On the first stability eigenvalue of surfaces with constant weighted mean curvature

TL;DR

This paper derives upper bounds for the first eigenvalue of the weighted Jacobi operator on surfaces with constant weighted mean curvature, leading to the conclusion that stable self-shrinkers do not exist in mean curvature flow.

Contribution

It provides new bounds for the first eigenvalue of the weighted Jacobi operator on such surfaces, connecting stability properties to curvature conditions.

Findings

Upper bounds for the first eigenvalue in terms of curvature and mean curvature

No stable self-shrinkers exist under the given conditions

Implications for stability analysis in mean curvature flow

Abstract

Let $\Sigma$ be a compact immersed surface with constant weighted mean curvature $H_f$ in a weighted manifold $(M^3,g,f)$. In this paper we obtain upper bounds for the first eigenvalue of the weighted Jacobi operator on $\Sigma$ in terms of $H_f$ and the curvature of the ambient. As consequence we obtain that there is no stable self-shrinker of the mean curvature flow.