Daniel Stern : Spectral shape optimization and new behaviors for free boundary minimal surfaces

Though the study of isoperimetric problems for Laplacian eigenvalues dates back to the 19th century, the subject has undergone a renaissance in recent decades, due in part to the discovery of connections with harmonic maps and minimal surfaces. By the combined work of several authors, we now know that unit-area metrics maximizing the first nonzero Laplace eigenvalue exist on any closed surface, and are realized by minimal surfaces in spheres. At the same time, work of Fraser-Schoen, Matthiesen-Petrides and others yields analogous results for the first eigenvalue of the Dirichlet-to-Neumann map on surfaces with boundary, with maximizing metrics induced by free boundary minimal immersions into Euclidean balls. In this talk, I'll describe a series of recent results characterizing the (perhaps surprising) asymptotic behavior of these free boundary minimal immersions (and associated Steklov-maximizing metrics) as the number of boundary components becomes large. (Based on joint work with Mikhail Karpukhin.)

• Category: Geometry and Topology
• Duration: 01:14:50
• Date:  March 28, 2022 at 3:10 PM
• Views: 259
• Tags:   seminar, Geometry and Topology Seminar