Extremal Steklov eigenvalues, free boundary minimal surfaces, and graph realizations
This talk explores the relationship between extremal Steklov eigenvalues of compact surfaces and the construction of free boundary minimal immersions into the unit ball. It details computational methods for deriving genus zero and genus one surfaces, connecting these concepts to analogous problems in graph theory.
Braxton Osting (University of Utah)
Beyond Graph Laplacians
Maximizing the first nontrivial eigenvalue of a weighted graph Laplacian over the edge weights is a classical convex problem. Its dual is a geometric one: the stationarity certificate states that the optimal eigenvectors draw the graph as a unit-distance realization of maximal variance. This problem has continuous counterparts in spectral geometry. Fraser and Schoen showed that a metric maximizing the first Steklov eigenvalue of a compact surface with boundary, normalized by boundary length, is induced by a free boundary minimal immersion into the unit ball: the first eigenfunctions map the surface to one with zero mean curvature that meets the sphere orthogonally. In this talk I will describe computational methods that use the Steklov correspondence to compute free boundary minimal surfaces of genus zero and genus one. The ingredients are conformal uniformization, layer potential methods for boundary-weighted Steklov problems, and nonsmooth eigenvalue optimization with a stationarity certificate that yields an explicit parameterization of the surface. I will present a large collection of computed surfaces, discuss what the computations say about the maximizers in each genus, and close with the analogous problem for Steklov eigenvalues on graphs with boundary.