Uniform end estimates indexed only by existing admissible slices

Fix one pair $(q,\mathrm{\Sigma })$ and let $G:C\to {\mathrm{\Sigma }}^{\circ }$ be its stationary graph diffeomorphism. Choose the adapted $y$ coordinates from F4, and put $s(G(z))=\mathrm{\mid }y(z)\mathrm{\mid }$. Keep $r=\mathrm{\mid }x\mathrm{\mid }$ for the separate physical end radius, so $r_q=qr$ there. On that end, with $v=\tilde{u}$, the induced metric is $h_{ij}=(g_q{)}_{ij}+(g_q{)}_{0i}{v}_j+(g_q{)}_{0j}{v}_i+(g_q{)}_{00}{v}_i{v}_j$ in $x_q$ coordinates. Individual slope decay and F1 give $h_{ij}={\delta }_{ij}+O_q(r_q^{-1})$. Hence $\mathrm{\mid }dr{\mathrm{\mid }}_h=q^{-1}\mathrm{\mid }d{r}_q{\mathrm{\mid }}_h$ is bounded by a finite constant $L_q$ on a sufficiently distant physical graph tail, before any lapse bound is used. This says nothing about $\mathrm{\mid }ds{\mathrm{\mid }}_h$.

For each sufficiently large physical radius $a$, the set $H_a=G(\{r\ge a\})$ is closed in $\mathrm{\Sigma }$. Indeed its inner sphere and each finite annulus $G(\{a\le r\le b\})$ are compact and closed by F5. If $z\in {\overline{H_a}}^{\mathrm{\Sigma }}\setminus H_a$, take a small relatively compact section coordinate ball or boundary half-ball $V$ about $z$, disjoint from the inner sphere. Choose $w\in H_a\cap V$ and the coordinate straight segment $\lambda :[0,1]\to V$ from $w$ to $z$. Its $h$-speed is bounded by a finite $M$ on the closed interval, by smoothness and F5. While the segment stays in the open graph tail, F7 and step 1.1 give $\mathrm{\mid }(r\circ \lambda {)}^{\mathrm{\prime }}\mathrm{\mid }\le L_{q}M$. The mean value theorem bounds its radius by $r(w)+L_{q}M$. At its first exit, including an exit at parameter one, compactness and closedness of the finite graph annulus put the exit point back in that annulus; it misses the inner sphere and thus still lies in the open tail, a contradiction.

Choose a compact smooth collar $W=c(S\times [0,a_0])$ in $\mathrm{\Sigma }$ using F4 and F5. The complements of the nested closed sets $H_n$ for sufficiently large integer physical radii $n$ cover $W$: boundary points lie in none of them, and every interior point has finite physical radius or is outside the physical chart. A finite subcover gives one $n$ with $W\cap H_n=\mathrm{\varnothing }$. Every base point of $W\setminus S$ therefore belongs to the original compact physical truncation $B_0\setminus \{r>n\}$. The continuous adapted radius $\mathrm{\mid }y\mathrm{\mid }$ is bounded on this truncation by F5; hence $s\le B$ on $W\setminus S$ for some $B$. On the compact outer collar face $L=c(S\times \{a_0\})$, the continuous $s>1$ has a minimum $m>1$.

Choose $0<\eta <m-1$ and set $H=G(y^{-1}(\{1<\mathrm{\mid }y\mathrm{\mid }<1+\eta \}))$. Radial segments and paths in $S^2$ join any two points of this shell; composition with $G$ gives paths in $H$, so [F6] makes $H$ connected. In the interior manifold the boundary of the open collar portion $W^{\circ }=c(S\times (0,a_0))$ is exactly $L$: the closed collar $W$ is closed by compactness and the Hausdorff property, and the collar chart leaves only its outer face after removing $S$. Thus $H\cap W^{\circ }$ is clopen in $H$, since $H$ avoids $L$. It is nonempty. Indeed, fix $p\in S$ and consider $c(p,a_0\mathrm{/}n)$ for integers $n\ge 2$. These points converge to $p$ in $\mathrm{\Sigma }$ and have $s\le B$ by step 3.1. If they all had $s\ge 1+\eta$, they would lie in the compact graph image $G(y^{-1}(\{1+\eta \le \mathrm{\mid }y\mathrm{\mid }\le B\}))$, closed in the Hausdorff manifold $\mathrm{\Sigma }$ by [F5]; their boundary limit $p$ would then lie in that interior graph image, a contradiction. Therefore $H\cap W^{\circ }\mathrm{\ne }\mathrm{\varnothing }$, and [F6] gives $H\subset W^{\circ }$. No extension of $G^{-1}$ or of the stationary height to $S$ has been assumed.

Fix $R$ sufficiently large for the physical end and write $K_R=B_0\setminus \{r>R\}$, compact by F4. The set $K_R\cap \{\mathrm{\mid }y\mathrm{\mid }\ge 1+\eta \}$ is compact, contained in $C$, and has compact graph image by F5. Step 4.1 puts all remaining points of $G(K_R\cap C)$ in $W$. Thus $\mathrm{\Sigma }\setminus G(\{r>R\})$ is a closed subset of the compact union $W\cup G(K_R\cap \{\mathrm{\mid }y\mathrm{\mid }\ge 1+\eta \})$ and is compact. The same proof applies at $2R$. Since $r_q=qr$ in the physical chart, these are exactly the truncated complement and compact core required by the positive-lapse supplier, with $R_q=qR$. No equality between $r$ and $s$ was used.

The compact-core conclusion of step 5.1 and individual end slope limit verify the remaining hypotheses of [F1] separately for each pair. Its individual decay gives the required end height and slope limits, hence $0<N\le d_q\mathrm{/}(4q)\le C_{\delta }$. For $D=-g_q^{-1}(d(t_q-\tilde{u}),d(t_q-\tilde{u}))$, $N=(d_q\mathrm{/}(4q))D^{-1\mathrm{/}2}$ gives $D\ge 1$. The uniform inverse end metric has spatial block at least $I\mathrm{/}2$, so $\mathrm{\mid }D\tilde{u}{\mathrm{\mid }}^2\mathrm{/}2\le C{r}_q^{-1}+C{r}_q^{-2}\mathrm{\mid }D\tilde{u}\mathrm{\mid }$. Absorb the last term to obtain the stated common slope bound. This calculation uses no uniform radius for the initial individual end limits.

Directly normalize the conormal of the stationary graph and set its divergence equal to zero. Write $x=x_q$, $r=r_q$ and $u=\tilde{u}$ in this end calculation. The result is ${\mathrm{\partial }}_i{P}^i(x,Du)=0$, with $P^i=\sqrt{-\det g_q}(g_q^{ij}{p}_j-g_q^{i0})\mathrm{/}\sqrt{-g_q^{00}+2g_q^{0j}{p}_j-g_q^{jk}{p}_j{p}_k}$. The common slope bound and end metric derivatives give a common elliptic nondivergence equation $\mathcal{L}u=a^{ij}{u}_{ij}+b^j{u}_j=f$, with symmetric $a=I+O(r^{-1})$, $b=O(r^{-2})$, $\mathrm{\mid }f\mathrm{\mid }\le C_f{r}^{-3}$ and no zeroth-order term. Here the coefficients are evaluated along the supplied graph and then kept fixed when applying $\mathcal{L}$ to a comparison function. Fix $0<\mu <1$. Direct differentiation gives $\mathcal{L}(r^{-\mu })=-\mu (1-\mu )r^{-\mu -2}+O(r^{-\mu -3})\le -c_{\mu }{r}^{-\mu -2}$ beyond a common radius. Fix any $r_{\ast }$ beyond this radius, let $M(r_{\ast })={\max }_{\mathrm{\mid }x\mathrm{\mid }=r_{\ast }}\mathrm{\mid }u(x)\mathrm{\mid }$, and choose a common $K>0$ with $c_{\mu }K>C_f$. Set $w(x)=(M(r_{\ast })+K\mathrm{/}{r}_{\ast })(r_{\ast }\mathrm{/}r{)}^{\mu }$. Then $-\mathcal{L}w\ge c_{\mu }K(r\mathrm{/}{r}_{\ast }{)}^{1-\mu }{r}^{-3}>\mathrm{\mid }f\mathrm{\mid }$ for $r\ge r_{\ast }$.

For each fixed indexed pair and $\epsilon >0$, its individual limit $u\to 0$ gives an outer radius $R_{\epsilon }>r_{\ast }$ such that $\mathrm{\mid }u\mathrm{\mid }\le \epsilon$ for $r\ge R_{\epsilon }$. On the compact annulus $r_{\ast }\le r\le R_{\epsilon }$, both $z_{+}=u-w-\epsilon$ and $z_{-}=-u-w-\epsilon$ are nonpositive on the boundary, while $\mathcal{L}{z}_{\pm }=\pm f-\mathcal{L}w>0$ in its interior. If either function were positive somewhere, [F3] would give a positive interior maximum with zero gradient and Hessian $H\le 0$. Since $a>0$, [F3] applied to $P=a$ and $Q=-H$ gives $\mathcal{L}{z}_{\pm }=\mathrm{tr}(aH)\le 0$ there, a contradiction. Thus $\mathrm{\mid }u\mathrm{\mid }\le w+\epsilon$ on the annulus; outside it the same inequality holds because $w\ge 0$ and $\mathrm{\mid }u\mathrm{\mid }\le \epsilon$. Letting $\epsilon \downarrow 0$ at each fixed end point proves $\mathrm{\mid }u(r)\mathrm{\mid }\le (M(r_{\ast })+K\mathrm{/}{r}_{\ast })(r_{\ast }\mathrm{/}r{)}^{\mu }$ for all $r\ge r_{\ast }$. The outer radius can depend on the pair; no uniform initial decay radius or convergence rate is needed.

Fix a unit vector $\omega \in S^2$ and put $h(t)=u(t^2\omega )$ on $[\sqrt{r_{\ast }},2\sqrt{r_{\ast }}]$. The radial curve lies in the common end, where $u$ is smooth, so $h$ is continuous on this closed interval and differentiable on its interior. By the chain rule in [F7] and the slope bound of step 6.1, $h^{\mathrm{\prime }}(t)=2tDu(t^2\omega )\cdot \omega$ and $\mathrm{\mid }{h}^{\mathrm{\prime }}(t)\mathrm{\mid }\le 2t{C}_{\delta }(t^2{)}^{-1\mathrm{/}2}=2C_{\delta }$. The mean value theorem in [F7] therefore gives $\mathrm{\mid }u(4r_{\ast }\omega )-u(r_{\ast }\omega )\mathrm{\mid }\le 2C_{\delta }\sqrt{r_{\ast }}$. Combining this with step 8.1 at $4r_{\ast }\omega$ and taking the maximum over $\omega$ gives $M(r_{\ast })\le 4^{-\mu }(M(r_{\ast })+K\mathrm{/}{r}_{\ast })+2C_{\delta }\sqrt{r_{\ast }}$. Fix one common sufficiently large $r_{\ast }$ and absorb the factor $4^{-\mu }<1$. This bounds $M(r_{\ast })$ and hence the full height decay uniformly over the indexed pairs. No uniform individual convergence rate was assumed to justify annular comparison.

On fixed rescaled annuli, the explicit flux in step 7.1 has common ellipticity and joint C4-Holder bounds on a fixed slightly enlarged slope ball: the slopes are small by step 6.1 and the background has common five differentiated end bounds. Its zero-slope divergence is O($R^{-2}$) in the rescaled C2-Holder norm, since $P(x,0)=O_5(r^{-2})$. Apply [F2] with the common normalized height bound from step 9.1. It yields $\mathrm{\parallel }u(R\cdot ){\mathrm{\parallel }}_{C^{4,\alpha }(A_{\mathrm{i}\mathrm{n}})}\le C({\sup }_{RA}\mathrm{\mid }u\mathrm{\mid }+R^{-1})$. Taking $\mu =3\mathrm{/}4$ proves the asserted uniform differentiated and scaled Holder estimates. Every estimate is indexed by a pair actually present in $\mathcal{I}$.