Eigenvalue estimates on bakry–Émery manifolds

Abstract

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