Minimal surfaces and eigenvalue problems

TL;DR

This paper links extremal Steklov eigenvalues on surfaces with boundary to free boundary minimal surfaces in the unit ball, exploring their properties and how conformal transformations affect boundary volumes.

Contribution

It establishes a connection between maximizing Steklov eigenvalues and free boundary minimal surfaces, and analyzes volume reduction under conformal transformations.

Findings

Metrics maximizing the k-th Steklov eigenvalue relate to free boundary minimal surfaces.

Boundary volume decreases under conformal transformations for these minimal surfaces.

The paper surveys known results and recent advances on extremal metrics for Laplacian and Steklov eigenvalues.

Abstract

We show that metrics that maximize the k-th Steklov eigenvalue on surfaces with boundary arise from free boundary minimal surfaces in the unit ball. We prove several properties of the volumes of these minimal submanifolds. For free boundary minimal submanifolds in the ball we show that the boundary volume is reduced up to second order under conformal transformations of the ball. For two-dimensional stationary integer multiplicity rectifiable varifolds that are stationary for deformations that preserve the ball, we prove that the boundary length is reduced under conformal transformations. We also give an overview of some of the known results on extremal metrics of the Laplacian on closed surfaces, and give a survey of our recent results from [FS2] on extremal metrics for Steklov eigenvalues on compact surfaces with boundary.