Systole and small eigenvalues of hyperbolic surfaces

TL;DR

This paper establishes a lower bound on the eigenvalues of the Laplacian for hyperbolic surfaces with large systoles, extending known results to geometrically finite surfaces without cusps.

Contribution

It proves a new eigenvalue lower bound for hyperbolic surfaces with systole greater than 3.46, including geometrically finite surfaces without cusps, generalizing previous results.

Findings

Eigenvalue lower bound for systole > 3.46

Extension to geometrically finite surfaces without cusps

Improved understanding of spectral geometry of hyperbolic surfaces

Abstract

Let $S$ be a closed orientable hyperbolic surface with Euler characteristic$\chi$, and let $\lambda_k(S)$ be the $k$-th positive eigenvalue for the Laplacian on $S$. According to famous result of Otal and Rosas, $\lambda_{-\chi}>\frac14$. In this article, we prove that if thesystole of $S$ is greater than 3,46, then $\lambda_{-\chi-1}>\frac14$.This inequality is also true for geometrically finite orientable hyperbolic surfaces without cusps with the same assumption on the systole.