Intrinsic sound of anti-de Sitter manifolds

TL;DR

This paper explores unique spectral properties of the Laplacian on anti-de Sitter manifolds, revealing dense eigenvalue distributions and stability under deformations, contrasting with classical Riemann surface cases.

Contribution

It introduces new spectral phenomena in pseudo-Riemannian geometry, specifically for anti-de Sitter manifolds, including dense eigenvalue distribution and deformation stability of certain eigenvalues.

Findings

L^2-eigenvalues can be densely distributed in R.

Existence of countably many stable L^2-eigenvalues under deformations.

Contrast with classical eigenvalue behavior on compact Riemann surfaces.

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.