Mikhail Karpukhin University College London

Eigenvalue optimisation and minimal surfaces

Abstract:

Given a Riemannian surface, the study of sharp upper bounds for Laplacian eigenvalues under the area constraint is a classical problem of spectral geometry. The particular interest in this problem stems from the surprising fact that the optimal metrics for such bounds arise as metrics on minimal surfaces in spheres. For surfaces with boundary a similar story connects Steklov eigenvalues with free boundary minimal surfaces in balls. In this talk I will describe the general idea behind this correspondence and survey recent exciting developments in the field.