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Geometry-Topology Seminar

Thursday, February 17, 2022 - 5:15pm

Daniel Stern

University of Chicago


University of Pennsylvania

via Zoom

The Zoom link for this talk is available via e-mail from After the talk, we will stay online for a while to socialize.

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 large 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 Laplacian eigenvalue exist on any closed surface, and are induced by (branched) minimal immersions into round 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 yielding 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. If time permits, I'll also discuss some open problems suggested by these results. (Based on joint work with Mikhail Karpukhin.)