6)\,\mathop{}\!\mathrm dV_b+O(D^{3-2j_0}), \tag{138}\] uniformly in the approximants. The scalar integrands in (136) are dominated by a fixed multiple of \(V_0(R_g+6+k)\), which is integrable; they converge pointwise to \(V_0(R_g+6)\). The same divergence calculation with \(V_i\), using \(|V_i|\le V_0\) and the corresponding derivative bounds, proves existence of all fluxes for the approximants. At a fixed \(D\) their surface fluxes converge by local \(C^2\) convergence. Dominated convergence in (138), followed by \(D\to\infty\), proves the asserted time-component convergence. This is also the scalar linearization underlying the mass definition in Chruściel and Herzlich (2003). Finally, the two-sided metric comparison compares the areas of every admissible intrinsic boundary with the same multiplicative factors; taking infima proves convergence of the minimum enclosing areas. ◻

Proof. We give the construction and its regularity before establishing the stress and charge assertions.

Constitutive estimates and regularization. For \(w>0\) put \(Q(w)=we'(w)/e(w)\). Logarithmic differentiation of (140) gives \[ Q(w)^{-1}=\frac12+\frac{3e(w)^{3/2}}{4(1-e(w)^{3/2})}. \tag{147}\] Thus \(0<Q\le2\). At zero, inversion in the variable \(\sqrt e\) gives \[e(w)=\frac{16}{9}w^2+O(w^5);\] in particular \(e(w)/w^2\) extends smoothly and positively to \(w=0\). At infinity, inversion in \(w^{-2}\) gives \[ e(w)=1-\frac{3}{8w^2}+O(w^{-4}),\qquad Q(w)=\frac{3}{4w^2}+O(w^{-4}),\qquad wQ'(w)=-\frac{3}{2w^2}+O(w^{-4}). \tag{148}\] Regularize only at zero by \[ e_\varepsilon(w)=w\frac{e(z)}z,\quad z=(w^2+\varepsilon^2)^{1/2},\quad Q_\varepsilon(w)=1+\frac{w^2}{z^2}(Q(z)-1),\quad 0<\varepsilon<1. \tag{149}\] The flux \(e_\varepsilon(|\xi|)\xi^\sharp/|\xi|\) is smooth and strictly elliptic on every finite gradient range, including zero. Its radial and transverse derivative eigenvalues are \(e_\varepsilon'\) and \(e_\varepsilon/w\), respectively. We have \(0<Q_\varepsilon\le2\); uniformly for sufficiently large \(w\), \[ c\le w^2Q_\varepsilon(w)\le C,\qquad -w^2\big(wQ_\varepsilon'(w)+Q_\varepsilon(w)\big)\ge c. \tag{150}\] These follow by substituting (148) in (149); the leading coefficient is \(3/4+\varepsilon^2\), bounded above and below independently of \(\varepsilon\).

Boundary and interior gradient bounds. On a connected smooth truncation \(\Omega_R\) prescribe \(u=-M_e\) on \(S\) and \(u=0\) on \(r=R\). The maximum principle gives \(-M_e\le u\le0\). At points with nonzero gradient the regularized equation is \[ P:\nabla^2u=0,\qquad P=g^{-1}+(Q_\varepsilon(w)-1)n\otimes n, \quad n=\nabla u/w. \tag{151}\] Let \(h\) be inward distance from \(S\). Since \(H(S)\le0\), smoothness gives \(\Delta h\le C h\) in a fixed collar. A supersolution above the inner boundary datum is \(-M_e+B(h)\), where \[B(h)=b_0^{-1}\log(1+Dh),\qquad B''=-b_0(B')^2.\] For \(p=B'\) sufficiently large, (150) gives \[P:\nabla^2B=Q_\varepsilon(p)B''+p\Delta h \le-cb_0+C/b_0,\] because \(hp\le1/b_0\). Choose \(b_0\) first to make this negative, then a collar width \(h_0\) so small that \((2b_0h_0)^{-1}\) is above the required gradient threshold, and finally \(D\) large so that \(B(h_0)>M_e\) and \(B'\ge(2b_0h_0)^{-1}\) there. Comparison with the constant lower barrier bounds the inward derivative on \(S\), independently of \(R\) and \(\varepsilon\). At the outer sphere, inward distance may be replaced by \(h=\eta_R-\eta\) in a uniform collar. For gradients parallel to \(d\eta\), asymptotic hyperbolicity and \(Q_\varepsilon\le2\) give \(P:\nabla^2h\le-1\) for sufficiently large \(R\). The linear lower barrier \(-C h\), together with \(u\le0\), bounds the outer derivative with \(C\) depending on \(M_e\) and the fixed collar width only.

For the interior estimate use an orthonormal frame with first vector \(n\), and write transverse indices as \(a,b\). Differentiating (151), commuting one derivative, and differentiating the norm of \(du\) gives \[\begin{align*} P:\nabla^2\log w ={}&\mathop{\mathrm{Ric}}(n,n) +(1-Q_\varepsilon)\frac{|d_\perp w|^2}{w^2} +\frac{\sum_{a,b}u_{ab}^2}{w^2} \\ &-\big(wQ_\varepsilon'+Q_\varepsilon\big)\frac{w_1^2}{w^2}. \tag{152}\end{align*}\] To specify the cancellation, the differentiated coefficient term is \(Q_\varepsilon'w_1^2+2(Q_\varepsilon-1)|d_\perp w|^2/w\); the differentiated norm contributes \([Q_\varepsilon\sum_a u_{1a}^2+\sum_{a,b}u_{ab}^2]/w\) before division by \(w\). Subtracting \(P(dw,dw)/w^2\) produces exactly (152). At a positive interior maximum of \(w\exp(B_0u)\) we have \(d_\perp w=0\) and \(w_1/w=-B_0w\). The left side of (152) is nonpositive, since \(P:\nabla^2u=0\). The global lower Ricci bound and (150) make the right side at least \(-C+cB_0^2\) when \(w\) is large. A fixed sufficiently large \(B_0\) excludes such a maximum. Combining the remaining interior maxima with the boundary estimates and the height bound gives \[ \sup_{\Omega_R}|du|_g\le C(M_e), \tag{153}\] uniformly in large \(R\) and \(0<\varepsilon<1\).

Existence on truncations and passage to the limit. For fixed \(R,\varepsilon\), vary the inner Dirichlet value from zero to \(-M_e\). The derivative of the divergence equation is a strictly elliptic divergence operator without zeroth-order term and is invertible with zero Dirichlet data by the maximum principle and linear elliptic theory. The bounds already proved hold on this entire path. On its bounded gradient range, ordinary uniformly elliptic quasilinear gradient Hölder estimates apply. For the boundary estimate subtract its constant value and reflect oddly in boundary normal coordinates, reflecting the metric as well. The reflected metric is Lipschitz; its mixed normal–tangential entries vanish on the face. The reflected equation is weakly valid by the zero trace. Equivalently, difference quotients give uniformly elliptic equations for the first derivatives with bounded divergence forcing. Scalar Hölder estimates followed by Schauder estimates on the original smooth side give closedness of the continuation. Here constants may depend on the fixed \(R,\varepsilon\); Gilbarg and Trudinger (2001) supplies the ordinary elliptic estimates being used. Thus smooth regularized solutions exist.

A decay bound is uniform in both regularizations. On the end put \(v=C(e^{-\eta}-e^{-2\eta})\). It is positive for large \(\eta\), and \[\frac{v''}{-v'}=\frac{1-4e^{-\eta}}{1-2e^{-\eta}} \le1-e^{-\eta}.\] Also \(|d\eta|_g^2=1+O(e^{-\tau\eta})\) and \(P:\nabla^2\eta=2+O(e^{-\min\{\tau,2\}\eta})\) for the radial test direction, uniformly for \(0<Q_\varepsilon\le2\). Since \(\tau>1\), these inequalities imply \(P:\nabla^2v\le0\) sufficiently far out, independently of \(C\). Comparison of the nonnegative solution \(-u\) with \(v\), choosing \(C\) on a fixed inner sphere and using \(u=0\) on the truncating edge, gives \(|u|\le Ce^{-\eta}\).

Take a subsequence with \(R\to\infty\) and \(\varepsilon\downarrow0\). The uniform Lipschitz bound gives local uniform convergence and weak gradient convergence to a locally Lipschitz function with the required traces. To pass to the equation, let \(\chi\) be a compactly supported cutoff and test the regularized equation with \(\chi(u_j-u)\). The terms containing \(d\chi\) tend to zero by uniform convergence. Subtracting the limiting-law flux evaluated at \(du\) gives \[\int\chi\, \big(A(du_j)-A(du)\big)\cdot(du_j-du)\,\mathop{}\!\mathrm dV_g\longrightarrow0, \qquad A(\xi)=e(|\xi|)\xi^\sharp/|\xi|.\] Uniform convergence of the constitutive functions on bounded ranges justifies replacing the regularized flux here. The law \(A\) is strictly monotone. On the fixed compact gradient range its monotonicity pairing has a positive minimum whenever the two arguments are separated by a prescribed positive distance. Thus \(du_j\to du\) in measure and strongly in every finite local \(L^p\). This proves the limiting weak equation.

Gradient regularity and compactification. We apply the scalar \(p=3\) gradient theorem recorded in Section 2, namely Theorem 1.1 and estimate (1.8) of Araújo and Zhang (2020). The coordinate flux must satisfy globally \[\begin{align*} \lambda|\xi|\,|\zeta|^2 &\le \mathcal A_\xi(x,\xi)[\zeta,\zeta], &|\mathcal A|+|\mathcal A_\xi|\,|\xi|&\le C|\xi|^2, \tag{154}\\ |\mathcal A(x,\xi)-\mathcal A(y,\xi)| &\le C|x-y|\,|\xi|^2. \end{align*}\] It must also be \(C^1\) in \(\xi\) and the solution must have bounded local \(W^{1,3}\) energy. The Lipschitz spatial bound supplies a \(C^{0,1/2}\) modulus, so \(n=p=3\) and zero forcing meet all the parameters of that theorem. Throughout this argument choose only \(0<\beta<\min\{\alpha_m,1/4\}\) as in Section 2; the decay and boundary limits below require a positive exponent, not an optimal one.

We verify global, rather than merely bounded-gradient, structure before applying this theorem. On the gradient range in (153), \(e(w)/w^2\) and \(Q(w)\) have positive upper and lower bounds. Thus \(e'(w)\) is comparable to \(w\). Choose a smooth cutoff \(\chi=1\) below the known range and zero above twice a larger fixed bound, and replace \(e'\) outside that range by \[\widetilde e'(w)=\chi(w)e'(w)+(1-\chi(w))c_1w, \qquad \widetilde e(0)=0,\] where \(c_1>0\). Then \(\widetilde e=e\) on the range attained by the solution, \(\widetilde e'\asymp w\), and \(\widetilde e\asymp w^2\) globally. In coordinates with metric matrix \(G\) the density flux is \[ \mathcal A^i(x,\xi)=\sqrt{\det G}\, \frac{\widetilde e(|\xi|_G)}{|\xi|_G}\,G^{ij}\xi_j, \qquad |\xi|_G=(G^{ij}\xi_i\xi_j)^{1/2}. \tag{155}\] Uniform positive metric bounds give (154); Lipschitz metric coefficients give the spatial bound. The flux is \(C^1\) at zero because it is \(O(|\xi|^2)\) and its derivative is \(O(|\xi|)\). Its derivative is symmetric, as is also apparent from its convex radial primitive. The solution has finite local cubic energy by its Lipschitz bound. The quoted theorem therefore proves local \(C^{1,\beta}\) regularity.

The same argument works at \(S\). In orthogonal boundary coordinates with reflection matrix \(J=\operatorname{diag}(-1,1,1)\), extend the metric matrix by \(G(-h,x)=JG(h,x)J\), and extend \(u+M_e\) oddly. The mixed metric entries vanish on the face, so this extension is Lipschitz. The reflected flux satisfies \(\mathcal A_-(\xi)=J\mathcal A_+(J\xi)\). Evaluated on the odd solution, its normal component is even and its tangential components are odd. A test on the reflected ball consequently reduces to a test on the original half-ball with zero trace. Such tests are allowed: cutoff at distance \(h=\delta\) creates an error tending to zero since the test is \(O(h)\), the flux is bounded, and the strip volume tends to zero. Thus the reflected weak equation satisfies all the same hypotheses, proving the asserted boundary regularity.

To obtain gradient decay, take a uniform metric ball far out with central radius \(r_*\), and put \(s_*=C/r_*\), so that \(|u|\le s_*\) on a slightly larger ball. Apply the preceding fixed global extension to the equation for \(u/s_*\); its flux is \[\mathcal A_{s_*}(x,\xi)=s_*^{-2}\mathcal A(x,s_*\xi).\] The cubic structure constants and spatial modulus are unchanged. The required energy bound does not presume a bound on the normalized gradient. Testing with \(\chi^3(u/s_*)\), and using coercivity and Young’s inequality, gives \[ \int\chi^3|d(u/s_*)|^3 \le C\int |u/s_*|^3|d\chi|^3\le C. \tag{156}\] The Hölder estimate for the gradient, together with this energy bound, gives its supremum on a smaller ball: some point has gradient bounded by the cubic mean, and the Hölder bound controls its difference from every other point. Therefore \(|du|_g\le C/r\).

Now let \(g_c=\rho^2g\) on the end. It extends to a positive Lipschitz metric up to \(\rho=0\), with vanishing mixed normal–tangential entries there. In the conformal end at hand this follows directly from \[g_c=\phi^4\left(d\rho^2+\tfrac14(1-\rho^2)^2\sigma\right)\] and (134). The estimate just obtained gives \(|du|_{g_c}\le C\), and the height estimate gives the zero trace. Writing \(\bar w=|du|_{g_c}\), the coordinate divergence equation in this metric has radial law \[ q_\rho(\bar w)=\rho^{-2}e(\rho\bar w) =\bar w^2 F(\rho\bar w), \qquad F(t)=e(t)/t^2. \tag{157}\] On the known bounded range of \(\bar w\), this satisfies uniform cubic structure and Lipschitz spatial dependence, including at \(\rho=0\). For an explicit uniform extension, set \(H(t)=2F(t)+tF'(t)>0\), choose a fixed cutoff equal to one below that range and zero above twice a larger bound, and define \[\widetilde q_\rho'(w) =w\big[\chi(w)H(\rho w)+(1-\chi(w))c_1\big], \qquad \widetilde q_\rho(0)=0.\] It agrees with \(q_\rho\) on the attained range, is comparable to \(w^2\), has derivative comparable to \(w\), and has the required uniform Lipschitz dependence in position. Reflect \(g_c\) as above, use \(|\rho|\) in this coefficient, and reflect \(u\) oddly. The gradient bound ensures \(W^{1,3}\) across the face. The same zero-trace test argument proves the reflected weak equation. The scalar theorem now gives \(C^{1,\beta}\) in compactified coordinates. No theorem about degenerate boundary equations is being assumed in addition to the stated interior result.

Stress and asymptotic coefficients. Convexity shows that the weak solution minimizes the integral of \(L(|du|)\) against compactly supported Lipschitz variations: integrate the convex tangent inequality and use (141). Apply this fact to compactly supported changes of independent variable. For the flow of a smooth vector field \(X\), differentiating the local energy of the transported potential gives \[\int\big(L(w)g-du\otimes E^\flat\big):\nabla X\,\mathop{}\!\mathrm dV_g=0.\] The calculation is justified by bounded gradients on the support; no finite total energy of \(L\) is needed. This is \(\mathop{\mathrm{div}}_g\Pi=0\) in distributions. The stress eigenvalues in the gradient and transverse directions are \(t\) and \(l=L=t+we\), respectively, with \(\Pi=g\) at zero gradient. They are positive at every finite gradient. Moreover, \[(t+2l)^2-2(t^2+2l^2)=4tl-t^2 =3t^2+4twe=3(1-e^{3/2})+3e^{3/2}=3,\] proving (143).

Since \(u\) has constant trace at conformal infinity, all its tangential derivatives in compactified coordinates vanish there. Its scaled gradient consequently has a continuous radial limit. Together with \(r\rho\to1/2\) and \(e(w)=(16/9)w^2+O(w^5)\), this gives a continuous \(e_0\ge0\) with \(r^2e(w)\to e_0\) uniformly. At a point where \(e_0>0\) the gradient direction approaches the radial direction, so \(\Pi(\nu,\nu)=t+o(r^{-3})\). At a point where \(e_0=0\), both \(t-1\) and \(l-t=we\) are \(O(e^{3/2})=o(r^{-3})\), and the same conclusion holds. Uniformity follows by splitting into the sets where \(e_0\) is larger or smaller than an arbitrary fixed positive threshold. Since \(t=1-e^{3/2}/2+O(e^3)\), this proves (144), as well as \(E=O(r^{-2})\) and \(\Pi-g=O(r^{-3})\) in metric norm.

Flux saturation. Continuous distributionally divergence-free fields have the usual flux identities on smooth compact domains: use smooth cutoffs approaching their boundaries and continuity in the shrinking collars. This proves independence in (145). The asymptotics and \(dA_g/r^2\to dA_\sigma\) give \(F_{M_e}\le\int e_0\,dA_\sigma\). Integration by parts in the electrostatic equation gives \[ M_eF_{M_e}=\int_\Omega w e(w)\,\mathop{}\!\mathrm dV_g. \tag{158}\] Indeed the infinity term tends to zero because \(u=O(r^{-1})\), \(E=O(r^{-2})\), and the sphere area is \(O(r^2)\). The inner term has the indicated sign: the outward normal of the exterior at \(S\) is \(-\nu\), while \(u=-M_e\) there. The integral is finite since \(we=O(r^{-3})\) and \(dV_g\asymp r\,dr\,dA_\sigma\).

Fix a large \(R_0\), choose \(4/3<s_0<2\), and set \[ \zeta=1\quad(r\le R_0),\qquad \zeta=(R_0/r)^{s_0}\quad(r>R_0),\qquad C_\zeta=\int_\Omega(1-t(\zeta))\,\mathop{}\!\mathrm dV_g<\infty. \tag{159}\] The last integrability follows from \(1-t(\zeta)=O(\zeta^{3/2})\) and \(s_0>4/3\). For every smooth function \(v\) equal to \(-M_e\) near \(S\) and zero outside a compact set, each regular level \(v=q\), \(-M_e<q<0\), obeys \[ \int_{v=q}\zeta\,\mathop{}\!\mathrm dA_g \ge(1-o_{R_0}(1))A_*. \tag{160}\] Here is the topological and geometric comparison for this assertion. Take the closure \(D\) of the component of \(\{v>q\}\) containing the end. It is a connected closed end-side domain, with boundary a subset of the level; it misses a neighborhood of all of \(S\). Choose a sphere \(r=R_1\), with \(R_1\in[R_0/2,R_0]\), transverse to its boundary, and adjoin to \(D\) the whole region \(r\ge R_1\). The union is still connected because the two domains have a common far-end region. Each newly exposed sphere-cap point lies on a radial ray meeting the old boundary at some \(r\ge R_1\): initially the ray is outside \(D\), and eventually it lies in \(D\). Radial projection onto \(r=R_1\) has upper two-dimensional Jacobian \[(1+o_{R_0}(1))(R_1/r)^2 \le(1+o_{R_0}(1))\zeta(r).\] The first estimate is uniform over tangent two-planes by the asymptotic tensor bounds; the second uses \(s_0<2\) and \(R_1\le R_0\). The area formula bounds the new caps by the weighted old boundary outside the sphere. Inside, the weight is one. The transverse corners can be rounded toward the side enlarging this union, in shrinking tubular neighborhoods of their intersection curves; the added area tends to zero. The rounded exterior is connected and its full intrinsic boundary is an admissible cut in the definition of \(A_*\). This proves (160) without any connectedness assertion about the level itself.

Coarea therefore gives \(\int\zeta|dv|\ge M_e(1-o_{R_0}(1))A_*\). The same holds for \(u\). To justify approximation, first replace \(u\) by its constant trace in a shrinking collar of \(S\), interpolating in a collar of comparable width. The error in its weighted gradient integral tends to zero by the bounded gradient and constant trace. Next multiply by a cutoff tending to zero in \(R<r<2R\) and set it equal to zero beyond \(2R\). The height and gradient decay make both the lost tail and cutoff error \(O(R^{1-s_0})\), tending to zero. Smooth the remaining compactly supported transition while retaining the constant boundary neighborhoods. This gives convergence in the required weighted \(W^{1,1}\) norm, and coarea applies to the smooth approximants.

Finally, the definition of \(L\) gives \(L(w)\ge\zeta w+t(\zeta)\), whereas \(L(w)-1\le we(w)\) since \(t(e)\le1\). Combining these facts with (158) yields \[ F_{M_e}\ge(1-o_{R_0}(1))A_*-C_\zeta/M_e. \tag{161}\] Let \(M_e\to\infty\) at fixed \(R_0\) and then \(R_0\to\infty\). This proves (146) and completes the proposition. ◻

Proof. Choose an electrostatic height \(M_e\) large enough that the preceding proposition gives \[ \langle e_0\rangle\ge a^2-o(1),\qquad \langle f\rangle=(4\pi)^{-1}\int_{\mathbb S^2}f\,\mathop{}\!\mathrm dA_\sigma, \tag{165}\] where the error can be arbitrarily small. On the whole exterior first define a continuous tensor \(K^{(0)}\) by \[ K^{(0)}-(\mathop{\mathrm{tr}}_gK^{(0)})g=2\Pi, \qquad K^{(0)}=2\Pi-(\mathop{\mathrm{tr}}_g\Pi)g. \tag{166}\] Its eigenvalues are \(t-2l=-t-2we<0\) in the gradient direction and \(-t<0\) transversely. Directly from (143), \[ (\mathop{\mathrm{tr}}_gK^{(0)})^2-|K^{(0)}|_g^2=6, \qquad \mathop{\mathrm{div}}_g\big(K^{(0)}-(\mathop{\mathrm{tr}}_gK^{(0)})g\big)=0 \tag{167}\] in distributions. For example, if \(s_\Pi=\mathop{\mathrm{tr}}_g\Pi\), then \(\mathop{\mathrm{tr}}_gK^{(0)}=-s_\Pi\) and \(|K^{(0)}|^2=4|\Pi|^2-s_\Pi^2\), giving the first identity.

On a coordinate sphere of radius \(r\), the induced metric is \(\phi^4r^2\sigma\), hence its Gauss curvature is positive for large \(r\). The conformal mean-curvature formula gives \[\begin{align*} H_i &=\phi^{-2}\left(\frac{2\sqrt{1+r^2}}r +4\sqrt{1+r^2}\,\partial_r\log\phi\right)\\ &=2+r^{-2}-16\alpha r^{-3}+o(r^{-3}). \tag{168}\end{align*}\] The relation (166) gives exactly \(\mathop{\mathrm{tr}}_{\mathcal C}K^{(0)}=-2\Pi(\nu,\nu)\), so (144) yields \[ k_T^{(0)}=-2+e_0^{3/2}r^{-3}+o(r^{-3}). \tag{169}\] All these remainders are uniform in angle. Consequently \(H_i>|k_T^{(0)}|\) on every sufficiently large sphere, and \[ \sqrt{H_i^2-(k_T^{(0)})^2} =\frac2r+\frac{-16\alpha+e_0^{3/2}}{r^2}+o(r^{-2}). \tag{170}\] Indeed the expression under the root is \(4r^{-2}+(-64\alpha+4e_0^{3/2})r^{-3}+o(r^{-3})\).

We explain carefully the smoothing used after fixing such a sphere \(\mathcal C\). On the compact \(\Omega_i\), continuity and strict positivity give positive lower bounds for the eigenvalues of \(\Pi\), the absolute negative eigenvalues of \(K^{(0)}\), and \(R_g+6\). There are smooth tensors \(\Pi_h\) on \(\Omega_i\), up to its boundary, such that \[ \|\Pi_h-\Pi\|_{C^0}\longrightarrow0, \qquad \|\mathop{\mathrm{div}}_g\Pi_h\|_{C^0}\longrightarrow0. \tag{171}\] Here is the needed commutator proof, including the boundary. In an interior chart convolve the tensor components with a smooth kernel \(\eta_h\). A divergence term has form \(a(x)\partial_jT+b(x)T\), with smooth coefficients. For the principal part, integration by parts in the commutator produces \((a(y)-a(x))\partial\eta_h(x-y)\) and a term with \((\partial a)(y)\eta_h(x-y)\). Together these cancel on constant \(T\). After subtracting \(T(x)\) their integral is bounded by a fixed constant times the modulus of continuity of \(T\) on a ball of radius \(O(h)\), and hence tends uniformly to zero. The lower-order terms commute up to the same type of error.

In a boundary chart evaluate the convolution instead at \(x+hc_0n_0\), with \(n_0\) a constant inward coordinate direction and \(c_0\) large enough that the kernel stays on the original side. The same proof applies: \(|a(y)-a(x)|=O(h)\) throughout its support, and the combined commutator still vanishes on a constant tensor. It therefore extends the uniform estimate to the boundary without extending the original stress as a divergence-free tensor. Take these local approximations on slightly larger charts and recombine with a smooth partition of unity. The extra differentiated partition terms tend to zero, because their limiting sum is \((\sum d\chi)\Pi=0\). This proves (171).

Set \(K_h=2\Pi_h-(\mathop{\mathrm{tr}}_g\Pi_h)g\). Then \(\mathop{\mathrm{div}}K_h-d\mathop{\mathrm{tr}}K_h=2\mathop{\mathrm{div}}\Pi_h\to0\) uniformly, and the algebraic part \(R_g+(\mathop{\mathrm{tr}}K_h)^2-|K_h|^2\) tends uniformly to \(R_g+6>0\). For sufficiently small \(h\), (162) follows uniformly for \(|v|\le1\), as does \(K_h<0\). The strict gap \(H_i>|k_T^{(0)}|\) at the now fixed sphere also persists. Write \(K\) for this smooth tensor. The error in any integral involving \(\sqrt{H_i^2-k_T^2}\) can be made arbitrarily small at this fixed sphere, because the square root is smooth on its positive gap. This order of choices is essential; we require no smoothing estimate uniform as the sphere tends to infinity.

The positive-Gauss metric on \(\mathcal C\simeq\mathbb S^2\) has a smooth strictly convex Euclidean isometric embedding by the smooth Weyl embedding theorem (Nirenberg 1953). This is precisely the spherical positive-curvature version also stated in the introduction of Shi and Tam (2002). Denote its outward Euclidean mean curvature by \(H_E\). We will construct the Shi–Tam scalar-flat extension with the smooth positive prescribed boundary curvature \(H_o=\sqrt{H_i^2-k_T^2}\). The required estimate is \[ m_o\le\frac1{8\pi}\int_{\mathcal C}(H_E-H_o)\,\mathop{}\!\mathrm dA. \tag{172}\] Before proving this extension assertion, we evaluate its right side and record the exact coefficient of the charge term.

Scale the induced metric on \(\mathcal C\) by \(r^{-2}\). Its area is \(4\pi+O(r^{-3})\) and its Gauss curvature is \(1+O(r^{-3})\), with smooth angular control. For its convex Euclidean embedding choose an interior point as origin and use Gauss-map coordinates \(\omega\). Let \(l(\omega)>0\) be the support function. Then the embedding is \(l\omega+\nabla_\sigma l\), its inverse shape operator is \(\nabla_\sigma^2l+l\sigma\), and \[ \int H_E^{\mathrm{scaled}}\,\mathop{}\!\mathrm dA =\int_{\mathbb S^2}\mathop{\mathrm{tr}}_\sigma(\nabla_\sigma^2l+l\sigma)\,\mathop{}\!\mathrm dA_\sigma =2\int_{\mathbb S^2}l\,\mathop{}\!\mathrm dA_\sigma. \tag{173}\] The scaled enclosed volume is \(\frac13\int l\,dA\), and \(dA=(1+O(r^{-3}))dA_\sigma\) by the Gauss curvature estimate. Euclidean isoperimetry bounds this volume by \(4\pi/3+O(r^{-3})\). Positivity of \(l\) now implies \(\int l\,dA_\sigma\le4\pi+O(r^{-3})\). Scaling back in (173) gives \[ \int_{\mathcal C}H_E\,\mathop{}\!\mathrm dA\le8\pi r+O(r^{-2}). \tag{174}\] This uses no quantitative stability theorem for the Weyl embedding.

Before smoothing the stress, (170) and \(dA=\phi^4r^2dA_\sigma\) give \[\int_{\mathcal C}H_o\,\mathop{}\!\mathrm dA =8\pi r+\int_{\mathbb S^2}(-16\alpha+e_0^{3/2})\,\mathop{}\!\mathrm dA_\sigma+o(1).\] The smoothing error in this equality can be made arbitrarily small after fixing \(r\). Moreover, direct substitution of \(g-b=(4\alpha r^{-3}+O(r^{-3-\xi}))b\) in the mass flux gives \[ p_0(g)=8\langle\alpha\rangle. \tag{175}\] For clarity, if \(e=fb\), the flux one-form is \(-2(V_0\,df-f\,dV_0)\); for \(f=4\alpha r^{-3}\) its radial sphere integral tends to \(32\int\alpha\,dA_\sigma\). Consequently (174) gives \[\begin{align*} \frac1{8\pi}\int_{\mathcal C}(H_E-H_o)\,\mathop{}\!\mathrm dA &\le p_0(g)-\tfrac12\langle e_0^{3/2}\rangle+o(1)\\ &\le p_0(g)-\tfrac12\langle e_0\rangle^{3/2}+o(1) \le p_0(g)-\tfrac12a^3+o(1). \tag{176}\end{align*}\] The second inequality is Jensen’s inequality. If \(a=0\), its charge term is simply nonnegative. In all cases first choosing \(M_e\), then \(r\), then the compact smoothing tolerance makes the final error smaller than the prescribed \(\varepsilon_0\).

The scalar-flat extension and its mass. We include the extension argument to fix both the existence hypotheses and the normalization in (172). It is the quasi-spherical construction of Shi and Tam (2002, sec. 2, Theorem 2.1, and Lemma 4.2). The input here is exactly a smooth strictly convex Euclidean sphere and a smooth positive prescribed mean curvature; no compact fill-in is used.

Let \(t\ge0\) be Euclidean distance along the outward normals to the convex embedding. The parallels foliate its entire exterior. Write the Euclidean metric as \(dt^2+\gamma_t\), and denote the Euclidean mean curvature and surface scalar curvature by \(H_E(t)\) and \(R_t\). Both are positive. If \(\lambda_1,\lambda_2>0\) are the initial principal curvatures, then those at time \(t\) are \(\lambda_j/(1+t\lambda_j)\), whence \[ \frac{R_t}{2H_E(t)} =\frac1{2t+c(\omega)},\qquad c=\lambda_1^{-1}+\lambda_2^{-1}>0. \tag{177}\] Solve the scalar parabolic equation \[ 2H_E\partial_tU =2U^2\Delta_{\gamma_t}U+(U-U^3)R_t, \qquad U(0,\cdot)=H_E(0,\cdot)/H_o. \tag{178}\] Smooth positive initial data give a local solution by strictly parabolic theory. The reaction term is nonpositive above one and nonnegative below one. The maximum principle therefore keeps \(U\) between fixed positive constants. On every finite time interval the equation is uniformly parabolic with smooth background geometry. Scalar parabolic Hölder estimates for bounded solutions, followed by Schauder estimates and bootstrapping, give continuation for all finite times.

For the asymptotic estimate, choose positive constants respectively above and below the entire initial range and one. Solve, for each of these constants as initial value, \[ \dot z=(z-z^3)/(2t+c_{\max}),\qquad c_{\max}=\max c. \tag{179}\] By (177) these are respectively upper and lower barriers: the sign of \(z-z^3\) reverses the inequality in the upper case. Since \[z^{-2}-1=(z(0)^{-2}-1)\frac{c_{\max}}{2t+c_{\max}},\] comparison gives \(U-1=O((1+t)^{-1})\). Put \(s=\log(1+t)\) and \(\widehat\gamma_t=(1+t)^{-2}\gamma_t\). In the Gauss-map parametrization the parallel embedding is \(X(\omega)+t\omega\), so these scaled metrics and all their derivatives have bounded geometry. Equation (178) in \(s\) is uniformly parabolic. Its local Hölder and then Schauder estimates on fixed \(s\)-strips first bound all derivatives of \(U\) uniformly. Next apply the linear estimates to \(U-1\) with the actual coefficients of this solution; its equation is homogeneous, with a bounded smooth zeroth-order coefficient. The supremum bound just proved gives the same \(O(e^{-s})\) decay at every fixed derivative order. This proves the stated ordinary AF derivative bounds in the Euclidean coordinates \(x'=X(\omega)+t\omega\).

Set \[ g_o=U^2dt^2+\gamma_t. \tag{180}\] It has the required induced metric and inner mean curvature \(H_E/U=H_o\). Its scalar curvature is, by the hypersurface Gauss and mean-variation formulas, \[ R_{g_o}=R_t(1-U^{-2})+2H_EU^{-3}\partial_tU -2U^{-1}\Delta_{\gamma_t}U=0, \tag{181}\] where the final equality is exactly (178). The positive bounds on \(U\) and the Euclidean completeness of the parallel exterior give completeness with the boundary included. There is precisely one end, and \(g_o-\delta=(U^2-1)dt^2\) has the stated decay.

Define \[\mathcal M(t)=\int_{\mathcal C_t}H_E(1-U^{-1})\,\mathop{}\!\mathrm dA_{\gamma_t}.\] The Euclidean identities \(\partial_tH_E=-|\mathrm{II}_E|^2\), \(\partial_t dA=H_EdA\), and \(H_E^2-|\mathrm{II}_E|^2=R_t\), together with (178), give \[\begin{align*} \mathcal M'(t) &=\int_{\mathcal C_t}\left[ R_t(1-U^{-1})+H_EU^{-2}\partial_tU\right]\,\mathop{}\!\mathrm dA\\ &=-\frac12\int_{\mathcal C_t}R_t(U+U^{-1}-2)\,\mathop{}\!\mathrm dA\le0. \tag{182}\end{align*}\] Here the integral of \(\Delta_{\gamma_t}U\) vanishes. The asymptotic bounds show that \(\mathcal M\) is bounded and that its derivative is integrable at infinity, so it has a finite limit.

We identify the limit with the ADM mass in the normalization (12). For the purely normal Euclidean metric error \(e'=(U^2-1)dt^2\), the ADM flux through the parallel surface is exactly \[\int_{\mathcal C_t} (\mathop{\mathrm{div}}_\delta e'-d\mathop{\mathrm{tr}}_\delta e')(\partial_t)\,\mathop{}\!\mathrm dA =\int_{\mathcal C_t}H_E(U^2-1)\,\mathop{}\!\mathrm dA.\] The normal derivatives of \(U^2-1\) cancel, and \(\nabla_{\partial_t}\partial_t=0\) in the Euclidean metric, leaving only the tangential divergence \(H_E\). The scalar-curvature linearization and \(R_{g_o}=0\) show that the divergence of this Euclidean flux is \(O(|x'|^{-4})\). Its integral between a large parallel and a comparable large round sphere tends to zero; thus it computes the ordinary ADM mass. Finally \[(U^2-1)-2(1-U^{-1})=O((U-1)^2)\] and \(H_E\,\mathop{\mathrm{Area}}(\mathcal C_t)=O(t)\), so \[ 16\pi m_o =\lim_{t\to\infty}\int_{\mathcal C_t}H_E(U^2-1)\,\mathop{}\!\mathrm dA =2\lim_{t\to\infty}\mathcal M(t) \le2\mathcal M(0). \tag{183}\] As \(H_E/U(0)=H_o\), this proves (172). Combining it with (176) gives (164).

Lastly, convexity gives \(\gamma_t(V,V)=\gamma_0((\mathop{\mathrm{Id}}+t\mathrm{II}_E^{\sharp})V, (\mathop{\mathrm{Id}}+t\mathrm{II}_E^{\sharp})V)\ge\gamma_0(V,V)\). The metric in (180) has no mixed normal–tangential terms. Its projection along the parallels to \((\mathcal C,\gamma_0)\) therefore has operator norm at most one, and hence upper two-dimensional Jacobian at most one. This is the area comparison required in the later transfer argument. ◻

Proof. Away from \(\mathcal C\), the lapse equation is a smooth metric Laplace equation with bounded divergence forcing and bounded scalar forcing. Its local \(W^{1,p}\) bounds hold for some \(p>3\), also up to \(S\) with its natural current flux. The prescribed constant flux at \(S\) can be put in the current by a smooth bounded extension before reflection. Differentiating the divergence height equation gives scalar uniformly elliptic equations for the coordinate derivatives of \(f\). Their divergence forcing contains first derivatives of \(N\), bounded in \(L^p\), and the bounded scalar height forcing, put into the corresponding component of the differentiated current. In particular differentiating that scalar forcing in the weak equation does not require a bound for \(\mathop{}\!\mathrm dT\). The inhomogeneous scalar divergence estimate gives Hölder bounds for \(\mathop{}\!\mathrm df\). At \(S\) subtract the constant height and reflect it oddly, the lapse evenly, and the metric with its tensor parities in an orthogonal collar. The reflected metric is Lipschitz. The height flux satisfies the weak reflected equation, since its normal component has the required even parity. Difference quotients justify differentiation there. These arguments prove the assertion away from the seam.

At the seam we first bound energy weighted by inverse distance. A modified Newton current then gives an estimate relative to compatible constant states. The weighted bound supplies a uniformly positive scale of small excess; a blowup to an explicit linear transmission problem gives excess decay and hence Hölder continuity.

Weighted energy. Set \[ E=\begin{cases}|\nabla N_i|^2+|\nabla^2 f|^2,&t<0,\\ |\nabla N_o|^2,&t>0, \end{cases} \qquad G=-\log s,\qquad Y=s^{-k_0}, \tag{211}\] where \(k_0\) is sufficiently large for Lemma 13. On the inside its divergence matrix is \(\rho\mathcal P\), where \(\rho=(1+y)/d\), and on the outside it is the base cometric. After including the volume densities, these give a bounded symmetric uniformly elliptic matrix \(a\) in the joined chart. \(Y\) is continuous across the face because \(N_o=s_i\) there. The energy identity and the boundary identity give \[ \partial_j(a^{jk}\partial_kY) \geq cE-C-C\delta_{\mathcal C}. \tag{212}\] This is a distributional inequality. Indeed \(\rho q=d\) on a Dirichlet-height face, and the inner normal \(Y\)-flux is \(-k_0YB_n/s+O(1)\), whereas the outer one is \(-k_0YB_n/s\). The bounded discrepancy is precisely the surface term in (212).

Fix concentric chart balls strictly contained in a larger joined chart. Let \(\theta_\varepsilon\) be the zero Dirichlet Green potential, for \(-\partial(a\partial)\) on the larger ball, of a nonnegative normalized bump at a point \(z_0\) in the smaller ball. If \(Y_H\) is the homogeneous replacement of \(Y\) with its boundary values, then \[ \big\langle\partial(a\partial Y),\theta_\varepsilon\big\rangle =\int(Y_H-Y)\,\chi_\varepsilon\leq C. \tag{213}\] The maximum principle bounds \(Y_H\), and \(Y\) is bounded by (209). For each classical solution the identity follows first with regular potentials and then by Sobolev approximation. The surface term is well defined by the trace theorem. The divergence Green bounds for symmetric measurable uniformly elliptic matrices in dimension three give an upper bound \(C|z-z_0|^{-1}\) and a comparable lower bound on a sufficiently small interior ball (Littman et al. 1963, Theorem 7.1 and pp. 66–67). The surface integral of \(|z-z_0|^{-1}\) is uniformly finite. Letting the bump shrink, using (212) and Fatou’s lemma, proves \[ \int_{B_{h_0}(z_0)}\frac{E(z)}{|z-z_0|}\,\mathop{}\!\mathrm dz\leq C. \tag{214}\] Both \(h_0>0\) and \(C\) are uniform on the finite collection of seam charts.

Comparison with a constant state. We next prove an energy bound relative to a compatible constant state. Such a state is \[ \mathcal V_*=(n_*,a_*\mathop{}\!\mathrm dt,s_*),\qquad s_*=s(n_*,|a_*|),\qquad \mathcal V=(N_i,\mathop{}\!\mathrm df,N_o). \tag{215}\] The pairs \((n_*,a_*)\) range over a compact rectangle containing the state ranges with a margin, with \(n_*>0\). Since \(|\mathop{}\!\mathrm dt|_g=1\), \(s_*\) is constant in this Fermi chart. Norms of the state mean the sum of the inner pair’s squared norm on the inner half and the outer lapse’s squared norm on the outer half.

To derive the bound, put \(\mathcal T_f=\nabla^2f-(\Delta f)g\). The Newton-current identity in Lemma 14 says \[ \rho\mathcal P\nabla G =-\frac{\nabla N_i}{s} +\mathcal T_f\left(\frac{y\nabla f}{d|\mathop{}\!\mathrm df|^2}\right)+O(1). \tag{216}\] The coefficient vector extends smoothly through \(\mathop{}\!\mathrm df=0\). In the current \(a\mathop{}\!\mathrm dY\), add \(k_0Y_*B/s_*\) on both sides, where \(Y_*=s_*^{-k_0}\), and on the inside subtract the Newton term in (216) with its full coefficient \(k_0Y,y\nabla f/(d|\mathop{}\!\mathrm df|^2)\) frozen at the reference. Include metric densities in all three currents. Call the resulting joined current \(\mathcal J_*\). The cancellation of the coefficients of \(\nabla N\) and \(\nabla^2f\) at the reference gives \[ |\mathcal J_*|\leq C\bigl(1+|\mathcal V-\mathcal V_*|\sqrt E\bigr). \tag{217}\] Moreover \(\mathop{\mathrm{div}}\mathcal T_f=\mathop{\mathrm{Ric}}(\nabla f,\cdot)\), so the divergence of the subtracted term is bounded by \(C(1+\sqrt E)\); its frozen vector is extended smoothly using \(\partial_t\). The added \(B\)-current has bounded divergence and no seam jump. The frozen Newton vector is normal at the seam. Its normal contraction there involves \(-\mathop{\mathrm{tr}}_{\mathcal C}\nabla^2f\), which is bounded since \(f\) is constant along that face and \(|\mathop{}\!\mathrm df|\) is bounded. Young’s inequality therefore gives, after lowering \(c\), \[ \mathop{\mathrm{div}}\mathcal J_*\geq cE-C-C\delta_{\mathcal C}. \tag{218}\] Test by \(\chi^2\), with \(\chi=1\) on \(B_h\), supported in \(B_{2h}\) and \(|\mathop{}\!\mathrm d\chi|\leq C/h\). Absorb half the energy using (217). The volume, constant current, and surface contributions are at most \(Ch^3\), \(Ch^2\), and \(Ch^2\), respectively. We obtain \[ h^{-1}\int_{B_h}E\leq C\left(\frac{1}{|B_{2h}|}\int_{B_{2h}} |\mathcal V-\mathcal V_*|^2+h\right). \tag{219}\] On a pure inner ball the identical construction uses any constant covector reference. The coefficients frozen against this reference may vary smoothly with the background metric; their derivatives give only \(C(1+\sqrt E)\) errors. On a pure outer ball the analogous assertion is the usual lapse Caccioppoli inequality.

A scale of small excess. Fix \(0<b_0<1\) and define the seam excess \[ \mathcal E(z_0,h)= \inf_{\mathcal V_*}\frac{1}{|B_h|}\int_{B_h(z_0)} |\mathcal V-\mathcal V_*|^2+h^{b_0}. \tag{220}\] There is a uniformly positive scale at which this excess is arbitrarily small. To see the quantitative content, put \(h_j=2^{-j}h_0\). For each \(z\ne z_0\) the sum of \(h_j^{-1}\) over the indices with \(|z-z_0|<h_j\) is at most \(C|z-z_0|^{-1}\). Thus (214) bounds \(\sum_j h_j^{-1}\int_{B_{h_j}}E\) uniformly. Given \(\eta>0\), first make the starting ceiling small enough that \(h_0^{b_0}+h_0^2<\eta\). Among a fixed finite number of its dyadic descendants there is a scale with \(h^{-1}\int_{B_h}E<\eta\). The number depends only on the Green bound and \(\eta\), so the chosen scale has a uniform positive lower bound.

Here is why this is also a small excess scale. Scaled Poincaré and the half-ball trace inequality bound both the volume deviations and the mean-square face deviations from each side mean by \[ C h^{-1}\int_{B_h}E+Ch^2. \tag{221}\] The additional \(Ch^2\) allows for Christoffel terms when replacing covariant Hessians by coordinate derivatives of \(\mathop{}\!\mathrm df\). The tangential trace of \(\mathop{}\!\mathrm df\) is zero; hence the tangential components of its mean are small. The trace relation \(N_o=s(N_i,\partial_t f)\) and the uniform Lipschitz bound for \(s\) on the state range show that the outer mean differs from the value obtained from the two inner means by at most the square root of (221). This provides a compatible reference with excess \(C\eta\). All references obtained this way lie in a compact subrange with room for the subsequent small updates.

Excess decay. We claim that, for fixed sufficiently small \(\vartheta>0\), whenever \(h\) and \(\mathcal E(z_0,h)\) are sufficiently small, \[ \mathcal E(z_0,\vartheta h) \leq 2\bigl(C\vartheta^2+\vartheta^{b_0}\bigr) \mathcal E(z_0,h). \tag{222}\] The constant \(C\) is independent of \(\vartheta\). We give the compactness argument, including the value trace.

If the claim fails, choose violating balls with \(\sigma_j^2=\mathcal E(z_j,h_j)\to0\) and best references \((n_j,a_j\mathop{}\!\mathrm dt,s_j)\). Translate and dilate these balls to unit size using \(z=z_j+h_j\zeta\), and rotate tangential coordinates to be orthonormal at their centers. Define the lapse differences divided by \(\sigma_j\), and the height difference by \[ u_{i,j}=\frac{N_i-n_j}{\sigma_j},\qquad u_{o,j}=\frac{N_o-s_j}{\sigma_j},\qquad \phi_j(\zeta)= \frac{f(z_j+h_j\zeta)-a_jh_j\zeta_3}{h_j\sigma_j}. \tag{223}\] The trace of \(\phi_j\) is zero. Their state has bounded \(L^2\) norm on the unit ball. Applying (219) on smaller balls gives locally bounded \(H^1\) norms for \(u_{i,j},u_{o,j}\) and \(\nabla\phi_j\). Indeed its error after normalization is \(h_j/\sigma_j^2\leq h_j^{1-b_0}\to0\). The \(L^2\) norm of \(\phi_j\) is controlled by its gradient and its zero trace. Rellich compactness and compactness of the \(H^1\) trace into \(L^2\) on smaller face patches give strong local \(L^2\) convergence of the state in the bulk and on the face, and weak \(H^1\) convergence.

Pass to a convergent reference \((n_*,a_*,s_*)\). The derivative of the height flux with respect to the gradient is \(s_*^2\mathcal P_*\); its lapse derivative is \(s_*^2(\beta/n_*)a_*\partial_t\). Taylor remainders divided by \(\sigma_j\) have \(L^1\) norm tending to zero, since the unnormalized state difference is \(\sigma_j\) times an \(L^2\)-bounded function. The same assertion holds on the face by the trace bounds. Smooth metric errors are \(O(h_j/\sigma_j)\), which tends to zero. The height source and lapse drift enter weak tests with this same factor; the scalar lapse source has the smaller factor \(O(h_j^2/\sigma_j)\). Thus no derivative of a source is being passed to the limit. Joined tests have a common value on the face, so lapse current matching passes to equality of the limiting normal derivatives. We arrive at \[\begin{align*} \Delta u_i&=\Delta u_o=0, & (q\partial_t^2+\Delta_{z'})\phi+k\partial_tu_i&=0, & k&=\beta a_*/n_*,\tag{224}\\ \phi&=0, &\partial_tu_i&=\partial_tu_o, &u_o&=r_0u_i+r_0c\partial_t\phi \quad(t=0), \tag{225}\end{align*}\] where \[ r_0=dI>0,\qquad c=-\frac{n_*y}{a_*},\qquad ck=q-1,\qquad r_0c=-\frac{s_*J}{a_*}. \tag{226}\] All expressions have their continuous interpretations at \(a_*=0\); in particular \(c=k=0\) there. Differentiating \(s^2+|\mathop{}\!\mathrm df|^2s^4=N_i^2\) gives \(\dot s=dI\dot N_i-sJ\partial_t\dot f/a_*\), which verifies the value relation directly.

The elementary harmonic estimate proved just below gives uniform smooth bounds for these limits on smaller closed half-balls in terms of their \(L^2\) state norms. Consequently their deviations from the center values have mean square at most \(C\vartheta^2\). The center values satisfy the linearized compatibility relation. Replace the old reference by \[ n_j'=n_j+\sigma_j u_i(0),\qquad a_j'=a_j+\sigma_j\partial_t\phi(0),\qquad s_j'=s(n_j',|a_j'|). \tag{227}\] The nonlinear compatibility error in the outer value is \(O(\sigma_j^2)\), and the tangential height gradient is zero at the center. Strong local \(L^2\) convergence therefore proves (222), contradicting its violation. The added radius term contributes at most \(\vartheta^{b_0}\sigma_j^2\), as required.

For completeness, the harmonic estimate used here is as follows. Reflect \(u_o\) into \(t<0\) and put \(U=u_i+u_o^{\rm refl}\). It is harmonic with zero Neumann data, so it is smooth with controlled local norms by even reflection. On a smaller inner cylinder choose a harmonic primitive \(D\) with \(D_t=u_i\). An explicit choice is \[ D(z',t)=\int_{t_0}^{t}u_i(z',v)\,\mathop{}\!\mathrm dv+a(z'), \qquad \Delta_{z'}a=-\partial_tu_i(z',t_0), \tag{228}\] where \(t_0<0\) is a fixed interior slice. The slice data are controlled by interior harmonic estimates. This formula gives \(D\in H^2\) locally up to the face: the mixed and double normal derivatives come from \(u_i\in H^1\), and tangential Laplace estimates applied to \(\Delta_{z'}D=-\partial_tu_i\) give its remaining second derivatives. The function \(V=D+c\phi\) satisfies \((q\partial_t^2+\Delta_{z'})V=0\), since \(ck=q-1\), and \(V=D\) on the face. Set \(\widetilde V(z',t)=V(z',\sqrt q\,t)\). It is harmonic and, with \(a_0=r_0/\sqrt q>0\), \[ (\widetilde V-D)|_{t=0}=0, \qquad \partial_t(D+a_0\widetilde V)|_{t=0}=U. \tag{229}\] Scalar harmonic Dirichlet and Neumann estimates control both combinations. Their coefficient matrix is invertible because \(1+a_0>0\); in fact \(a_0=1/\sqrt{1+y}\). They control \(D\), then \(u_i=D_t\) and \(u_o=U-u_i^{\rm refl}\). Finally the scalar Dirichlet equation in (224) controls \(\phi\). This last step avoids division by \(c\), so all estimates are uniform through zero slope. The normal rescaling only reduces the patch by a uniform factor on the state range.

Choose \(\vartheta\) so small, and then \(\gamma>0\) so small, that \(2(C\vartheta^2+\vartheta^{b_0})\leq\vartheta^{2\gamma}\). Begin iteration at the small-excess scales with uniform positive lower bound constructed above. The references stay in the permitted range because their successive changes are bounded by a convergent geometric sum. This yields a mean-square oscillation bound \(Ch^{2\gamma}\) at every seam center. On a pure ball near the seam, the first scale comparable to its distance from the seam is controlled by a seam ball. The corresponding interior excess argument, with an arbitrary constant gradient reference, then gives the bound at all smaller scales. Its limit equations are a harmonic lapse and a constant uniformly elliptic height equation with the known lapse coupling; scalar interior estimates give the same decay. Campanato’s characterization on each closed half-chart proves (210). Finitely many charts and the earlier ordinary boundary estimates complete the proof. ◻

Proof. It suffices first to work on a flat seam patch. Subtract scalar particular solutions of the two lapse Poisson equations, with \(C^{2,\alpha}\) control, and a \(C^{3,\alpha}\) particular solution of the height row, including its \(C^{3,\alpha}\) boundary trace. The height forcing includes the first derivative of the particular inner lapse, hence belongs to \(C^{1,\alpha}\) as required. Such particular solutions can be made on smooth domains having flat face portions containing the smaller patch, after extending localized data; ordinary scalar Dirichlet estimates give their stated norms. We are left with harmonic lapses and the homogeneous height row, zero height trace, and residuals \[ \partial_tu_i-\partial_tu_o=h_1, \qquad u_o-r_0u_i-r_0c\partial_t\phi=h_2, \tag{231}\] where \(h_1\in C^{1,\alpha}\) and \(h_2\in C^{2,\alpha}\). The reflected sum \(U=u_i+u_o^{\rm refl}\) is harmonic with Neumann data \(h_1\), and has a controlled \(C^{2,\alpha}\) norm on a smaller patch. Construct \(D\) by (228) and use its same local supremum bound. The two harmonic combinations in (229) now have Dirichlet datum zero and Neumann datum \(U-h_2\), respectively. The latter belongs to \(C^{2,\alpha}\), so both combinations, and therefore \(D\), are controlled in \(C^{3,\alpha}\). This controls both lapses in \(C^{2,\alpha}\). Scalar Dirichlet estimates for the height row finish the \(C^{3,\alpha}\) estimate for \(\phi\).

For the conformal-factor block, the frozen inner matrix in orthonormal face coordinates is a positive multiple of \(\operatorname{diag}(1,1,d^2)\), and the outer matrix is isotropic. Subtract particular solutions, rescale the two normal coordinates, and reflect the outer field. The resulting fields are harmonic; their difference has the prescribed value datum and a positive weighted sum has the prescribed Neumann datum. This gives their \(C^{2,\alpha}\) bounds. More generally, if the value relation is \(v_o=r v_i+h\) with \(r>0\), and the flux relation after reflection is \(a\partial_tv_i+b\partial_tv_o^{\rm refl}=k\), then the harmonic combinations \(v_o^{\rm refl}-rv_i\) and \(av_i+bv_o^{\rm refl}\) have Dirichlet and Neumann data. Their coefficient determinant is \(a+br>0\). It stays positive if \(b\) decreases to zero with \(a>0\). Variable fixed \(C^{2,\alpha}\) ratios are treated by freezing, as in the next paragraph. Localization differentiates the value row at most twice and the flux row at most once; the constants depend only on the corresponding coefficient norms. The ordinary boundary and interior estimates follow from the scalar estimates, with the lapse estimated before the height row.

For fixed regular variable coefficients, localization gives the same estimate with lower-order terms. In a sufficiently small chart the leading coefficients differ little in supremum norm from their frozen values. Apply the constant-coefficient estimate to cutoff fields on that chart. In a product such as \((a-a_*)D^2v\), the coefficient supremum multiplies the top Hölder seminorm and can be absorbed. Its fixed Hölder seminorm multiplies the second-derivative supremum, which is bounded by an arbitrarily small multiple of the full high norm plus a fixed multiple of the field supremum, by interpolation. The same argument applies to the order-\(1\) height coupling and to the differentiated value relation. Cutoff commutators have strictly lower orders. A finite cover and interpolation for separated-point Hölder quotients give the claimed local or global variable-coefficient estimate. This argument requires only fixed coefficient bounds and is uniform along compact families of such coefficients. ◻

Proof. Discard compact lower-order terms in the linearization. This is legitimate with the column orders used above. For example, a variation of the trace target in the expanded lapse row contains at most two derivatives of the height variation or one of the other variations, and therefore maps compactly to its \(C^{0,\alpha'}\) residual. Variations of \(F/d_S\) lose one height derivative by (233), which still leaves this compactness. In contrast, the first differential of \(N_o-s(N_i,\mathop{}\!\mathrm df)\) in the \(C^{2,\alpha'}\) seam residual is principal and is retained. The principal height coupling to a lapse derivative in its \(C^{1,\alpha'}\) residual is also retained.

In this principal operator scale the background \(\mathop{}\!\mathrm df\) continuously to zero and recompute \(q,\mathcal P,\beta,s,A\), the height coupling, and the value-differential coefficients from that scaled background. This scales coefficient fields, not the variations. In particular \(ck=q-1\) in (226) holds all along the path. The states remain in a fixed positive ellipticity range. The local estimates of Lemma 36, followed by localization, give along this path \[ \|v\|_X\leq C\bigl(\|L_\lambda v\|_{\mathscr Y} +\|v\|_0\bigr), \tag{240}\] where \(\|v\|_0\) is a sum of field supremum norms and \(C\) is uniform in the path parameter \(\lambda\).

At zero background gradient the height decouples, and the lapse and \(w\) blocks have scalar value and positive conormal transmission. Decrease each outer derivative coefficient in its flux relation to zero, leaving the inner coefficient positive. The determinant \(a+br\) in the proof of Lemma 36 stays positive, so (240) remains valid. At the endpoint each scalar block is triangular: solve the inner Neumann problem, then the outer Dirichlet problem with the trace prescribed from the inner solution. On each connected inner component the scalar Neumann problem has equal kernel and compatibility-cokernel dimensions, both one; the outer Dirichlet and height Dirichlet problems have index zero. Dividing by positive scalar principal factors and discarding connection terms reduces this assertion to ordinary Poisson theory. Thus the endpoint index is zero.

We justify index propagation using only (240). A sufficiently fine fixed finite grid of field evaluations defines \(\mathfrak p:X\to\mathbb R^l\) with \(\|v\|_0\leq\eta\|v\|_X+C_\eta|\mathfrak p v|\). Choose \(\eta\) small in (240). The augmented operators \[ \mathfrak A_\lambda=(L_\lambda,\mathfrak p): X\longrightarrow\mathscr Y\times\mathbb R^l \quad\hbox{satisfy}\quad \|v\|_X\leq C\|\mathfrak A_\lambda v\|. \tag{241}\] They are injective with closed range. At an index-zero Fredholm parameter their range has codimension \(l\). If parameters \(\lambda_j\) of this kind tend to \(\lambda\), and the limit range had codimension at least \(l+1\), choose an \((l+1)\)-dimensional subspace \(Z_0\) disjoint from that range. At each \(\lambda_j\) there is a unit vector \(z_j\in Z_0\cap\operatorname{ran} \mathfrak A_{\lambda_j}\). Its preimages \(v_j\) are bounded by (241). A subsequence of \(z_j\) converges in the finite-dimensional space to a unit vector \(z\), and operator-norm continuity gives \(\mathfrak A_\lambda v_j\to z\). Closedness of the limit range is a contradiction. Its codimension is therefore finite.

The kernel of \(L_\lambda\) is finite dimensional because \(\mathfrak p\) is injective on it. Its range is closed and has finite codimension: add the finite-dimensional factor \(\{0\}\times\mathbb R^l\) to the closed finite-codimensional augmented range, and then quotient by that factor. This sum is closed, and the resulting quotient is exactly \(\operatorname{ran}L_\lambda\). Thus \(L_\lambda\) is Fredholm also at the limit. Openness and local constancy of the Fredholm index, together with this closedness argument starting from the endpoint, propagate index zero along the whole path. Restoring compact lower-order terms does not change it.

For the independent scalar density problem take the fixed positive \(C^{2,\alpha'}\) ratio and fixed \(C^{1,\alpha'}\) drift as coefficients. Its local principal boundary estimate is the scalar positive-ratio estimate already proved, and its lower-order drift is compact at the scalar column order two. Localization uses only the stated coefficient norms, so no approximation in a Hölder norm is needed. The same outer-flux deformation and augmentation argument applies without reference to a nonlinear zero or to an a priori estimate for such zeros. For its adjoint the value ratio and positive derivative weight exchange their roles; the same scalar combinations apply. This proves both independent index assertions. ◻

Proof. We first control the normalized lapse without a slope bound. Height compactness and a sublevel argument then exclude positive limiting heights and a seam at bounded scaled distance. Only after this do we use the gradient estimate to obtain the flat graph limit. Finally a truncated Harnack argument controls the conformal factor with a constant independent of the fixed scaled radius.

It is enough to examine an arbitrary sequence \(M\to\infty\) and marked centers \(P\in\mathcal U_M\). Write \(r=r_P\). Use smooth base charts, with Fermi charts at a boundary, and rescale their lengths by \(r\); equivalently use the base metric \(g_r=r^{-2}g\). On each fixed scaled compact set the base metrics converge smoothly to the Euclidean metric, or to the Euclidean metric on a half-space. The two compact boundary faces \(S\) and \(\mathcal C\) have positive base separation, so at most one can remain at a bounded scaled distance.

Lapse bounds before slope bounds. We first obtain lapse estimates without using height-gradient bounds. Testing the height equation on nested base balls with a cutoff and a height function bounded in absolute value by \(M\) gives, in scaled volume units, \[ \int_{B_H}(W^2-1)\,\mathop{}\!\mathrm dV_{g_r} \le C_H\frac{M}{r} \int_{B_{H'}}N_i\,\mathop{}\!\mathrm dV_{g_r},\qquad H<H'. \tag{255}\] For a ball reaching a Dirichlet face, subtract its constant boundary height before testing; the boundary term then vanishes. To verify the estimate, use \(p\cdot\mathop{}\!\mathrm df=W^2-1\), \(|p|\le N_i\), \(|T|\le C\), and a cutoff derivative of size \(C_H/r\). These are also the only estimates required if the ball meets a face.

Put \(U=N_i/N_0\), and let \(A_{H'}\) be a supremum of this normalized lapse on a larger fixed scaled patch. The scalar forcing in its rescaled equation obeys \[\begin{align*} \left\|\frac{r^2}{N_0}(\Sigma_++\Sigma_-)\right\|_{L^2(B_H,g_r)} &\le C_H\left(r^2+ M^{1/2}r^{3/2}N_0^{-1/2}A_{H'}^{1/2}\right) \tag{256}\\ &\le C_H\left(r^2+M^{-7/40}A_{H'}^{1/2}\right). \tag{257}\end{align*}\] Here \(\Sigma_+\le LW\), \(|\Sigma_-|\le C\delta_0\), \(N_0\ge\delta_0\), and (255) were used. The divergence drift has size \(O(r)\). The natural flux error at \(S\) has size \(O(rb_*/N_0)\); a bounded extension of the normal datum incorporates it in the divergence forcing.

If no seam is present on the patch, the inhomogeneous Harnack estimate for \(U\), including natural-flux reflection at \(S\), bounds the smaller supremum by a constant times its infimum plus the right side of (257). Choose the smaller patch to contain the center, or use a boundary-foot ball containing it, so that the infimum is at most \(U(P)=1\). Thus, on finitely many nested patches, \[ A_H\le C_H\left(1+r^2+M^{-7/40}A_{H'}^{1/2}\right). \tag{258}\] The crude bound \(N_i\le C(L)M\) gives \(A_{H'}=O(M)\). The exponent of \(M\) improves by \(a\mapsto\max(0,a/2-7/40)\). Two applications give \(1\mapsto13/40\mapsto0\). We have therefore obtained local boundedness by a finite iteration, before asserting that the forcing tends to zero.

There is a slightly different normalization argument if the seam has bounded scaled distance. Reflect the exterior lapse into the inner side in the two base Fermi collars, and set \(V=U+N_o^{\rm refl}/N_0\). Use the inner density-cometric matrix for the summed equation. The common induced metric implies that the two matrices differ by \(O(r|t|)\) at scaled distance \(|t|\) from the face. Positivity and the interior gradient estimate for the harmonic exterior lapse on balls of radius comparable to \(|t|\) give \[ |(a_i-a_o^{\rm refl})\nabla(N_o^{\rm refl}/N_0)| \le Cr(N_o^{\rm refl}/N_0). \tag{259}\] The exterior is harmonic on the fixed omitted collar, which contains every such scaled patch for large \(M\). The inner drift obeys the same bound with \(U\). Consequently the total correction is a bounded divergence drift of size \(Cr\) times \(V\), with zero natural flux at the face, and the scalar forcing is still bounded by (257) with a larger supremum of \(V\).

The infimum in Harnack for \(V\) must be anchored to the inner center value. Here is that step explicitly. Let \(m\) be the infimum of \(V\) on a fixed smaller half-patch containing the center and a central disk in the face. On the face, \(N_o=dN_i\), \(0<d\le1\), whence \(U\ge m/2\). Choose a fixed smooth half-domain with that flat disk in its boundary and containing the center. The harmonic measure of a smaller disk at the center is bounded below by a fixed positive number; this remains true as the center approaches that disk. Write \(U=H+e\), where \(H\) is the harmonic replacement with the same boundary values. The zero-Dirichlet estimate for \(e\), using divergence forcing in \(L^p\), \(3<p<6\), and scalar forcing in \(L^2\), gives \[ \|e\|_\infty\le C_H\left(rA_{H'}+r^2+M^{-7/40}A_{H'}^{1/2}\right). \tag{260}\] The remaining boundary values of \(H\) are nonnegative, and \(U(P)=1\). Harmonic measure therefore bounds \(m\) by a constant times one plus this error. Combining this with the Harnack estimate for the reflected sum yields \[ A_H\le C_H\left(1+rA_{H'}+r^2+ M^{-7/40}A_{H'}^{1/2}\right). \tag{261}\] Starting again from \(O(M)\), the exponent improves by \(a\mapsto\max(0,a-9/20,a/2-7/40)\). Three nested applications give \(1\mapsto11/20\mapsto1/10\mapsto0\). Thus the sum is locally bounded. This argument uses no derivative of the seam trace ratio and no separate trace compactness for either lapse.

The forcing in (257) now tends to zero. Local Laplace estimates give \(W^{1,p}\) bounds, \(p>3\), and Hölder compactness, including reflected natural boundary estimates when appropriate. Without a seam, every subsequential limit is a positive harmonic function on all of \(\mathbb R^3\), or on a half-space with zero Neumann data. Positive harmonic Liouville, after reflection in the latter case, and the center normalization show that the limit is one. At a seam the reflected sum has instead a positive constant limit. Its constant is positive because it is at least the inner center value. Its trace lower bound, the relation \(U\ge V/2\) on the face, and harmonic replacement with the error (260) tending to zero give a positive lower bound for the inner lapse up to the face. On compact open inner sets it converges, in a smaller Hölder exponent, to a smooth positive harmonic function. These conclusions hold on every fixed patch by diagonal extraction. In particular both upper and positive lower lapse bounds are now available in every case.

Height compactness. We turn to the height, setting \[ \mathcal F=\frac{N_0(M-f)}{r},\qquad \mathcal U=\frac{N_i}{N_0},\qquad \widetilde s=\frac{s}{N_0}. \tag{262}\] It is nonnegative and \(\mathcal F(P)\le M^{-1}\) by the closed cutoff inequality. Its flux in the metric \(g_r\) is \[ \mathfrak p=\widetilde s^2\nabla_{g_r}\mathcal F =-\frac r{N_0}p,\qquad \mathop{\mathrm{div}}_{g_r}\mathfrak p=-rT\mathcal U=o(1). \tag{263}\] The constitutive rule is unchanged: \(\widetilde s^2+|\mathfrak p|_{g_r}^2=\mathcal U^2\). If \(\mathcal U\) is frozen at its actual value, the flux is strictly monotone in its gradient argument \(\xi\), and the positive upper and lower lapse bounds give \[ \langle\mathfrak p(\xi),\xi\rangle \ge c\min(|\xi|^2,|\xi|),\qquad |\mathfrak p(\xi)|^2 \le C\langle\mathfrak p(\xi),\xi\rangle. \tag{264}\] For example, in an orthonormal frame \(\mathfrak p(\xi)=a\xi\), where \(a+a^2|\xi|^2=\mathcal U^2\); these assertions follow directly by differentiation and by considering \(|\xi|\le1\) and \(|\xi|\ge1\).

We require a compactness argument for possibly very steep graphs. In the product of a scaled base chart with a height line, the bounded vector field \((-\mathfrak p,1)\) has bounded divergence and its pairing with the upward unit graph normal is bounded below by a positive constant, by (264). Gauss–Green applied to the subgraph and supergraph in product balls gives, for either side’s volume function \(V(a)\), an upper bound \(C(V'(a)+V(a))\) for the graph perimeter in that ball. It also gives local perimeter bounds by using fixed surrounding balls. The four-dimensional isoperimetric inequality, at sufficiently small radii to absorb the volume term, yields \(V'(a)\ge cV(a)^{3/4}\). Thus at every interior graph point both sides have volume at least \(ca^4\). These constants are uniform on each of the fixed patches under consideration.

At \(S\), \(\mathcal F=0\). For product balls at heights bounded away above zero, extend the subgraph as the empty set across the face. There is no new face perimeter, because its trace is empty there. The same subgraph density estimate is therefore valid even when the ball reaches \(S\). BV compactness on open inner product patches and vertical nesting now give a subsequence such that all bounded height truncations converge in local measure to those of an extended measurable function \(\mathcal F_\infty\ge0\), whose value is allowed initially to be \(+\infty\).

We next prove that a nonzero limiting height forces a positive lower bound. On a ball \(B_H\) contained in the inner side, test (263) with \(\chi^2(k-\mathcal F)^+\), where \(\chi=1\) on \(B_h\). By (264) and Young’s inequality, \[ \int_{B_h}\min(|\mathop{}\!\mathrm d(k-\mathcal F)^+|^2, |\mathop{}\!\mathrm d(k-\mathcal F)^+|) \le C\left(\frac{k^2}{(H-h)^2}+\varepsilon_M k\right) |\{\mathcal F<k\}\cap B_H|, \tag{265}\] where \(\varepsilon_M\to0\) on the fixed patch. The same test is allowed on a half-patch with extension by zero across a Dirichlet face whose height exceeds \(k\). On the portion with derivative at most one, use Cauchy–Schwarz; on its complement use the linear part of the energy. The \(W^{1,1}\) Sobolev inequality for an intermediate cutoff then gives, for \(0<l<k\), \[ (k-l)|\{\mathcal F<l\}\cap B_h|^{2/3} \le C\left(\frac{k}{H-h}+ \frac{k^2}{(H-h)^2}+o(1)\right) |\{\mathcal F<k\}\cap B_H|. \tag{266}\] The error here includes \((\varepsilon_M k)^{1/2}\) and \(\varepsilon_M k\), which tend to zero for fixed levels.

For completeness, take radii decreasing from \(H\) to \(H/2\) and levels decreasing from \(k\) to \(k/2\) geometrically. First take \(M\) sufficiently large at the fixed \(H,k\) to absorb the forcing into the first cutoff term. This also absorbs it at every later step, since the level remains at least \(k/2\) and the cutoff derivatives only increase. After dividing volumes by \(H^3\), (266) has the form \[ v_{j+1}^{2/3}\le C^{j+1}(1+k/H)v_j. \tag{267}\] For \(0<k\le H\), a sufficiently small initial fraction, with a threshold independent of \(k/H\), therefore forces \(v_j\to0\). If \(\mathcal F_\infty\) is not zero almost everywhere, choose a density ball of one of its positive superlevel sets and then a small fixed \(k>0\). Convergence of truncations supplies that small initial fraction. Consequently \(\mathcal F\ge k/2\) on a smaller open inner ball for all sufficiently large indices.

This lower bound propagates to compact subsets of the connected limiting inner domain. To justify propagation for the capped flux, choose a fixed smooth connecting subdomain with a small nonnegative Dirichlet bump on a boundary portion inside the known positive ball and zero data elsewhere. On this subdomain the actual lapse fields converge in a Hölder exponent to smooth positive coefficients. The flux is smooth and uniformly elliptic near zero gradient. The Dirichlet inverse of its linearization at zero maps divergence \(C^\gamma\) data to \(C^{1,\gamma}\); contraction in a small \(C^{1,\gamma}\) ball therefore supplies small-gradient barriers, uniformly for the sequence, with divergence a positive constant tending to zero that dominates the right side of (263). Their limiting solutions have positive interior values by the strict maximum principle. The constructed boundary values are below those of \(\mathcal F\). Strict gradient-flux monotonicity, testing the positive part of the barrier minus \(\mathcal F\), proves comparison in the required direction. A finite chain of such domains and a finite covering prove propagation on every compact inner subset.

In the \(S\) alternative, the lapse convergence is also Hölder up to that face. The connecting subdomains may then reach a flat boundary patch, with zero data on that portion. The limiting barrier is smooth there and has strictly positive inward derivative by Hopf’s lemma. The \(C^1\) convergence of the barriers gives a uniform lower bound by a positive constant times inward scaled distance. This contradicts the closed unsafe condition \[ \frac{\mathcal F(P)}{d_S(P)/r}\le M^{-1}, \tag{268}\] with the normal derivative interpretation if \(P\in S\). An interior limiting center is already excluded by \(\mathcal F(P)\le M^{-1}\). Thus a nonzero limiting height is impossible in these two alternatives.

Excluding seam limits. If the center approaches the seam, the boundary height is instead \(N_0M/r\to\infty\). A positive limiting height propagates through the open half-space as above. On a fixed ball reaching the face, omit first a thin strip adjacent to the face. The lower bound on the remaining compact set, followed by a small choice of \(k\), makes the starting sublevel fraction arbitrarily small. Apply (267) with the negative part extended by zero across the seam. It supplies a positive lower bound up to the central face patch and contradicts center touchdown. This step does not require Hölder convergence of the individual lapse traces.

The zero almost everywhere alternative is also impossible at a seam. On a fixed half-patch choose a smooth nonnegative function \(b\) with amplitude \(k\), derivative \(O(k)\), positive by \(ck\) on a central face disk and zero near the other edges. The test \((b-\mathcal F)^+\) has zero boundary trace, since the seam height tends to infinity. Monotonicity and (264) give \[ \int_{\{\mathcal F<b\}}\min(|\mathop{}\!\mathrm d\mathcal F|^2, |\mathop{}\!\mathrm d\mathcal F|) \le Ck^2+o(1)k. \tag{269}\] The function \(v=\min(\mathcal F,b)\) has trace \(b\) on that disk and tends to zero in measure in the interior. On almost every line normal to a smaller central disk, the fundamental theorem of calculus, followed by averaging over interior slices of a fixed thin collar \(D_\varepsilon\), gives \(\int_{D_\varepsilon}|\mathop{}\!\mathrm dv|\ge ck-o(1)\). Here convergence in measure suffices because \(0\le v\le k\). Since \(\min(t^2,t)\ge t-1/4\) for \(t\ge0\), the left side of (269) is at least \[ ck-Ck|D_\varepsilon|-\tfrac14|D_\varepsilon|-o(1). \tag{270}\] Choose first \(k>0\) small enough that \(ck\) dominates \(Ck^2\), and then the collar thin enough. This contradicts (269). All possible seam limits have now been excluded. Consequently \[ \mathop{\mathrm{dist}}_g(P,\mathcal C)/r_P\longrightarrow\infty, \qquad \mathcal F_\infty=0 \quad\hbox{almost everywhere in the remaining limits}. \tag{271}\]

Uniform height and slope convergence. The height convergence is in fact uniform on every scaled compact set, including up to \(S\). Otherwise continuity and local convergence in measure produce graph points at some fixed positive height in a slightly larger patch. A fixed small product ball at such a point has uniformly positive subgraph volume by the density bound proved above. But that volume tends to zero because the limiting height is zero. If the points approach \(S\), use the empty extension across the face; alternatively omit a thin boundary strip before taking the limit. The strip has too little product volume to account for the density lower bound. This excludes positive spikes as well as bounded positive plateaus.

We can now apply the differential gradient estimate Lemma 12. Under the rescaling above the synthetic endomorphism and trace target become \(-rK\) and \(-rT\), and the lapse current becomes \((r^2/N_0)B\), so the same identity applies. Maximize \[ -\log\widetilde s+j(\mathcal F)+k_0\log\chi, \tag{272}\] where \(k_0\ge2\), \(\chi\) is a fixed local cutoff, and \(j',j''\) are positive, bounded, and bounded away from zero on the now bounded height range. If \(m=|\mathop{}\!\mathrm d\mathcal F|_{g_r}\), the constitutive relations give \[ qm^2=\frac{J}{\widetilde s^2} =\frac{JW^2}{\mathcal U^2},\qquad (1-y)^2(j'm)^2=\frac{(j')^2z^2}{\mathcal U^2}. \tag{273}\] At an interior maximum with \(y\ge1/2\), the positive leading term therefore controls \(W^2\). The source term has size \(r^2\Sigma_+/N_i\le CW\), and localization costs \(C\chi^{-2}\). Hence \[ cW^2\le C(1+W+\chi^{-2}). \tag{274}\] At \(S\), (36), \(B_n=-b_*\), \(k_T\hat z\ge0\), and \(H_g\le0\) give \(\partial_n(-\log\widetilde s)\ge0\) for the inward normal. Also \(\partial_n\mathcal F=m\ge0\). The derivative condition at a boundary maximum in (272) therefore gives \(j'm\le C/\chi\), which controls that case. When \(y<1/2\), \(W\le\sqrt2\) directly. The maximum test and \(k_0\ge2\) thus bound \(W\) on a smaller patch.

After this finite slope bound, differentiating the height divergence equation gives uniformly elliptic scalar divergence equations for its first derivatives. Their divergence forcing is controlled in \(L^p\), \(p>3\), by the normalized lapse estimates. Odd reflection of \(\mathcal F\), with even reflection of lapse and the Fermi metric, gives the same estimate at \(S\). The resulting gradient Hölder estimates and the uniform height convergence imply \[ \mathcal U\longrightarrow1,\qquad \mathop{}\!\mathrm d\mathcal F\longrightarrow0,\qquad \widetilde s\longrightarrow1 \tag{275}\] on every fixed scaled compact set.

The conformal factor. It remains to control the conformal factor uniformly in the chosen fixed radius. Put \(w'=w/w(P)\). On every fixed scaled patch, the height just proved gives \(f\ge9M/10\), hence \(w\ge T_1\) with \(T_1=M^{9/10}\). The lapse is eventually at least \(\delta_0/2\); choose the subsequent cutoff \(\varepsilon<\delta_0/8\). Also \(x=sw\to\infty\) there. The volume source and the low-lapse penalty are consequently absent. The positive function \(w'\) is a supersolution for its uniformly elliptic divergence principal part, while \(v'=(w'-a)^+\), \(a=4T_1/w(P)\), is a subsolution: above that truncation the remaining height penalty vanishes as well. Weak Harnack for \(w'\), combined with the local supremum bound for \(v'\) with the same positive mean exponent, gives on balls with fixed nesting ratios \[ \sup_{B_R}w' \le a+C\left(\frac1{|B_{2R}|}\int_{B_{2R}}(w')^p\right)^{1/p} \le a+C'\inf_{B_R}w'\le(4+C')\inf_{B_R}w'. \tag{276}\] The last step uses \(w'\ge T_1/w(P)=a/4\). Natural zero flux permits reflection at \(S\).

These constants can be chosen independently of the fixed radius. Indeed, on each arbitrarily large fixed patch the normalized principal matrices tend to the identity by (275). If the boundary is far enough, use concentric balls about the center with universal radius ratios. Otherwise use comparable larger balls about its boundary foot in a single flattened chart, and reflect. The smaller ball contains both the center and the requested region. Uniform ellipticity with fixed constants and dilation invariance of the homogeneous estimates give one common constant in (276). The index from which that constant applies may depend on the radius. Since \(w'(P)=1\), it follows that \(C^{-1}\le w'\le C\).

In the rescaled coordinates the metric is \[ h=\frac{x}{x(P)} \left(g_r+\widetilde s^2\mathop{}\!\mathrm d\mathcal F^2\right), \qquad \frac{x}{x(P)}=\frac{\widetilde s}{\widetilde s(P)}w'. \tag{277}\] Here, as usual, the expression denotes its pullback by the coordinate rescaling. Equations (275) and (276) prove (253). At \(S\) the mixed metric entries vanish, so reflection preserves this comparison. If the uniform statement of the lemma failed, a sequence violating it would admit precisely the subsequences just analyzed, a contradiction. ◻

Proof. By Lemma 39 the closed unsafe set is disjoint from the seam for large \(M\). Glue the two metrics using their own Gaussian normal collars. In the resulting smooth manifold structure the joined metric is continuous and has the form \(\mathop{}\!\mathrm dt^2+\lambda(t)\), piecewise smooth, with the inner side at \(t<0\). The common induced metric and the mean-curvature inequality in (206) give \[ \frac12\mathop{\mathrm{tr}}_{\lambda(0)} \big(\lambda'(0^+)-\lambda'(0^-)\big)\le0. \tag{285}\] This change of collar identification retains a smooth identification of each original closed piece with its image; comparisons with the base metric will always use those identifications.

Mollify \(\lambda\) in \(t\) with a nonnegative smooth even kernel of radius \(a\), and interpolate back to the original smooth metrics in \(2a\le|t|\le3a\). The resulting \(h_a\) converges uniformly to \(h\); its first normal derivatives and fixed tangential derivatives are bounded. Its second normal derivative is the jump of \(\lambda'\) times the mollifier plus a uniformly bounded remainder. The interpolation has the same bound, since on those smooth one-sided regions the mollification error is \(O(a^2)\) before taking the two cutoff derivatives. In the hypersurface scalar-curvature formula the only potentially unbounded term is \(-\mathop{\mathrm{tr}}_\lambda\lambda''\). Replacing the inverse metric in its jump term by \(\lambda(0)^{-1}\) costs \(O(a)O(a^{-1})=O(1)\); the leading term has favorable sign by (285). Thus \(\mathop{\mathrm{Scal}}_{h_a}\ge-C(h)\) in the smoothing collar, uniformly as \(a\downarrow0\). Choose a smooth \(c_a\ge0\), bounded by a constant depending on the fixed metric, supported in \(|t|<4a\), and majorizing the negative scalar part created in that collar.

Each orientable component of \(S\) bounds a compact handlebody. Attach such caps, extend the metric smoothly through \(S\), and denote their union by \(O\). This gives a smooth complete one-ended AF manifold on which to solve the correction. Its homogeneous Sobolev inequality \[ \|v\|_{L^6}\le C\|\nabla v\|_{L^2}, \qquad v\in C_c^\infty(X), \tag{286}\] has a constant uniform for small \(a\). Indeed all these metrics are uniformly equivalent to one fixed smooth AF metric. The Euclidean Sobolev inequality on the end and compact local Sobolev inequalities leave only a compact \(L^2\) term. Integration along radial rays, followed by Cauchy–Schwarz with the integrable weight \(r^{-2}\), controls the trace at a fixed end sphere by the end gradient energy. Poincaré’s inequality with this trace controls the remaining compact term, proving (286).

Solve in the completion for the gradient norm \[ -8\Delta_{h_a}u_a=c_a(1+u_a). \tag{287}\] Since \(\|c_a\|_{3/2}\to0\), the bilinear form \(8\int|\nabla v|^2-\int c_av^2\) is coercive by (286). The right side is a bounded functional with norm tending to zero, because \(\|c_a\|_{6/5}\to0\). Lax–Milgram gives a solution with \(\|u_a\|_6+\|\nabla u_a\|_2=o(1)\). Testing with its negative part and using coercivity proves \(u_a\ge0\). Local boundedness for scalar divergence equations, treating the bounded \(c_a u_a\) term as a zeroth-order coefficient and the \(L^2\)-small \(c_a\) as forcing, gives \(\|u_a\|_\infty=o(1)\) on a fixed compact set. On the unchanged end, \(u_a\) is harmonic and tends to zero: this follows first from its \(L^6\) membership and scaled local estimates. Exterior comparison then gives the global supremum bound. Elliptic estimates imply smooth convergence to zero on compact sets off the seam. The same end barriers as before give \(u_a=O_2(|x'|^{-\beta})\).

Set \(h'=(1+u_a)^4h_a\) and restrict to the original side of the caps. The conformal scalar formula is \[ \mathop{\mathrm{Scal}}_{h'}=(1+u_a)^{-5} \big(-8\Delta_{h_a}(1+u_a)+\mathop{\mathrm{Scal}}_{h_a}(1+u_a)\big) =(1+u_a)^{-4}(\mathop{\mathrm{Scal}}_{h_a}+c_a). \tag{288}\] It is nonnegative off the unsafe set. On that set the metric was not mollified, \(c_a=0\), and \(\mathop{\mathrm{Scal}}_h\ge-1\); since \(u_a\ge0\), the lower bound remains at least \(-1\). In particular the claimed fixed lower bound \(-2\) holds. The end scalar decay is preserved by the conformal formula.

The smoothing itself changes no asymptotic metric. The conformal ADM flux, followed by integration of (287) on the capped manifold, gives \[ m_{h'}-m_h =-\frac1{2\pi}\lim_{R\to\infty} \int_{|x'|=R}\partial_nu_a\,\mathop{}\!\mathrm dA_{h_a} =\frac1{16\pi}\int_X c_a(1+u_a)\,\mathop{}\!\mathrm dV_{h_a}=o(1). \tag{289}\] The decay \(\beta>1/2\) justifies replacing conormal-area products in this flux as well. Uniform metric convergence gives the quadratic-form comparison. Finally \(H_h(S)<0\) by (206); the smooth convergence of the correction near this disjoint compact surface preserves that strict inequality. Each smallness requirement here is imposed after the fixed solution has been selected. ◻

Proof. Extend the smooth inner base geometry across \(S\) in a fixed collar for the purpose of defining balls. Mark every closed base ball \[ \overline B_{g_{\rm ext}}(P,R_*r_P),\qquad P\in\mathcal U_M, \tag{290}\] and include their union together with \(O\) as an obstacle. For fixed parameters the union is compact: the center set is compact and its positive radius function is continuous. By Lemma 39, for each fixed \(R_*\) these balls and the fixed larger multiples used below lie off the seam once \(M\) is sufficiently large. All lie inside the comparator \(\mathcal S_1\).

Minimize \(h'\)-perimeter among precompact finite-perimeter sets containing this obstacle modulo null sets. A minimizer exists by BV compactness and lower semicontinuity inside a sufficiently large smooth truncation. This is also a global minimizer in the stated class. To see that the outer truncation causes no restriction, all sufficiently far AF spheres have strictly positive outward mean curvature. The unit normal field of their foliation has positive divergence. For almost every cutting radius, Gauss–Green on the removed part of a competitor shows that cutting inward saves perimeter if the removed volume is positive. The same argument excludes contact with the outer constraint. The fixed comparator and the small metric change give a uniform bound, after enlarging \(C_*\) once, \[ \mathop{\mathrm{Per}}_{h'}(E)\le C_*. \tag{291}\]

We show that its perimeter support cannot meet any marked ball. Suppose it meets the ball marked at \(P\), at a point \(q\). Use scaled smooth base coordinates with length unit \(r_P\) on a fixed multiple of radius \(R_*\). The base metric and the comparison metric are equivalent to Euclidean metrics with constants independent of \(R_*\), by Lemma 39 and the chosen small smoothing. If this region reaches \(S\), reflect in a base collar for these comparisons. Mixed entries of the graph metric vanish on the face. No comparison for the arbitrary cap metric is needed: perimeters of sets containing \(O\) are independent of its interior extension, and a reflected continuous metric can be used there in the comparison argument.

The complement of \(E\) has uniform lower volume density at \(q\) in these scaled coordinates. Indeed filling a coordinate ball preserves the obstacle condition. Minimality and slicing therefore bound its interior perimeter by a constant times the complement’s trace area on the ball. If \(V(a)=|B_a(q)\setminus E|\), Euclidean isoperimetry gives \[ V(a)^{2/3}\le C V'(a) \quad\hbox{for almost every }a,\qquad V(a)\ge ca^3. \tag{292}\] The second inequality follows by integrating the first from zero; the volume is positive at every radius because \(q\) is in the perimeter support. Apply this on a ball of radius comparable to \(R_*\). On a slightly larger common coordinate ball, \(E\) itself has volume at least \(cR_*^3\), since it contains the whole marked base ball modulo null sets. The uniform comparison of the scaled base metric with Euclidean metric justifies this statement even when the marked ball crosses \(S\). Relative isoperimetry on that common ball now gives \[ \mathop{\mathrm{Per}}_{h'}(E)\ge cR_*^2. \tag{293}\] Choose \(R_*\) so that this exceeds \(C_*\), and only then choose \(M\) large enough for all the scaled charts and their fixed multiples. This contradicts (291). Compactness of the center set also covers contacts with the closure of the union of marked balls. This is where the radius-independent constant in (254) is essential.

There is no contact with \(S\) either. The parallel outward normal field in a sufficiently thin collar of \(O\) has strictly negative divergence, since \(H_{h'}(S)<0\). Adjoin that collar to a purported minimizer. Gauss–Green on the added portion bounds the newly exposed leaf area minus the removed perimeter by the integral of this negative divergence. The added volume is positive for every sufficiently small width at a contact point; otherwise that point would not be in the perimeter support. Slicing allows an almost every width for which all perimeter formulas hold. The resulting strict decrease contradicts minimality. Adjoining a collar is permitted whether or not marked balls are present.

The perimeter support is thus separated from both obstacles and the outer constraint. It is locally perimeter minimizing in a smooth three-manifold. Interior codimension-one minimizing boundary regularity gives a smooth compact embedded minimal surface. Take its open set representative for \(E\). Filling any bounded open exterior component would strictly reduce perimeter, so its open exterior is connected. Since every precompact enlargement is an admissible competitor, \(E\) is outward minimizing. Its boundary is disjoint from every marked ball, so the unsafe set lies in its interior. The metric on its closed exterior has nonnegative scalar curvature, and its completeness and unique AF end follow from smoothness of compact truncations and the AF estimates.

Now minimize with obstacle \(O\) alone. The same compactness, outer barriers, and strict inner mean-curvature barrier give a smooth free minimal boundary and a connected open exterior. Every component of this open minimizer meets \(O\), because an open component disjoint from \(O\) can be removed, strictly reducing perimeter. Choose a minimizer of maximal volume within the fixed truncation. It is a largest minimizer modulo null sets: the perimeter submodularity inequality implies that the union and intersection of any two minimizers are minimizers, and maximal volume forces the union to agree with this one. Denote its open representative by \(E_0\). Any proper precompact enlargement of equal perimeter would also be a minimizer and would contradict maximal volume. The outer barriers allow this reasoning without the truncation. Thus \(E_0\) is strictly outward minimizing. Its boundary avoids \(S\), so \(O\Subset E_0\). Finally \(E\) is itself an admissible enclosure of \(O\), proving (246). ◻

Proof of Theorem 3. The approximation of Proposition 28 preserves \(H(S)\le0\), converges uniformly in metric comparison, and preserves the limiting time component. The comparison of enclosing areas in Section 2 therefore allows us first to assume that \(R_g+6>0\) and that the end has the smooth conformal form used in Section 6. Fix this approximated metric for the constructions that follow, and put \[A_*=A_{\min}(S;g),\qquad a=\sqrt{A_*/(4\pi)}.\]

Given an arbitrarily small mass tolerance \(\varepsilon_m>0\), Proposition 30 produces a fixed compact inner piece, a smooth synthetic tensor on it, and a scalar-flat AF attachment with mass \(m_o\) satisfying \[ m_o\le p_0(g)-\frac{a^3}{2}+\varepsilon_m. \tag{296}\] The electrostatic height, attachment radius, and smoothing tolerances used to obtain this inequality have already been fixed. They are independent of the large graph height in the next construction.

Apply Proposition 31 and the comparison construction of Proposition 38. Their parameters are chosen in the proved order: the small inner flux and scalar cutoff costs first, then a desired relative-area tolerance \(0<\varepsilon_A<1\), a sufficiently small contraction threshold, a thin omitted exterior collar, and a sufficiently large fixed volume penalty; next the graph height is taken large, and finally the lapse clipping, finite penalty strengths, and corner-smoothing tolerances are chosen. The costs can be made small enough that the resulting smooth AF comparison metric satisfies \[ m_{\ensuremath{\mathrm{AF}}}(h')\le m_o+\varepsilon_m. \tag{297}\]

Let \(E\) be its minimizing enclosure containing the caps and the unsafe-region obstacles. Its free boundary is smooth, compact, minimal, and outer minimizing. The exterior is complete and has \(R_{h'}\ge0\), the AF decay required in Theorem 7, and one end. Let \(E_0\) instead be a largest perimeter-minimizing enclosure of the caps alone. Then \[ \mathop{\mathrm{Area}}_{h'}(\partial E)\ge\mathop{\mathrm{Area}}_{h'}(\partial E_0). \tag{298}\] The exterior of \(E_0\) has a scalar lower bound independent of the large graph height, even where it meets the unsafe region. The bad-volume estimates from Sections 7 and 9, together with the projection along the AF attachment, satisfy Proposition 19. Consequently, when \(A_*>0\), the parameters above can be chosen so that \[ \mathop{\mathrm{Area}}_{h'}(\partial E_0)\ge(1-\varepsilon_A)A_*. \tag{299}\] The small initial conformal change used only for the flow fixes this initial area. It is not made in applying the mass theorem.

Theorem 7 applies to the exterior of \(E\), not the possibly unsafe exterior of \(E_0\). Equations (296)–(299) give \[\begin{align*} p_0(g) &\ge \frac{a^3}{2}+m_o-\varepsilon_m\\ &\ge \frac{a^3}{2}+m_{\ensuremath{\mathrm{AF}}}(h')-2\varepsilon_m\\ &\ge \frac{a^3}{2}+\frac a2\sqrt{1-\varepsilon_A} -2\varepsilon_m. \end{align*}\] Let the two tolerances tend to zero, making fresh choices of the finite parameters as necessary. If \(A_*=0\), the same construction and the nonnegativity consequence of Theorem 7 give \(p_0(g)\ge-2\varepsilon_m\) without the area step. Thus (11) holds for every fixed approximated metric. Passing back through the preliminary approximation proves it for the original metric. ◻

Proof of the inequality in Theorem 2. The maximal constraint and (7) imply \[\int_{\mathrm{end}} V_0|R_g+6|\mathop{}\!\mathrm dV_g<\infty.\] Use Proposition 6 to choose a rest chart for the original covector. All hypotheses remain valid in that chart and \(p_0(g)=m_{\ensuremath{\mathrm{AH}}}(g)\).

Define \[F(A)=\frac12\left(\sqrt{\frac A{4\pi}} +\left(\frac A{4\pi}\right)^{3/2}\right).\] This function is continuous, nonnegative, and nondecreasing on \([0,\infty)\). Theorem 3 supplies the time-symmetric input of Theorem 20, including comparison metrics whose covectors are not timelike. The reduction therefore gives \[p_0(g)\ge F\bigl(A_{\min}(S;g)\bigr).\] Since the chart was fixed before any construction, this is precisely (10). The one-sided time-component budgets suffice; there is no further assertion about the spatial components of intermediate comparison metrics. ◻