On largeness and multiplicity of the first eigenvalue of hyperbolic surfaces

TL;DR

This paper investigates the properties of the first non-zero eigenvalue of the Laplacian on hyperbolic surfaces, revealing unboundedness and topological disconnection in the moduli space for genus two surfaces.

Contribution

It demonstrates that for genus two hyperbolic surfaces, the set of surfaces with first eigenvalue greater than 1/4 is unbounded and disconnects the moduli space, using topological methods.

Findings

The set of genus two surfaces with λ₁ > 1/4 is unbounded.

This set disconnects the moduli space of genus two surfaces.

Topological methods are effective in spectral analysis of hyperbolic surfaces.

Abstract

We apply topological methods to study the smallest non-zero number $\lambda_1$ in the spectrum of the Laplacian on finite area hyperbolic surfaces. For closed hyperbolic surfaces of genus two we show that the set $\{S \in {\mathcal{M}_2}: {\lambda_1}(S) > 1/4 \}$ is unbounded and disconnects the moduli space ${\mathcal{M}_2}$.