Metric upper bounds for Steklov and Laplace eigenvalues

TL;DR

This paper establishes new upper bounds for Steklov and Laplace eigenvalues on Riemannian manifolds with boundary, linking geometric properties to spectral bounds using measure concentration and volume growth techniques.

Contribution

It introduces two novel upper bounds for Steklov eigenvalues based on volume, boundary geometry, and concentration inequalities, extending to Laplace eigenvalues for certain manifolds.

Findings

Upper bounds relate eigenvalues to volume and boundary geometry.

Large Steklov eigenvalues imply small extrinsic boundary radius.

Bounds apply to cylinders over closed manifolds, connecting to classical spectral results.

Abstract

We prove two upper bounds for the Steklov eigenvalues of a compact Riemannian manifold with boundary. The first involves the volume of the manifold and of its boundary, as well as packing and volume growth constants of the boundary and its distortion. Its proof is based on a metric-measure space technique that was introduced by Colbois and Maerten. The second bound is in terms of the extrinsic diameter of the boundary and its injectivity radius. It is obtained from a concentration inequality, akin to Gromov-Milman concentration for closed manifolds. By applying these bounds to cylinders over closed manifold, we obtain bounds for eigenvalues of the Laplace operator, in the spirit of Grigor'yan-Netrusov-Yau and of Berger-Croke. For a family of manifolds that has uniformly bounded volume and boundary of fixed intrinsic geometry, we deduce that a large first nonzero Steklov eigenvalue implies that each boundary component is contained in a ball of small extrinsic radius.