The Steklov eigenvalues are the eigenvalues of the Dirichlet-to-Neumann operator on the boundary of a Riemannian manifold. In this talk, we study the spectral gap of the Steklov problem. When the manifold has only one boundary component, it is known that the geometry of the boundary collar alone can provide a lower bound for the spectral gap. We therefore focus on the case of manifolds with at least two boundary components and show how the interior geometry can affect the spectral gap. We review known results and then present a new connection between Steklov spectral gaps and the Neumann eigenvalues of interior subdomains.
This talk is based on joint work with Tirumala Chakradhar and Bruno Colbois.