Spectra of Lorentzian quasi-Fuchsian manifolds

Abstract

A three-dimensional quasi-Fuchsian Lorentzian manifold $M$ is a globally hyperbolic spacetime diffeomorphic to $Σ\times (-1,1)$ for a closed orientable surface $Σ$ of genus $\geq 2$. It is the quotient $M=Γ\backslash Ω_Γ$ of an open set $Ω_Γ\subset {\rm AdS}_3$ by a discrete group $Γ$ of isometries of ${\rm AdS}_3$ which is a particular example of an Anosov representation of $π_1(Σ)$. 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 $Γ$ extend meromorphically to $\mathbb{C}$. This is then used to prove that there is a natural notion of resolvent of the pseudo-Riemannian Laplacian $\Box$ 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.