Ailana Fraser, University of British Columbia
"Minimal surfaces and eigenvalue problems"
Finding sharp eigenvalue bounds and characterizing the extremals is a basic problem in geometric analysis. We will describe the structure of metrics which are obtained by maximizing the first eigenvalue of the Dirichlet-to-Neumann map over all metrics on a surface with boundary. It turns out that the extremals are related to minimal surfaces in the ball with a natural boundary condition, and in some cases it is possible to use minimal surface theory to characterize the extremal metrics. This is joint work with R. Schoen.