Spectral Distribution of Twisted Laplacian on Typical Hyperbolic Surfaces of High Genus

TL;DR

This paper studies the spectral distribution of a twisted Laplacian on typical high-genus hyperbolic surfaces, showing that harmonic forms have small supremum norms and establishing a uniform Weyl law for the spectrum.

Contribution

It introduces a new approach to estimate spectral distribution using harmonic form norms and proves a uniform Weyl law for typical hyperbolic surfaces of high genus.

Findings

Harmonic forms have small supremum norms on typical high-genus surfaces.

Spectral distribution can be estimated by harmonic form norms.

A uniform Weyl law for the spectrum is established.

Abstract

We investigate the spectral distribution of the twisted Laplacian associated with uniform square-integrable bounded harmonic 1-form on typical hyperbolic surfaces of high genus. First, we estimate the spectral distribution by the supremum norm of the corresponding harmonic form. Subsequently, we show that the square-integrable bounded harmonic form exhibits a small supremum norm for typical hyperbolic surfaces of high genus. Based on these findings, we prove a uniform Weyl law for the distribution of real parts of the spectrum on typical hyperbolic surfaces.