Inverse spectral positivity for surfaces

TL;DR

This paper investigates the geometric implications of the non-negativity of a differential operator involving Laplacian, Gaussian curvature, and a potential function on complete non-compact surfaces, providing new proofs and improvements of classical theorems.

Contribution

It offers a new proof of Huber's theorem and Cohn-Vossen's inequality, and extends previous results for specific cases of the operator involving curvature and potential functions.

Findings

Derived conditions linking operator non-negativity to surface geometry.

Provided new proofs of classical theorems in differential geometry.

Improved results for cases with non-positive W and specific values of a.

Abstract

Let $(M,g)$ be a complete non-compact Riemannian surface. We consider operators of the form $\Delta + aK + W$, where $\Delta$ is the non-negative Laplacian, $K$ the Gaussian curvature, $W$ a locally integrable function, and $a$ a positive real number. Assuming that the positive part of $W$ is integrable, we address the question "What conclusions on $(M,g)$ and $W$ can one draw from the fact that the operator $\Delta + aK + W$ is non-negative ?" As a consequence of our main result, we get a new proof of Huber's theorem and Cohn-Vossen's inequality, and we improve earlier results in the particular cases in which $W$ is non-positive and $a = 1/4$ or $a \in (0,1/4)$.