Short geodesics and multiplicities of eigenvalues of hyperbolic surfaces

TL;DR

This paper establishes upper bounds on Laplacian eigenvalue multiplicities for hyperbolic surfaces, linking them to the count of short geodesics and genus, advancing a longstanding conjecture.

Contribution

It provides new bounds connecting short geodesics and eigenvalue multiplicities, improving understanding of spectral geometry of hyperbolic surfaces.

Findings

Eigenvalue multiplicities are sublinear in genus when short geodesics are sublinear

Progress on Colin de Verdière's conjecture from the 1980s

Upper bounds depend on the number of short geodesics

Abstract

In this paper, we obtain upper bounds on the multiplicity of Laplacian eigenvalues for closed hyperbolic surfaces in terms of the number of short closed geodesics and the genus $g$. For example, we show that if the number of short closed geodesics is sublinear in $g$, then the multiplicity of the first eigenvalue is also sublinear in $g$. This makes new progress on a conjecture by Colin de Verdi\`ere in the mid 1980s.