Conformal optimization of eigenvalues on surfaces with symmetries

TL;DR

This paper investigates the maximization of Laplace and Steklov eigenvalues on symmetric Riemann surfaces within conformal classes, providing existence, regularity results, and complete solutions for specific symmetric cases.

Contribution

It introduces simplified methods for proving existence and regularity of eigenvalue maximizers and solves the equivariant maximization problem for symmetric surfaces, addressing open questions.

Findings

Complete solution for Laplace eigenvalues on the sphere with symmetry.

Complete solution for Steklov eigenvalues on the disk with symmetry.

Resolved open questions on eigenvalue maximization and inequalities.

Abstract

Given a conformal action of a discrete group on a Riemann surface, we study the maximization of Laplace and Steklov eigenvalues within a conformal class, considering metrics invariant under the group action. We establish natural conditions for the existence and regularity of maximizers. Our method simplifies previously known techniques for proving existence and regularity results in conformal class optimization. Finally, we provide a complete solution to the equivariant maximization problem for Laplace eigenvalues on the sphere and Steklov eigenvalues on the disk, resolving open questions posed by Arias-Marco et al. (2024) regarding the sharpness of the Hersch-Payne-Schiffer inequality and the maximization of Steklov eigenvalues by the standard disk among planar simply connected domains with $n\text{-rotational}$ symmetry.