Estimates for low Steklov eigenvalues of surfaces with several boundary components

TL;DR

This paper provides computable lower bounds for the first non-zero Steklov eigenvalue of surfaces with multiple boundary components, linking geometric properties to spectral estimates, and extends results to hyperbolic surfaces with geodesic boundaries.

Contribution

It introduces new geometric bounds for Steklov eigenvalues on surfaces with boundary and relates these to the geometry of hyperbolic surfaces, extending classical spectral results.

Findings

Lower bounds for Steklov eigenvalues involving boundary geometry

Estimates for hyperbolic surfaces with geodesic boundaries

Connection to classical Laplace eigenvalue results

Abstract

In this article, we give computable lower bounds for the first non-zero Steklov eigenvalue $\sigma_1$ of a compact connected 2-dimensional Riemannian manifold $M$ with several cylindrical boundary components. These estimates show how the geometry of $M$ away from the boundary affects this eigenvalue. They involve geometric quantities specific to manifolds with boundary such as the extrinsic diameter of the boundary. In a second part, we give lower and upper estimates for the low Steklov eigenvalues of a hyperbolic surface with a geodesic boundary in terms of the length of some families of geodesics. This result is similar to a well known result of Schoen, Wolpert and Yau for Laplace eigenvalues on a closed hyperbolic surface.