Upper bounds for Steklov eigenvalues on surfaces

TL;DR

This paper establishes explicit isoperimetric upper bounds for Steklov eigenvalues on compact orientable surfaces, relating them to topological and geometric features, and generalizing previous results.

Contribution

It provides new explicit bounds for Steklov eigenvalues on surfaces, extending prior inequalities and unifying classical and recent results.

Findings

Explicit bounds depend on genus, boundary length, and components

Generalizes Fraser-Schoen's recent results

Builds on classical inequalities by Hersch-Payne-Schiffer

Abstract

We give explicit isoperimetric upper bounds for all Steklov eigenvalues of a compact orientable surface with boundary, in terms of the genus, the length of the boundary, and the number of boundary components. Our estimates generalize a recent result of Fraser-Schoen, as well as the classical inequalites obtained by Hersch-Payne-Schiffer, whose approach is used in the present paper.