Eigenvalue comparisons in Steklov eigenvalue problem and some other eigenvalue estimates

TL;DR

This paper establishes eigenvalue comparison theorems for the Steklov problem on manifolds with curvature bounds and provides sharper bounds for Wentzell eigenvalues, extending classical results in spectral geometry.

Contribution

It introduces new comparison theorems for Steklov eigenvalues and improves bounds for Wentzell eigenvalues, generalizing classical spectral estimates.

Findings

Eigenvalue comparison theorems for Steklov problem

Sharper bounds for Wentzell eigenvalues

Extension of classical eigenvalue estimates

Abstract

In this paper, two interesting eigenvalue comparison theorems for the first non-zero Steklov eigenvalue of the Laplacian have been established for manifolds with radial sectional curvature bounded from above. Besides, sharper bounds for the first non-zero eigenvalue of the Wentzell eigenvalue problem of the weighted Laplacian, which can be seen as a natural generalization of the classical Steklov eigenvalue problem, have been obtained.