Exterior Schrödinger maximum principles, Hopf growth and Harnack

On a connected smooth Riemannian exterior $(\mathrm{\Sigma },h)$ with compact smooth boundary $S$, let $V\ge 0$ be smooth and let $u\in C^2({\mathrm{\Sigma }}^{\circ })\cap C_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Sigma })$ satisfy $({\mathrm{\Delta }}_h-V)u=0$, $u{\mathrm{\mid }}_S=0$, and $u\to c>0$ uniformly at infinity. Then $0<u\le c$ in the interior; when $S\mathrm{\ne }\mathrm{\varnothing }$, $u<c$ there. The metric inward unit normal derivative is strictly positive at $S$. On a sufficiently small fixed compact boundary collar, $u\ge c_0\rho$ for $c_0>0$, where $\rho$ is the collar coordinate or the metric distance to $S$.

Positive solutions admit multiplicative interior Harnack comparison on fixed compact connected interior subsets. The constants are uniform for families with a common finite chart cover and uniform metric ellipticity, metric coefficient and bounded-potential controls there. A uniform absolute lower bound additionally requires a positive uniform anchor value. No uniformity is asserted when the metric degenerates, the cover loses control, or the only normalization is an individual end limit.

The prior Laplacian-only Hopf supplier and additive-forcing Harnack supplier are not used outside their interfaces. The boundary and maximum arguments are supplied locally in the explicit helper chain. The outstanding bounded-potential Harnack theorem is an honestly labeled independent external input in the Harnack chain; its underlying iteration proof is not claimed as locally reconstructed. This resolution concerns a generated prerequisite draft, and asserts no defect in the original article or its literature.