Eigenvalue estimates and applications on weighted manifolds

TL;DR

This paper provides eigenvalue estimates for weighted manifolds using Bakry-Émery Ricci curvature and applies these results to stability conditions of h-minimal hypersurfaces.

Contribution

It introduces new eigenvalue bounds based on curvature and applies them to geometric stability problems in weighted manifolds.

Findings

First eigenvalue estimates in terms of Bakry-Émery Ricci curvature

Stability conditions for h-minimal hypersurfaces derived from eigenvalue bounds

Application to Dirichlet and Neumann eigenvalue problems

Abstract

We will present an estimate for the first eigenvalue of the Dirichlet and Neumann problems in terms of the Bakry-\'Emery Ricci curvature for a compact weighted manifold. As an application we will establish a stability condition for a h-minimal hypersurface.