Curvature at Infinity Governs the Topology of Complete Non-Compact Surfaces Admitting Schrödinger Operators of Finite Index

In this article, we investigate the global topology of a complete non-compact Riemannian $2$-manifold $Σ$ admitting a Schrödinger operator with non-negative potential and finite Morse index. While classical results of Fischer-Colbrie classify such manifolds under the assumption of vanishing index or geometric stability as immersed minimal surfaces in a Riemannian $3$-manifold, we show that the curvature at infinity $λ_\infty^*(Σ)$ of the Fischer-Colbrie metric $g^*$---a complete conformal metric determined by a positive function furnished by Fischer-Colbrie's theorem---governs the global topology and geometric rigidity of $Σ$ without imposing either assumption. More precisely, we derive a fundamental identity relating $λ_\infty^*(Σ)$ to the area growth of $(Σ,g^*)$, show that all critical points of the distance function $d_p^*$ from a fixed base point $p$ are confined to a bounded region, and, as a corollary, obtain a quantitative bound for the number of ends. We further distinguish two complementary geometric viewpoints. On the one hand, a quantitative condition on $λ_\infty^*(Σ)$ forces $Σ$ to be diffeomorphic to the Euclidean plane $\mathbb{R}^2$. On the other hand, when $Σ$ has exactly one end, another condition on $λ_\infty^*(Σ)$ guarantees that every Busemann function on $(Σ,g^*)$ is an exhaustion. By clarifying the relationship between these two regimes---the critical-point structure of distance functions relative to a base point and the global behavior of Busemann functions at infinity---we exhibit two complementary manifestations of how $λ_\infty^*(Σ)$ controls the global geometry and topology of $Σ$.