Geometric spectral optimization on surfaces

TL;DR

This paper proves the existence of optimal metrics for Laplace eigenvalues on closed surfaces, relating them to minimal immersions and extending previous results to combinations of eigenvalues across all surface topologies.

Contribution

It establishes the existence of optimal metrics for eigenvalue combinations on any genus surface, generalizing prior work and removing the need for equivariant conditions.

Findings

Existence of optimal metrics for Laplace eigenvalues on all closed surfaces.

Explicit relation between optimal metrics and minimal eigenmaps into ellipsoids.

Extension of previous results to combinations of eigenvalues, not just the first.

Abstract

We prove the existence of optimal metrics for a wide class of combinations of Laplace eigenvalues on closed orientable surfaces of any genus. The optimal metrics are explicitely related to Laplace minimal eigenmaps, defined as branched minimal immersions into ellipsoids parametrized by the eigenvalues of the critical metrics whose coordinates are eigenfunctions with respect to these eigenvalues. In particular, we prove existence of maximal metrics for the first Laplace eigenvalue on orientable surfaces of any genus. In this case, the target of eigenmaps are spheres. This completes a broad picture, first drawn by J. Hersch, 1970 (sphere), M. Berger 1973, N. Nadirashvili 1996 (tori). Our result is based on the combination of accurate constructions of Palais-Smale-like sequences for spectral functionals and on techniques by M. Karpukhin, R. Kusner, P. McGrath, D. Stern 2024, developped in the case of an equivariant optimization of the first Laplace and Steklov eigenvalues. Their result is significantly extended for two reasons: specific equivariant optimizations are not required anymore to obtain existence of maximizers of the first eigenvalue for any topology and our technique also holds for combinations of eigenvalues.