Differential Geometry Seminar: Handle attachment and the normalized first eigenvalue

Seminar | September 30 | 3-4 p.m. | 939 Evans Hall

 Henrik Matthiesen, University of Chicago

 Department of Mathematics

I will discuss asymptotic lower bounds of the first eigenvalue for two constructions of attaching degenerating handles to a given closed Riemannian surface. One of these constructions is relatively simple but often fails to strictly increase the first eigenvalue normalized by area. Motivated by this negative result, we then give a much more involved construction that always strictly increases the first eigenvalue normalized by area. As a consequence we obtain the existence of a metric that maximizes the first eigenvalue among all unit area metrics on a given closed surface.

This is joint work with A. Siffert.