Eigenvalues of the Laplacian and min-max for the energy functional.

The Laplacian is a canonical second order elliptic operator defined on any Riemannian manifold. The study of optimal upper bounds for its eigenvalues is a classical problem 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 and harmonic maps. In the present talk we survey recent developments in the field. In particular, we will discuss a min-max construction for the energy functional and its applications to eigenvalue inequalities, including the regularity theorem for optimal metrics. The talk is based on the joint work with D. Stern.