Closedness and smooth limiting maximal sections

Choose a countable compact exhaustion of ${\mathrm{\Sigma }}_1^{\circ }$ by coordinate patches with compact closure and a finite subcover at each stage. On each patch, [F1] bounds derivatives through order four and makes those derivatives equicontinuous. Apply [F4] componentwise to the finitely many derivatives through order four, and diagonalize over the patches and exhaustion. To identify these componentwise limits, write $f_{\alpha }$ for the uniform limit of $D^{\alpha }{u}_j$ on a smaller coordinate box. For every coordinate segment remaining in that box, [F9] gives $D^{\alpha }{u}_j(x+t{e}_{\mathrm{\ell }})-D^{\alpha }{u}_j(x)={\int }_0^t{D}^{\alpha +e_{\mathrm{\ell }}}{u}_j(x+s{e}_{\mathrm{\ell }})ds$. The integrability and uniform integral-error bound in [F9] permit passage to $f_{\alpha }(x+t{e}_{\mathrm{\ell }})-f_{\alpha }(x)={\int }_0^t{f}_{\alpha +e_{\mathrm{\ell }}}(x+s{e}_{\mathrm{\ell }})ds$. After division by nonzero $t$, continuity of $f_{\alpha +e_{\mathrm{\ell }}}$ makes the right-hand average tend to $f_{\alpha +e_{\mathrm{\ell }}}(x)$ as $t\to 0$, identifying the corresponding partial derivative. Induction over derivative order therefore identifies all the extracted limits with derivatives of $u_{\ast }=f_0$. The resulting subsequence therefore converges in $C^4$ on every compact subset. The uniform fourth-derivative bounds imply convergence in $C^{3,\alpha }$ there for every $0<\alpha <1$: the difference of third derivatives tends uniformly to zero and has a common Lipschitz bound, so its $\alpha$-Hölder seminorm also tends to zero. Denote the limit by $u_{\ast }$.

For the graph level function ${\tau }_q=t-u_q(x)$, the future normal is $n_q=-{\mathrm{grad}}_{g_q}{\tau }_q\mathrm{/}\sqrt{-g_q^{-1}(d{\tau }_q,d{\tau }_q)}$. Since $T={\mathrm{\partial }}_t$ and $d{\tau }_q(T)=1$, the lapse is $N_q=-g_q(T,n_q)=1\mathrm{/}\sqrt{-g_q^{-1}(d{\tau }_q,d{\tau }_q)}$. Fix a compact subset of the reference interior. By [F1] its candidate points and normals are uniformly bounded in fixed compact exterior coordinates. The smooth stationary field $T$ and the jointly smooth metrics $g_q$ have bounded coordinate components on that compact region and the compact parameter interval. Consequently the contraction $N_q=-g_q(T,n_q)$ has a uniform upper bound $C>0$ there. Thus $-g_q^{-1}(d{\tau }_q,d{\tau }_q)=N_q^{-2}\ge C^{-2}$. Joint metric convergence and step 1.1 pass this inequality to $q_{\ast },u_{\ast }$, making the limiting level covector timelike and the graph spacelike. Its mean curvature is a continuous expression in the metric and its first derivatives and the graph derivatives through order two, so it is zero by passage to the limit.

Use the compact supplier's explicit full stationary $C^{m,\alpha }$ estimates in [F1] at every finite $m$. The same complete compact, height/barrier, horizon, normalized-end and positive-lapse hypotheses are required in the Statement, with the actual reference section and selected family matched in [F6]. The quantitative coordinate transfer is supplied by that result, rather than inferred here from pointwise transversality alone. Apply [F4] to successive finite collections of these stationary derivative fields and diagonalize over the exhaustion and derivative orders, retaining a subsequence of step 1.1. Use the coordinate-segment argument in step 1.1 with [F9] to identify every new derivative limit as a derivative of that same $u_{\ast }$. Hence $u_{\ast }$ is smooth in the interior.

On the common normal collar, express the heights in the fixed coordinate domain $(R,y)\in [0,b)\times S$ using the jointly smooth maps in [F6]. The uniform finite-order bounds in [F2] remain uniform under these smooth background coordinate changes. Apply [F4] in finitely many closed collar charts and diagonalize over derivative orders and smaller closed target collars, retaining the interior subsequence from step 3.1. Identify the componentwise derivative limits by the coordinate-segment argument in step 1.1 using [F9], first in boundary half-box interiors and then on their closed faces by continuity. Write the resulting heights as $w_j\to w_{\ast }$ smoothly up to $R=0$ on every smaller closed collar, and set $G_j(R,y)={\mathrm{\Psi }}_{q_j}(w_j(R,y),R,y)$ and $G_{\ast }(R,y)={\mathrm{\Psi }}_{q_{\ast }}(w_{\ast }(R,y),R,y)$. Then $G_j\to G_{\ast }$ smoothly there. The zero boundary values give $G_{\ast }(0,y)=y$, and the uniform tangent spacelikeness in [F2] makes $G_{\ast }$ a smooth spacelike embedded normal graph with boundary $S$. It remains to compare its punctured image with the stationary limit.

Fix $0<R<b$ and $y\in S$. By [F7], $G_j(R,y)={\varphi }_{s_j}^K(c_{q_j}({\rho }_j,{\eta }_j))$ with $\mathrm{\mid }{s}_j\mathrm{\mid }\le C_{\delta }$ and $R\mathrm{/}{c}_2\le {\rho }_j\le R\mathrm{/}{c}_1<a\mathrm{/}2$. In the fixed identification $S\cong S^2$, apply [F8] successively to the five bounded real coordinates $s_j,{\rho }_j,{\eta }_j^1,{\eta }_j^2,{\eta }_j^3$. Along a further subsequence for this fixed point they converge to $s_{\mathrm{\infty }},{\rho }_{\mathrm{\infty }},{\eta }_{\mathrm{\infty }}$; the closed interval inequalities and sphere equation pass to the limit, so ${\rho }_{\mathrm{\infty }}\ge R\mathrm{/}{c}_2>0$ and ${\eta }_{\mathrm{\infty }}\in S$. Smoothness of the reference collars and complete flow in [F6] gives $G_{\ast }(R,y)={\varphi }_{s_{\mathrm{\infty }}}^K(c_{q_{\ast }}({\rho }_{\mathrm{\infty }},{\eta }_{\mathrm{\infty }}))\in \mathcal{E}$. Thus these candidate points converge to an exterior point, and their stationary projections lie eventually in one compact interior coordinate neighborhood. Write their stationary coordinates as $(t_j,x_j)\to (t_{\ast },x_{\ast })$. Since $t_j=u_{q_j}(x_j)$, uniform convergence on that compact neighborhood in step 1.1 and continuity of $u_{\ast }$ imply $t_{\ast }=u_{\ast }(x_{\ast })$. Consequently every punctured normal-limit point belongs to the stationary limit. The positive lower bound for ${\rho }_j$ and bounded flow times supply precisely the control that prevents its stationary projections from escaping to the horizon.

Conversely, choose a smaller common ambient neighborhood of $S$ inside the capture neighborhood in [F7] and the Gaussian domains in [F6], with all its target radial coordinates less than some $b^{\mathrm{\prime }}<b$. Let $P=(u_{\ast }(x),x)$ be any stationary-limit point there. Its candidate approximants $P_j=(u_{q_j}(x),x)$ converge to $P$ by step 1.1 and are eventually in the capture neighborhood, so $P_j=G_j(R_j,y_j)$ with $0<R_j<b$. Joint smoothness of the Gaussian inverses gives $({\tau }_j,R_j,y_j)\to ({\tau }_{\ast },R_{\ast },y_{\ast })$, where ${\tau }_j=w_j(R_j,y_j)$, $0\le R_{\ast }<b$, and $P={\mathrm{\Psi }}_{q_{\ast }}({\tau }_{\ast },R_{\ast },y_{\ast })$. The cone inequality in [F7] passes to $\mathrm{\mid }{\tau }_{\ast }\mathrm{\mid }\le \theta R_{\ast }$. If $R_{\ast }=0$, this forces ${\tau }_{\ast }=0$ and hence $P=y_{\ast }\in S$, contradicting $P\in \mathcal{E}$. Therefore $R_{\ast }>0$. Smooth convergence of the normal heights on smaller closed collars in step 4.1 gives ${\tau }_{\ast }=w_{\ast }(R_{\ast },y_{\ast })$, so $P=G_{\ast }(R_{\ast },y_{\ast })$. This proves the reverse inclusion near $S$. Together with step 5.1 it identifies the two images on an ambient overlap, and the embedded spacelike normal graph from step 4.1 extends the stationary limit smoothly with boundary exactly $S$.

In the end coordinates the rescalings depend smoothly on $q$ and remain invertible for $q\in [\delta ,1]$. Write $y=x_q$. The normalized height definition gives ${\tilde{u}}_q(y)=d_q{u}_q(y\mathrm{/}q)\mathrm{/}(4q)$. For fixed $q$, the map $y\mapsto y\mathrm{/}q$ is linear with derivative $q^{-1}I$. Applying [F5] successively to $u_q$ and its partial derivatives gives $D_y^{\beta }{\tilde{u}}_q(y)=d_q(D_x^{\beta }{u}_q)(y\mathrm{/}q)\mathrm{/}(4q^{1+\mathrm{\mid }\beta \mathrm{\mid }})$ for every multi-index $\beta$ with $0\le \mathrm{\mid }\beta \mathrm{\mid }\le 4$; order zero is the defining identity. These applications are valid because each candidate is smooth, and the linear rescaling has no higher derivatives. At any fixed point $y$ on a sufficiently distant common normalized end, $y\mathrm{/}{q}_j\to y\mathrm{/}{q}_{\ast }$ inside a compact neighborhood of the original end. Step 1.1 gives uniform convergence of the derivatives through order four on that neighborhood. The displayed identity and continuity of the limit derivatives and scalar factors therefore give convergence of normalized derivatives through order four at $y$. Passing to the uniform inequalities in [F3] yields $\mathrm{\mid }{D}_{x_{q_{\ast }}}^k{\tilde{u}}_{\ast }\mathrm{\mid }\le C_k{r}_{q_{\ast }}^{-3\mathrm{/}4-k}$ for $0\le k\le 4$. The case $k=0$ gives the required zero limit at infinity. Together with steps 2.1–6.1 this proves the Statement.