Degenerating hyperbolic surfaces and spectral gaps for large genus

TL;DR

This paper investigates the spectral gaps of the Laplacian on hyperbolic surfaces of large genus, showing that the maximum spectral gap approaches at least 1/4 as genus increases, and introduces a min-max principle for degenerating surfaces.

Contribution

It provides new bounds on spectral gaps for hyperbolic surfaces of large genus and establishes a min-max principle for eigenvalues on degenerating surfaces.

Findings

Spectral gaps' supremum approaches at least 1/4 as genus increases

Established a min-max principle for eigenvalues on degenerating hyperbolic surfaces

Spectral gap differences are studied up to the (2g-2)-th eigenvalue

Abstract

In this article we study the differences of two consecutive eigenvalues $\lambda_{i}-\lambda_{i-1}$ up to $i=2g-2$ for the Laplacian on hyperbolic surfaces of genus $g$, and show that the supremum of such spectral gaps over the moduli space has infimum limit at least $\frac{1}{4}$ as genus goes to infinity. A min-max principle for eigenvalues on degenerating hyperbolic surfaces is also established.