Mathematical Physics Seminar
The Laplacian is a canonical second order elliptic operator defined on any Riemannian manifold. The study of upper bounds for its eigenvalues is a classical subject of spectral geometry going back to J. Hersch, P. Li and S.-T. Yau. It turns out that the optimal isoperimetric inequalities for Laplacian eigenvalues are closely related to minimal surfaces in spheres, a classical object of geometric analysis. Recently, this connection was used to obtain a variety of new exciting results. In the present talk we will survey some recent advances in the field, including the optimal isoperimetric inequality for all Laplacian eigenvalues on the sphere and the projective plane. The talk is based on joint works with N. Nadirashvili, A. Penskoi and I. Polterovich.