A note on lower bounds estimates for the Neumann eigenvalues of manifolds with positive Ricci curvature

TL;DR

This paper introduces new heat kernel estimates for manifolds with positive Ricci curvature and convex boundary, leading to improved lower bounds for Neumann eigenvalues that align with Weyl's asymptotic behavior.

Contribution

It provides novel heat kernel bounds and eigenvalue estimates for manifolds with positive Ricci curvature and convex boundary, extending existing spectral geometry results.

Findings

New heat kernel estimates for Neumann boundary conditions

Lower bounds for eigenvalues consistent with Weyl's law

Enhanced understanding of spectral properties of curved manifolds

Abstract

We study new heat kernel estimates for the Neumann heat kernel on a compact manifold with positive Ricci curvature and convex boundary. As a consequence, we obtain new lower bounds for the Neumann eigenvalues which are consistent with Weyl's asymptotics.