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Conformal optimization of eigenvalues on surfaces with symmetries

Denis Vinokurov

TL;DR

The paper develops a unified framework for conformal optimization of Laplace- and Steklov-eigenvalues on surfaces with finite-group symmetries, proving existence and regularity of Γ-invariant maximizers that arise from Γ-equivariant harmonic maps (or free-boundary harmonic maps). It provides a streamlined bubbling-tree approach in 2D that reduces maximizing metrics to a finite combination of harmonic maps and bubbles, and derives sharp upper and lower bounds for the eigenvalue functionals. The results yield complete, explicit classifications of equivariant maxima on the sphere and disk for all finite Γ, resolving open questions about the sharpness of the Hersch–Payne–Schiffer inequality and the maximization of Steklov eigenvalues by the standard disk under symmetry. Significantly, the work both simplifies prior methods in conformal eigenvalue optimization and delivers concrete configurations (disjoint unions of spheres or disks) that realize the maxima in symmetric settings. This advances understanding of optimal metrics under symmetry constraints and deepens the connection between spectral optimization and harmonic map theory in two dimensions.

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.

Conformal optimization of eigenvalues on surfaces with symmetries

TL;DR

The paper develops a unified framework for conformal optimization of Laplace- and Steklov-eigenvalues on surfaces with finite-group symmetries, proving existence and regularity of Γ-invariant maximizers that arise from Γ-equivariant harmonic maps (or free-boundary harmonic maps). It provides a streamlined bubbling-tree approach in 2D that reduces maximizing metrics to a finite combination of harmonic maps and bubbles, and derives sharp upper and lower bounds for the eigenvalue functionals. The results yield complete, explicit classifications of equivariant maxima on the sphere and disk for all finite Γ, resolving open questions about the sharpness of the Hersch–Payne–Schiffer inequality and the maximization of Steklov eigenvalues by the standard disk under symmetry. Significantly, the work both simplifies prior methods in conformal eigenvalue optimization and delivers concrete configurations (disjoint unions of spheres or disks) that realize the maxima in symmetric settings. This advances understanding of optimal metrics under symmetry constraints and deepens the connection between spectral optimization and harmonic map theory in two dimensions.

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 symmetry.

Paper Structure

This paper contains 29 sections, 33 theorems, 216 equations.

Table of Contents

  1. Introduction and main results
  2. Regularity and existence of the maximizers
  3. Applications
  4. Conformal actions on Lg
  5. Conformal actions on Lg
  6. Ideas of the proof of existence and regularity
  7. Acknowledgements
  8. Preliminaries
  9. Eigenvalues of measures
  10. Clarke subdifferentials
  11. Measures as bilinear forms
  12. Proof of existence and regularity
  13. Constructing a maximizing sequence
  14. Convergence and regularity near stable points
  15. Proof of Theorem \ref{['thm:main']} and Theorem \ref{['thm:main-stek']}
  16. ...and 14 more sections

Key Result

Proposition 1.1

Given a compact connected surface $(\Sigma, [g])$, there exists a smooth harmonic map $\Phi\colon \Sigma \to \mathbb{S}^n$ (the case $\Sigma = \varnothing$), or a smooth free boundary harmonic map $\Psi\colon \Sigma \to \mathbb{B}^n$ (the case $\Sigma \neq \varnothing$) such that one of the followin or where $\left|\partial_\nu\Psi\right|$ is extended arbitrarily in the interior. The quantities $

Theorems & Definitions (73)

  • Proposition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Remark 1.6
  • Corollary 1.7
  • Remark 1.8
  • Proposition 1.9
  • proof
  • ...and 63 more