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 \(\Omega _\Gamma \subset \textrm{AdS}_3\) by a discrete group \(\Gamma \) of isometries of \(\textrm{AdS}_3\) 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 \(\Box \) 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.