Large Steklov eigenvalues on hyperbolic surfaces

TL;DR

This paper investigates the behavior of large Steklov eigenvalues on hyperbolic surfaces, constructing sequences with unbounded eigenvalues and establishing growth bounds in high-genus moduli spaces.

Contribution

It constructs hyperbolic surfaces with unbounded first normalized Steklov eigenvalues and proves generic growth bounds in high-genus moduli spaces.

Findings

First normalized Steklov eigenvalue can tend to infinity on certain hyperbolic surfaces.

For large genus, a generic surface's eigenvalue exceeds a constant times boundary length sum.

Eigenvalues grow unboundedly as the boundary length norm increases.

Abstract

In this paper, we first construct a sequence of hyperbolic surfaces with connected geodesic boundary such that the first normalized Steklov eigenvalue $\tilde{\sigma}_1$ tends to infinity. We then prove that as $g\rightarrow \infty$, a generic $\Sigma\in \mathcal{M}_{g,n}(L_g)$ satisfies $\tilde{\sigma}_1(\Sigma)>C\cdot \|L_g\|_1$ where $C$ is a positive universal constant. Here $\mathcal{M}_{g,n}(L_g)$ is the moduli space of hyperbolic surfaces of genus $g$ and $n$ boundary components of length $L_g=(L_g^1,\cdots, L_g^n)$ endowed with the Weil-Petersson metric where $\|L_g\|_1\rightarrow\infty$ satisfies certain conditions.