DMS Applied Mathematics Seminar

Speaker: Chiu-Yen Kao, Claremont McKenna College

Title: Computational Approaches to Steklov Eigenvalue Problems and Free Boundary Minimal Surfaces 

Abstract: Recently Fraser and Schoen showed that the solution of a certain extremal Steklov eigenvalue problem on a compact surface with boundary can be used to generate a free boundary minimal surface, i.e., a surface contained in the ball that has (i) zero mean curvature and (ii) meets the boundary of the ball orthogonally (doi:10.1007/s00222-015-0604-x). In this talk, we discuss our new numerical methods that use this connection to realize free boundary minimal surfaces. Namely, on a compact surface, Σ, with genus γ and b boundary components, we maximize σj (Σ, g) L(∂Σ, g) over a class of smooth metrics, g, where σj (Σ, g) is the jth nonzero Steklov eigenvalue and L(∂Σ, g) is the length of ∂Σ. Our numerical method involves (i) using conformal uniformization of multiply connected domains to avoid explicit parameterization for the class of metrics, (ii) accurately solving a boundary-weighted Steklov eigenvalue problem in multi-connected domains, and (iii) developing gradient-based optimization methods for this non-smooth eigenvalue optimization problem. We numerically solve the extremal Steklov problem for the first eigenvalue and use corresponding eigenfunctions to generate a free boundary minimal surface, which we display in striking images. Many results are shown to demonstrate the accuracy and robustness of the proposed approaches.  

This is a joint work with Braxton Osting at Department of Mathematics, University of Utah, United States, and Edouard Oudet at LJK, Universit´e Grenoble Alpes, France.