Variation of Laplace spectra of compact "nearly" hyperbolic surfaces

TL;DR

This paper investigates how the Laplace spectra of negatively curved compact surfaces with fixed genus, area, and curvature bounds change smoothly, using Ricci flow real analyticity to provide quantitative estimates and insights into spectral properties.

Contribution

It extends spectral variation results for hyperbolic surfaces by applying Ricci flow real analyticity to derive quantitative spectral estimates and new spectral property insights.

Findings

Spectra vary in a controlled, quantifiable manner.

Real analyticity leads to unexpected spectral property conclusions.

Results apply to surfaces with fixed genus, area, and curvature bounds.

Abstract

We use the time real analyticity of Ricci flow proved by Kotschwar to extend a result in ~\cite{B}, namely, we prove that the Laplace spectra of negatively curved compact surfaces having same genus $\gamma \geq 2$, same area and same curvature bounds vary in a "controlled way", of which we give a quantitative estimate (Theorem 1.1 below). We also observe how said real analyticity can lead to unexpected conclusions about spectral properties of generic metrics on a compact surface of genus $\gamma \geq 2$ (Proposition 1.5 below).