Eigenvalue Estimates on Bakry-Emery Manifolds

TL;DR

This paper establishes lower bounds for eigenvalues on compact Bakry-Emery manifolds using maximum principles, heat kernel estimates, and Sobolev inequalities, advancing understanding of spectral properties in weighted Riemannian geometry.

Contribution

It introduces new lower bounds for all eigenvalues on Bakry-Emery manifolds by adapting maximum principles and heat kernel techniques to the weighted setting.

Findings

Lower bounds for the first eigenvalue using a generalized maximum principle.

Lower bounds for all eigenvalues via heat kernel estimates.

Application of Sobolev inequalities to spectral estimates.

Abstract

We demonstrate lower bounds for the eigenvalues of compact Bakry-Emery manifolds with and without boundary. The lower bounds for the first eigenvalue rely on a generalised maximum principle which allows gradient estimates in the Riemannian setting to be directly applied to the Bakry-Emery setting. Lower bounds for all eigenvalues are demonstrated using heat kernel estimates and a suitable Sobolev inequality.