The first Steklov eigenvalue, conformal geometry, and minimal surfaces

TL;DR

This paper explores the relationship between the first Steklov eigenvalue and the geometry of compact Riemannian manifolds with boundary, establishing bounds and characterizing extremal metrics, especially on annular surfaces and in conformal classes.

Contribution

It provides new bounds for Steklov eigenvalues on surfaces, characterizes extremal metrics like the critical catenoid, and extends conformal volume bounds to higher dimensions with minimal immersion conditions.

Findings

The upper bound for sigma_1L(∂Σ) is 2(2γ + k)π for surfaces with genus γ and k boundary components.

The extremal metric for annuli is achieved by the catenoid portion meeting a sphere orthogonally.

Any free boundary minimal surface in two dimensions has area at least π.

Abstract

We consider the relationship of the geometry of compact Riemannian manifolds with boundary to the first nonzero eigenvalue sigma_1 of the Dirichlet-to-Neumann map (Steklov eigenvalue). For surfaces Sigma with genus gamma and k boundary components we obtain the upper bound sigma_1L(\partial \Sigma) \leq 2(2gamma+k)\pi. We attempt to find the best constant in this inequality for annular surfaces (gamma=0 and k=2). For rotationally symmetric metrics we show that the best constant is achieved by the induced metric on the portion of the catenoid centered at the origin which meets a sphere orthogonally and hence is a solution of the free boundary problem for the area functional in the ball. For a general class of (not necessarily rotationally symmetric) metrics on the annulus, which we call supercritical, we prove that $\sigma_1(\sig)L(\p\Sigma)$ is dominated by that of the critical catenoid with equality if and only if the annulus is conformally equivalent to the critical catenoid by a conformal transformation which is an isometry on the boundary. We prove general upper bounds for conformal metrics on manifolds of any dimension which can be properly conformally immersed into the unit ball in terms of certain conformal volume quantities. We show that these bounds are only achieved when the manifold is minimally immersed by first Steklov eigenfunctions. We also use these ideas to show that any free boundary solution in two dimensions has area at least \pi.