A Spectral Gap for Spinors on Hyperbolic Surfaces

TL;DR

This paper constructs an explicit sequence of hyperbolic surfaces with increasing genus that have a uniform spectral gap for the Dirac operator, advancing understanding of spectral properties in geometric analysis.

Contribution

It provides a new explicit construction of hyperbolic surfaces with a uniform spectral gap for the Dirac operator, using a tower of covers with arithmetic surfaces.

Findings

Sequence of hyperbolic surfaces with increasing genus

Uniform spectral gap for Dirac operator established

Construction is explicit and based on towers of covers

Abstract

The purpose of this note is to construct a sequence of spin hyperbolic surfaces $\Sigma_n$ with genus going to infinity and with a uniform spectral gap for the Dirac operator. Our construction is completely explicit. In particular, the $\Sigma_n$ can be taken to be a tower of covers, with each $\Sigma_n$ an arithmetic hyperbolic surface.