Geometric Analysis and Topology Seminar
Existence of extremal metrics for Laplace eigenvalues and minimal surfaces of prescribed topology
Speaker: Daniel Stern, Cornell University
Location: Warren Weaver Hall 512
Date: Friday, May 2, 2025, 11 a.m.
Synopsis:
Classic results of Yang—Yau and Li--Yau show that, on every closed surface, the first nonzero eigenvalue of the Laplacian is bounded above by a constant depending only on the topology and area. Later work of Nadirashvili revealed that metrics are critical for the area-normalized first eigenvalue of the Laplacian if and only if they are induced by minimal immersions to spheres by first eigenfunctions, and about fifteen years ago, Fraser and Schoen discovered an analogous correspondence between intrinsic shape optimization problems for Steklov eigenvalues and free boundary minimal surfaces in the ball. These observations prompted a wave of developments in the existence theory for metrics maximizing various eigenvalue functionals, but even the basic question of existence for metrics maximizing the first Laplace eigenvalue on arbitrary closed surfaces has remained stubbornly open for some time. After surveying some of this history, I'll discuss recent progress on the existence and geometric characterization of metrics maximizing the first Laplace and Steklov eigenvalues under certain discrete symmetries, leading to the construction of many new families of embedded minimal surfaces in S^3 and free boundary minimal surfaces in B^3 with low area and prescribed (oriented) topological type. I'll then explain how a refinement of a key analytic ingredient in those results allows us to finally complete the existence theory for the classic \lambda_1-maximization problem on closed surfaces of any (orientable or non-orientable) topological type. Based on joint work with Karpukhin, Kusner, and McGrath, and forthcoming work with Karpukhin and Petrides.