Maximization of the first nontrivial eigenvalue on the surface of genus two

TL;DR

This paper proves that the maximum value of the first nontrivial Laplacian eigenvalue on genus 2 surfaces is 16π, confirming a conjecture and advancing understanding of spectral geometry on Riemann surfaces.

Contribution

It establishes the exact supremum of the first nontrivial eigenvalue for genus 2 surfaces, confirming a longstanding conjecture in spectral geometry.

Findings

Supremum of eigenvalue is 16π for genus 2 surfaces

Confirms the conjecture by Jakobson et al.

Advances understanding of eigenvalue optimization on Riemann surfaces

Abstract

The first nontrivial eigenvalue of the Laplacian can be considered as a functional on the space of all Riemannian metrics of unit volume on a fixed surface. In this paper we prove that for the surface of genus 2 the supremum of this functional is equal to $16\pi$. This provides a positive answer to the conjecture by Jakobson, Levitin, Nadirashvili, Nigam and Polterovich.