Isoperimetric control of the spectrum of a compact hypersurface

TL;DR

This paper establishes upper bounds for Laplace-Beltrami eigenvalues on hypersurfaces based on isoperimetric ratios, linking spectral properties to geometric measures and exploring implications for isometric embeddings.

Contribution

It introduces new bounds connecting eigenvalues with isoperimetric ratios, advancing understanding of geometric-spectral relationships in Riemannian geometry.

Findings

Eigenvalue bounds in terms of isoperimetric ratios

Applications to extrinsic geometry of embeddings

Insights into spectral geometry of hypersurfaces

Abstract

Upper bounds for the eigenvalues of the Laplace-Beltrami operator on a hypersurface bounding a domain in some ambient Riemannian manifold are given in terms of the isoperimetric ratio of the domain. These results are applied to the extrinsic geometry of isometric embeddings.