Spectra of Lorentzian quasi-Fuchsian manifolds

Abstract

A three-dimensional quasi-Fuchsian Lorentzian manifold $M$ is a globally hyperbolic spacetime diffeomorphic to $\Sigma \times (-1,1)$ for a closed orientable surface $\Sigma$ of genus $\ge 2$. It is the quotient $M=\Gamma \backslash {\Omega }_{\Gamma }$ of an open set ${\Omega }_{\Gamma }\subset {\mathrm{AdS}}_3$ by a discrete group $\Gamma$ of isometries of ${\mathrm{AdS}}_3$ which is a particular example of an Anosov representation of ${\pi }_1(\Sigma )$. We first show that the spacelike geodesic flow of $M$ is Axiom A, has a discrete Ruelle resonance spectrum with associated (co-)resonant states, and that the Poincaré series for $\Gamma$ extend meromorphically to $\mathbb{C}$. This is then used to prove that there is a natural notion of resolvent of the pseudo-Riemannian Laplacian $\square$ of $M$, which is meromorphic on $\mathbb{C}$ with poles of finite rank, defining a notion of quantum resonances and quantum resonant states related to the Ruelle resonances and (co-)resonant states by a quantum-classical correspondence. This initiates the spectral study of convex co-compact pseudo-Riemannian locally symmetric spaces.

Revised version. 87 pages