# Geometric Bounds for Low Steklov Eigenvalues of Finite Volume Hyperbolic Surfaces

## TL;DR

This paper establishes geometric lower bounds for low Steklov eigenvalues on hyperbolic surfaces with geodesic boundaries.

## Contribution

The paper introduces sharp geometric bounds for Steklov eigenvalues that depend on surface disconnection properties.

## Key findings

- Geometric lower bounds for Steklov eigenvalues are derived based on multi-geodesic disconnection lengths.
- The bounds remain valid even when curvature is bounded between two negative constants.
- The results improve and extend previous findings in the compact case.

## Abstract

We obtain geometric lower bounds for the low Steklov eigenvalues of finite-volume hyperbolic surfaces with geodesic boundary. The bounds we obtain depend on the length of a shortest multi-geodesic disconnecting the surfaces into connected components each containing a boundary component and the rate of dependency on it is sharp. Our result also identifies situations when the bound is independent of the length of this multi-geodesic. The bounds also hold when the Gaussian curvature is bounded between two negative constants and can be viewed as a counterpart of the well-known Schoen-Wolpert-Yau inequality for Laplace eigenvalues. The proof is based on analysing the behaviour of the corresponding Steklov eigenfunction on an adapted version of thick–thin decomposition for hyperbolic surfaces with geodesic boundary. Our results extend and improve the previously known result in the compact case obtained by a different method.