Isoperimetric inequalities for eigenvalues of the Laplace-Beltrami operator on surfaces

Lord Rayleigh asked in his famous book "Theory of Sound'' (1877-1878) the following question: what shape of a drum membrane provides the lowest possible sound among all membranes of a given fixed area? The answer (the disc) was obtained by Lord Rayleigh using physical heuristics and rigorously proven later by Faber and Krahn in 1921.

The contemporary analogue of this problem in Riemannian geometry is the following one: given a compact surface without boundary and a natural number k, find the supremum of the k-th eigenvalue of the Laplace-Beltrami operator (depending on a Riemannian metric) over the space of all Riemannian metrics of given fixed area. This difficult problem turns out to be very rich and related to such classical domains as Differential and Algebraic Geometry, Geometric Analysis, PDEs, Topology etc.