A positive global supersolution with the exact exterior decay

Write ${\mathrm{\Delta }}_h=h^{ij}{\mathrm{\partial }}_i{\mathrm{\partial }}_j+b^i{\mathrm{\partial }}_i$. The stated decay gives $h^{ij}={\delta }^{ij}+O(r^{-1})$ and $b^i=O(r^{-2})$. Direct radial differentiation gives ${\mathrm{\Delta }}_h{r}^{\beta }=\beta (\beta +1)r^{\beta -2}+O(r^{\beta -3})$. Since $\beta (\beta +1)<0$, choose a large $R$ so $-{\mathrm{\Delta }}_h{r}^{\beta }\ge c_0{r}^{\beta -2}$ on $r\ge R$.

Let $M_R$ be the end component truncated at $r=R$. It is compact with smooth boundary. It is connected: any excursion of a path into $r>R$ can be retracted radially onto $S_R$, which is connected. In particular it meets its boundary $S_R$, so the compact Poisson supplier applies. Solve $-{\mathrm{\Delta }}_{h}t=1$, $t=0$ on its entire boundary, including any original boundary, obtaining $t\in C^{4,\alpha }(M_R)$. If $t$ had a negative minimum, it would be interior, where its vanishing gradient and nonnegative Hessian imply ${\mathrm{\Delta }}_{h}t\ge 0$, contradicting ${\mathrm{\Delta }}_{h}t=-1$. Thus $t\ge 0$.

In each relatively compact interior or boundary chart the equation for $t$ is a strictly elliptic scalar equation with smooth coefficients and forcing $-1$. Positivity of the smooth metric gives uniform ellipticity on the smaller compact patch. The solution is already classical by step 2.1. Theorem 6.17 applies in interior charts. At a boundary chart, Theorem 6.19's local-boundary-portion conclusion applies to the smooth physical boundary with zero data; no regularity of values on artificial chart edges is required. For each integer $k\ge 0$ these inputs give $C^{k+2,\alpha }$ regularity on smaller patches. A finite chart cover proves $t\in C^{\mathrm{\infty }}(M_R)$ up to its entire boundary, including $S_R$. This is a local application of the imported theorems, not a reduction of the compact core to one Euclidean domain.

Set $B=R^{\beta }+ϵt$ on $M_R$ and $B=r^{\beta }$ outside. Both traces on the seam are $R^{\beta }$. There $\nu ={\mathrm{\nabla }}_{h}r\mathrm{/}\mathrm{\mid }{\mathrm{\nabla }}_{h}r{\mathrm{\mid }}_h$, so $\nu (r)=\mathrm{\mid }{\mathrm{\nabla }}_{h}r{\mathrm{\mid }}_h>0$ and the exterior derivative $\beta R^{\beta -1}\nu (r)$ is strictly negative. The Riemannian gradient is the metric dual of the differential Compactness of $S_R$ bounds it uniformly away from zero and bounds ${\mathrm{\partial }}_{\nu }t$. Choose $ϵ>0$ small enough that $J_{\nu }B=\beta R^{\beta -1}\nu (r)-ϵ{\mathrm{\partial }}_{\nu }t<0$ on the whole sphere. Inside, $-{\mathrm{\Delta }}_{h}B=ϵ$; outside use step 1.1. The positive smooth $\rho$ has finite maximum and positive minimum on compact sets and equals $r$ far out, so decreasing $c>0$ if necessary gives $-{\mathrm{\Delta }}_{h}B\ge c{\rho }^{\beta -2}$ on both pieces. The barrier is positive; on the core it lies between $R^{\beta }$ and $R^{\beta }+ϵ\max t$, proving $B\asymp 1$ there. Its asserted piecewise smoothness follows from step 3.1 and smoothness of $r^{\beta }$ on the end.

First consider a function $w$ with the stated continuity and one-sided regularity, with ${\mathrm{\Delta }}_{h}w>0$ on each piece, $J_{\nu }w>0$, negative original boundary values and ${\mathrm{lim\hspace{0.17em}sup}}_{r\to \mathrm{\infty }}w\le 0$. If $w(p)>0$ anywhere, choose $0<a<w(p)$. Outside a compact set $w\le a$, so $w$ attains a positive global maximum on a compact truncation, away from both its outer boundary and its original boundary. At an interior maximum off the seam, the gradient vanishes and the Hessian is negative semidefinite, giving ${\mathrm{\Delta }}_{h}w\le 0$, a contradiction. At a seam maximum, the interior derivative in direction $\nu$ is nonnegative and the exterior derivative in the same direction is nonpositive. Hence $J_{\nu }w\le 0$, again a contradiction. Thus $w\le 0$. On compact $M_T$ the same argument uses compactness and negative entire boundary values directly.

For $v$ satisfying the weak comparison hypotheses set $w=v-\eta B$ for any $\eta >0$. Then ${\mathrm{\Delta }}_{h}w={\mathrm{\Delta }}_{h}v-\eta {\mathrm{\Delta }}_{h}B>0$ off the seam, $J_{\nu }w=J_{\nu }v-\eta J_{\nu }B>0$, and $w<0$ on every boundary where $v\le 0$. Since $B=r^{\beta }\to 0$, the condition ${\mathrm{lim\hspace{0.17em}sup}}_{r\to \mathrm{\infty }}v\le 0$ also gives ${\mathrm{lim\hspace{0.17em}sup}}_{r\to \mathrm{\infty }}w\le 0$. Step 5.1 yields $v\le \eta B$; let $\eta \downarrow 0$ at each point to obtain $v\le 0$. For compact truncations the same perturbation has negative values on the entire boundary, so the compact version of step 5.1 applies. This proves the comparison and completes the Statement.