Frontiers in Mathematical Sciences 7th Conference University of Isfahan - January 1-3, 2020
 Title: A comparison between Steklov and Laplace eigenvalues on a Riemannian manifold. Speaker: Asma Hassannezhad, University of Bristol Date, Time, and Venue:  Thursday, January 2 | 14:00-14:45 | Hall 1 Abstract: The Dirichlet-to-Neumann operator 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. We show that for every $k\in N$ the $k$-th Steklov eigenvalue $\sigma_k$ is comparable to the square root of the $k$-th eigenvalue $\sqrt{\lambda_k}$ of the Laplacian on the boundary. Our results, in particular, give interesting geometric lower and upper bounds for Steklov eigenvalues. This is joint work with Bruno Colbois and Alexandre Girouard.