The Steklov and Laplacian spectra of Riemannian manifolds with boundary

The Dirichlet-to-Neumann map is a first order pseudodifferential operator acting on the smooth functions of the boundary of a compact Riemannian manifold M. Its spectrum is known as the Steklov spectrum of M. The asymptotic behaviour (as j tends to infinity) of the Steklov eigenvalues s_j is determined by the Riemannian metric on the boundary of M. Neverthless, each individual eigenvalue can become arbitrarily big if the Riemannian metric is perturbed adequately. This can be achieved while keeping the geometry of the boundary unchanged, but it requires modifications of the metric in arbitrarily small neighborhoods of the boundary. In our recent work with Bruno Colbois and Asma Hassannezhad, we impose constraints on the geometry of M on and near its boundary. This allows the comparison of each Steklov eigenvalue s_j with the eigenvalues l_j of the Laplace operator acting on the boundary. This control is uniform in the index j. In this talk I will discuss the proof of this result, which is based on the Pohozaev identity and on comparison results for the principal curvatures of hypersurfaces that are parallel to the boundary.