Extremal Eigenvalue Problems and Free Boundary Minimal Surfaces in the Ball

Abstract

The main theme of this chapter is the study of extremal eigenvalue problems and its relations to minimal surface theory. We describe joint work with R. Schoen on progress that has been made on the Steklov eigenvalue problem for surfaces with boundary, and in higher dimensions. For surfaces, the Steklov eigenvalue problem has a close connection to free boundary minimal surfaces in Euclidean balls. Specifically, metrics that maximize Steklov eigenvalues are characterized as induced metrics on free boundary minimal surfaces in \(\mathbb {B}^n\). We discuss the existence of maximizing metrics for surfaces of genus zero, and explicit characterizations of maximizing metrics for the annulus and Mobius band. We also give an overview of results on existence, uniqueness, and Morse index of free boundary minimal surfaces in \(\mathbb {B}^n\).