Intrinsic sound of anti-de Sitter manifolds

Abstract

As is well-known for compact Riemann surfaces, eigenvalues of the Laplacianbare distributed discretely and most of eigenvalues vary viewed as functions on the Teichmuller space. We discuss a new feature in the Lorentzian geometry, or more generally, in pseudo-Riemannian geometry. One of the distinguished features is that $L^2$-eigenvalues of the Laplacian may be distributed densely in R in pseudo-Riemannian geometry. For three-dimensional anti-de Sitter manifolds, we also explain another feature proved in joint with F. Kassel [Adv. Math. 2016] that there exist countably many $L^2$-eigenvalues of the Laplacian that are stable under any small deformation of anti-de Sitter structure.