On closed surfaces with nonnegative curvature in the spectral sense

TL;DR

This paper investigates closed orientable surfaces with a spectral curvature condition, deriving geometric inequalities and compactness results by analyzing associated conformal metrics with nonnegative curvature.

Contribution

It introduces a spectral condition involving the Laplacian and Gauss curvature, leading to new geometric inequalities and precompactness results for such surfaces.

Findings

Establishes isoperimetric inequalities under spectral conditions

Proves area growth and diameter bounds for these surfaces

Demonstrates H"older precompactness and almost rigidity results

Abstract

We study closed orientable surfaces satisfying the spectral condition $\lambda_1(-\Delta+\beta K)\geq\lambda\geq0$, where $\beta$ is a positive constant and $K$ is the Gauss curvature. This condition naturally arises for stable minimal surfaces in 3-manifolds with positive scalar curvature. We show isoperimetric inequalities, area growth theorems and diameter bounds for such surfaces. The validity of these inequalities are subject to certain bounds for $\beta$. Associated to a positive super-solution $\Delta\varphi\leq\beta K\varphi$, the conformal metric $\varphi^{2/\beta}g$ has pointwise nonnegative curvature. Utilizing the geometry of the new metric, we prove H\"older precompactness and almost rigidity results concerning the main spectral condition.