Cheeger type inequalities associated with isocapacitary constants on Riemannian manifolds with boundary

TL;DR

This paper establishes Cheeger-type inequalities relating the first Steklov eigenvalue and the spectrum of the Dirichlet-to-Neumann operator on Riemannian manifolds with boundary, using isocapacitary constants.

Contribution

It introduces new Cheeger-type inequalities for Steklov eigenvalues and spectrum estimates based on isocapacitary constants for both compact and non-compact manifolds.

Findings

Cheeger-type inequality for first Steklov eigenvalue on compact manifolds

Estimate of the spectrum bottom of Dirichlet-to-Neumann operator on non-compact manifolds

Use of isocapacitary constants to relate geometric and spectral properties

Abstract

In this paper, we study the Steklov eigenvalue of a Riemannian manifold (M, g) with smooth boundary. For compact M , we establish a Cheeger-type inequality for the first Steklov eigenvalue by the isocapacitary constant. For non-compact M , we estimate the bottom of the spectrum of the Dirichlet-to-Neumann operator by the isocapacitary constant.