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

Abstract

In this article, we investigate the global topology of a complete non-compact Riemannian $2$-manifold $\Sigma$ 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 ${\lambda }_{\infty }^{\ast }(\Sigma )$ of the Fischer-Colbrie metric $g^{\ast }$---a complete conformal metric determined by a positive function furnished by Fischer-Colbrie's theorem---governs the global topology and geometric rigidity of $\Sigma$ without imposing either assumption. More precisely, we derive a fundamental identity relating ${\lambda }_{\infty }^{\ast }(\Sigma )$ to the area growth of $(\Sigma ,g^{\ast })$, show that all critical points of the distance function $d_p^{\ast }$ 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 ${\lambda }_{\infty }^{\ast }(\Sigma )$ forces $\Sigma$ to be diffeomorphic to the Euclidean plane ${\mathbb{R}}^2$. On the other hand, when $\Sigma$ has exactly one end, another condition on ${\lambda }_{\infty }^{\ast }(\Sigma )$ guarantees that every Busemann function on $(\Sigma ,g^{\ast })$ 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 ${\lambda }_{\infty }^{\ast }(\Sigma )$ controls the global geometry and topology of $\Sigma$.

33 pages, 3 figures