Emily Dryden, Bucknell University

Upper Bounds on Eigenvalues of the Laplacian: Surfaces and Beyond

In 1970, Joseph Hersch showed that for the sphere, the product of its surface area and lowest nonzero eigenvalue is at most 8π, regardless of the metric. Moreover, the bound is realized only by the usual constant curvature metric. Following Hersch's work, attempts were made to generalize his result to other surfaces and to higher-dimensional manifolds. We will explore some of the successes, some of the surprises, and some open questions in this area.

Back to schedule