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 paper, we develop 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. For genus γ = 0 and b = 2, …, 9, 12, 15, 20 boundary components, we numerically solve the extremal Steklov problem for the first eigenvalue. The corresponding eigenfunctions generate a free boundary minimal surface, which we display in striking images. For higher eigenvalues, numerical evidence suggests that the maximizers are degenerate, but we compute local maximizers for the second and third eigenvalues with b = 2 boundary components and for the third and fifth eigenvalues with b = 3 boundary components.
Highlights
We develop computational methods for solving the extremal Steklov eigenvalue problem (1.3) and generating free boundary minimal surfaces via Theorem 1.1
The argument relies on two ingredients: 1. The uniformization result that for a smooth, compact, connected, genus-zero Riemannian surface with b boundary components, (Σ, g), there exists a conformal mapping f : (Σ, g) → (Ω, ρI), where Ω is a disk with b − 1 holes and ρI is a conformally flat metric
We developed computational methods to maximize the length-normalized jth Steklov eigenvalue, σj(Σ, g) := σj(Σ, g)L(∂Σ, g) over the class of smooth Riemannian metrics, g on a compact surface, Σ, with genus γ and b boundary components
Summary
Schoen discovered a rather surprising connection between an extremal Steklov eigenvalue problem and the problem of generating free boundary minimal surfaces in the Euclidean ball [14,15,16]. These findings have been further developed [12, 17, 19] and were recently reviewed in [28]. We develop numerical methods to further investigate this connection. We first briefly review some of these previous results before stating the contributions of the present work
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