Asymptotic maximal graph equation and coefficients

By [F8], the Harrison stationarity result applies to the supplied data on the end and gives ${\mathcal{L}}_T{g}_q=0$. The local flow of $T={\mathrm{\partial }}_t$ translates $t$ with the spatial coordinates fixed, so the flow definition of the metric Lie derivative makes every component of $g_q$ independent of $t$. Since ${\mathrm{\partial }}_{t_q}=(4q\mathrm{/}{d}_q)T$, the components in the normalized coordinates are independent of $t_q$ as well. Set $\tau =t_q-u(x_q)$ and $D=-g^{00}+2g^{0j}{u}_j-g^{jk}{u}_j{u}_k$. The spacelike graph has timelike normal covector $d\tau$, so $D>0$. Its future unit normal has components $n^{\mu }=-g^{\mu \nu }{\mathrm{\partial }}_{\nu }\tau \mathrm{/}\sqrt{D}$. Extend this normal off the graph using the same formula and the time-independent function $u$. The mean curvature vanishes exactly when ${\mathrm{\nabla }}_{\mu }{n}^{\mu }=0$: differentiating $g(n,n)=-1$ shows the normal-normal contribution to the ambient divergence vanishes. Applying [F4] to the smooth extended normal gives ${\mathrm{\nabla }}_{\mu }{n}^{\mu }=(\sqrt{-\det g}{)}^{-1}{\mathrm{\partial }}_{\mu }(\sqrt{-\det g}{n}^{\mu })$. Time independence removes the time derivative, and $n^i=(g^{ij}{u}_j-g^{i0})\mathrm{/}\sqrt{D}$. This proves the displayed divergence equation.

Write $p=(p_j)$ and define $D(x,p)=-g^{00}+2g^{0j}{p}_j-g^{jk}{p}_j{p}_k$ and $A^i(x,p)=\sqrt{-\det g}(g^{ij}{p}_j-g^{i0})\mathrm{/}\sqrt{D(x,p)}$. The metric estimates and matrix inversion give $g^{00}=-1+O_5(r^{-1})$, $g^{0i}=O_5(r^{-2})$, $g^{ij}={\delta }^{ij}+O_5(r^{-1})$, and $\sqrt{-\det g}=1+O_5(r^{-1})$. For $p=s\mathrm{\partial }u$, $0\le s\le 1$, the slope bound implies $D(x,p)=1+O(r^{-1})$ uniformly; hence it is bounded away from zero on a common larger end. Direct differentiation of the displayed rational expression, holding $p$ fixed for spatial derivatives, gives ${\mathrm{\partial }}_{p_j}{A}^i(x,p)={\delta }^{ij}+O(r^{-1})$ and ${\mathrm{\partial }}_{x_i}{\mathrm{\partial }}_{p_j}{A}^i(x,p)=O(r^{-2})$ uniformly on this segment. Also $A^i(x,0)=O(r^{-2})$ and ${\mathrm{\partial }}_{x_i}{A}^i(x,0)=O(r^{-3})$. These estimates use only the first spatial derivatives of the metric bounds and the explicit factors $p$, with $\mathrm{\mid }p{\mathrm{\mid }}^2=O(r^{-1})$.

Define $B^{ij}(x,p)={\int }_0^1{\mathrm{\partial }}_{p_j}{A}^i(x,sp)ds$. By step 2.1, for each fixed $x$ and $p=\mathrm{\partial }u(x)$, the curve $s\mapsto A^i(x,sp)$ is smooth on an open interval containing $[0,1]$. The chain rule [F6] gives its derivative as ${\mathrm{\partial }}_{p_j}{A}^i(x,sp)p_j$. Applying [F5] componentwise gives $A^i(x,p)=A^i(x,0)+B^{ij}(x,p)p_j$. The same identity holds for nearby $(x,p)$ wherever the segment has $D(x,sp)>0$.

Fix $x_0$ on the common end and $p_0=\mathrm{\partial }u(x_0)$. For each spatial index $i$, hold the other coordinates and $p=p_0$ fixed. Positivity of $D(x_0,s{p}_0)$ for $0\le s\le 1$ and smoothness give a sufficiently small closed interval $[a,b]$ containing $x_0^i$ in its interior on which $D(x,s{p}_0)>0$ for every $s\in [0,1]$. On this rectangle the integrand ${\mathrm{\partial }}_{p_j}{A}^i(x,s{p}_0)$ and its $x_i$ derivative are continuous, so [F7] applies with integration interval $[0,1]$. Consequently, with $p$ held fixed, ${\mathrm{\partial }}_{x_i}{B}^{ij}(x_0,p_0)={\int }_0^1{\mathrm{\partial }}_{x_i}{\mathrm{\partial }}_{p_j}{A}^i(x_0,s{p}_0)ds$. The uniform pointwise bound in step 2.1 makes this $O(r^{-2})$ after summing over $i$; the auxiliary interval need not be uniform in $q$.

Apply the chain rule [F6] to $x\mapsto A^i(x,\mathrm{\partial }u(x))$. The divergence equation becomes ${\mathrm{\partial }}_{p_j}{A}^i(x,\mathrm{\partial }u)u_{ij}+{\mathrm{\partial }}_{x_i}{A}^i(x,\mathrm{\partial }u)=0$, with the second derivative holding $p$ fixed. Differentiating the identity in step 3.1 at fixed $p$ and using step 4.1 yields $a^{ij}{u}_{ij}+b^j{u}_j=f$, with $a^{ij}={\mathrm{\partial }}_{p_j}{A}^i(x,\mathrm{\partial }u)$, $b^j={\mathrm{\partial }}_{x_i}{B}^{ij}(x,\mathrm{\partial }u)$ with $p$ held fixed in that derivative, and $f=-{\mathrm{\partial }}_{x_i}{A}^i(x,0)$. Steps 2.1 and 4.1 give $a^{ij}={\delta }^{ij}+O(r^{-1})$, $b^j=O(r^{-2})$ and $f=O(r^{-3})$, uniformly in $q$. This specifies one admissible choice of the coefficients; the divergence equation itself is independent of that choice.