Spectra of Lorentzian Quasi-Fuchsian Manifolds

Abstract A three-dimensional quasi-Fuchsian Lorentzian manifold M is a globally hyperbolic spacetime diffeomorphic to $$\Sigma \times (-1,1)$$ Σ × ( - 1 , 1 ) for a closed orientable surface $$\Sigma $$ Σ of genus $$\ge 2$$ ≥ 2 . It is the quotient $$M=\Gamma \backslash \Omega _\Gamma $$ M = Γ \ Ω Γ of an open set $$\Omega _\Gamma \subset \textrm{AdS}_3$$ Ω Γ ⊂ AdS 3 by a discrete group $$\Gamma $$ Γ of isometries of $$\textrm{AdS}_3$$ AdS 3 which is a particular example of an Anosov representation of $$\pi _1(\Sigma )$$ 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 $$\Gamma $$ Γ extend meromorphically to $$\mathbb {C}$$ 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 $$\mathbb {C}$$ 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.