On a relation between the spectra of the Laplacian and the Dirichlet-to-Neumann map on manifolds

The Dirichlet-to-Neumann operator is a first order elliptic pseudodifferential operator. It acts on smooth functions on the boundary of a Riemannian manifold and maps a function to the normal derivative of its harmonic extension. The eigenvalues of the Dirichlet-to-Neumann map are also called Steklov eigenvalues. It has been known that the geometry of the boundary has a strong influence on the Steklov eigenvalues. In this talk, we show that for every $k$, the $k$th Steklov eigenvalue is comparable to the square root of the $k$th Laplace eigenvalue. This result, in particular, gives a two-sided geometric bound for any Steklov eigenvalue which depends only on the geometry near the boundary. This is joint work with Bruno Colbois and Alexandre Girouard.