Sharp eigenvalue bounds and minimal surfaces in the ball

TL;DR

This paper establishes the existence and regularity of metrics on surfaces with boundary that maximize the first Steklov eigenvalue times boundary length, linking them to free boundary minimal surfaces in the unit ball and providing explicit solutions for specific topologies.

Contribution

It proves the existence and characterization of maximizers for the Steklov eigenvalue problem on surfaces with boundary, connecting them to free boundary minimal surfaces and providing explicit solutions for annuli and Mobius bands.

Findings

Maximizers correspond to free boundary minimal surfaces in the unit ball.

Unique solutions for annulus and Mobius band cases are identified.

As boundary components increase, the maximum approaches 4pi.

Abstract

We prove existence and regularity of metrics on a surface with boundary which maximize sigma_1 L where sigma_1 is the first nonzero Steklov eigenvalue and L the boundary length. We show that such metrics arise as the induced metrics on free boundary minimal surfaces in the unit ball B^n for some n. In the case of the annulus we prove that the unique solution to this problem is the induced metric on the critical catenoid, the unique free boundary surface of revolution in B^3. We also show that the unique solution on the Mobius band is achieved by an explicit S^1 invariant embedding in B^4 as a free boundary surface, the critical Mobius band. For oriented surfaces of genus 0 with arbitrarily many boundary components we prove the existence of maximizers which are given by minimal embeddings in B^3. We characterize the limit as the number of boundary components tends to infinity to give the asymptotically sharp upper bound of 4pi. We also prove multiplicity bounds on sigma_1 in terms of the topology, and we give a lower bound on the Morse index for the area functional for free boundary surfaces in the ball.