Asymptotically sharp bound for Wentzel-Laplace eigenvalues

TL;DR

This paper establishes asymptotically optimal upper bounds for Wentzel-Laplace eigenvalues on Riemannian manifolds, emphasizing the influence of boundary geometry alongside dimension and volume.

Contribution

It provides the first sharp asymptotic bounds for Wentzel-Laplace eigenvalues incorporating boundary geometric properties.

Findings

Bounds depend on boundary geometry, dimension, and volume.

Results are asymptotically sharp and optimal.

Applicable to manifolds with Ricci curvature bounded below.

Abstract

We prove asymptotically optimal upper bounds for the eigenvalues of the Wentzel-Laplace operator on Riemannian manifolds with Ricci curvature bounded below. These bounds depend highly on the geometry of the boundary in addition to the dimension and the volume of the manifold.