Katie Gittins

Upper bounds for Steklov eigenvalues, ratios and gaps

The interplay between the eigenvalues of the Steklov problem and the geometry of the underlying object is a key theme within Spectral Geometry. Substantial progress has been made in obtaining geometric upper bounds for the Steklov eigenvalues in various geometric settings. The Steklov spectral ratios and gaps have also received attention in recent years, and they too afford insights into this interplay.
In this talk we will first give an overview of some geometric upper bounds for Steklov eigenvalues of Riemannian manifolds of dimension at least 3. We will then present results regarding upper bounds for the Steklov eigenvalues, ratios and gaps on balls with revolution-type metrics based on joint work with Jade Brisson and Bruno Colbois. Time-permitting, we will also present upper bounds for the Steklov eigenvalues of warped product manifolds which are part of an ongoing project with Jade Brisson, Bruno Colbois and Alexandre Girouard.