The Klein quartic maximizes the multiplicity of the first positive eigenvalue of the Laplacian

TL;DR

This paper proves that the Klein quartic surface uniquely maximizes the multiplicity of the first positive Laplacian eigenvalue among genus 3 hyperbolic surfaces, and provides bounds for genus 2 and higher.

Contribution

It establishes the maximal eigenvalue multiplicity for the Klein quartic and extends bounds on eigenvalue multiplicities for all genus g hyperbolic surfaces.

Findings

Klein quartic has eigenvalue multiplicity 8 for genus 3.

Maximum multiplicity for genus 2 is between 3 and 6.

Bound on eigenvalue multiplicities in [0, 1/4 + δ_g] interval.

Abstract

We prove that Klein quartic maximizes the multiplicity of the first positive eigenvalue of the Laplacian among all closed hyperbolic surfaces of genus $3$, with multiplicity equal to $8$. We also obtain partial results in genus $2$, where we find that the maximum multiplicity is between $3$ and $6$. Along the way, we show that for every $g\geq 2$, there exists some $\delta_g>0$ such that the multiplicity of any eigenvalue of the Laplacian on a closed hyperbolic surface of genus $g$ in the interval $[0,1/4+\delta_g]$ is at most $2g-1$ despite the fact that this interval can contain arbitrarily many eigenvalues. This extends a result of Otal to a larger interval but with a weaker bound, which nevertheless improves upon the general upper bound of S\'evennec.