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Spectra of Lorentzian quasi-Fuchsian manifolds

Benjamin Delarue, Colin Guillarmou, Daniel Monclair

TL;DR

This work develops a discrete spectral framework for three-dimensional Lorentzian quasi-Fuchsian manifolds M = Γ\Ω_Γ, by extending the spacelike geodesic flow to a complete extended space and proving meromorphic continuations for its flow resolvent, yielding Ruelle resonances with finite-rank residues. It then constructs and meromorphically continues a Poincaré series D_λ via microlocal methods tied to the extended flow, establishing a direct link between the flow resolvent and the Poincaré series. A resolvent for the pseudo-Riemannian Laplacian □_g on M is defined through a Γ-periodization of the AdS_3 resolvent, and a quantum–classical correspondence is proved showing that quantum resonances (poles of R_{□_g}) live in translates of Ruelle resonances with corresponding resonant states given by pushforwards of Ruelle states. In the Fuchsian case, the authors provide a complete description of quantum resonances and resonant states, illustrating the robustness of the approach and its potential to yield a Selberg-type theory for pseudo-Riemannian locally symmetric spaces.

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.

Spectra of Lorentzian quasi-Fuchsian manifolds

TL;DR

This work develops a discrete spectral framework for three-dimensional Lorentzian quasi-Fuchsian manifolds M = Γ\Ω_Γ, by extending the spacelike geodesic flow to a complete extended space and proving meromorphic continuations for its flow resolvent, yielding Ruelle resonances with finite-rank residues. It then constructs and meromorphically continues a Poincaré series D_λ via microlocal methods tied to the extended flow, establishing a direct link between the flow resolvent and the Poincaré series. A resolvent for the pseudo-Riemannian Laplacian □_g on M is defined through a Γ-periodization of the AdS_3 resolvent, and a quantum–classical correspondence is proved showing that quantum resonances (poles of R_{□_g}) live in translates of Ruelle resonances with corresponding resonant states given by pushforwards of Ruelle states. In the Fuchsian case, the authors provide a complete description of quantum resonances and resonant states, illustrating the robustness of the approach and its potential to yield a Selberg-type theory for pseudo-Riemannian locally symmetric spaces.

Abstract

A three-dimensional quasi-Fuchsian Lorentzian manifold is a globally hyperbolic spacetime diffeomorphic to for a closed orientable surface of genus . It is the quotient of an open set by a discrete group of isometries of which is a particular example of an Anosov representation of . We first show that the spacelike geodesic flow of is Axiom A, has a discrete Ruelle resonance spectrum with associated (co-)resonant states, and that the Poincaré series for extend meromorphically to . This is then used to prove that there is a natural notion of resolvent of the pseudo-Riemannian Laplacian of , which is meromorphic on 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.
Paper Structure (42 sections, 58 theorems, 490 equations)

This paper contains 42 sections, 58 theorems, 490 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Spectral theory of the generator of the extended space-like geodesic flow
  4. Poincaré series
  5. Spectrum of the pseudo-Riemannian Laplacian
  6. Further previous results, comments and open questions
  7. Structure of the paper
  8. Geometry of anti-de Sitter space and its unit spacelike tangent bundle
  9. Pseudo-Riemannian geometry of anti-de Sitter space and its conformal boundary
  10. Isometries of anti-de Sitter space
  11. The torus coordinates
  12. The SL(2,R) coordinates
  13. The upper half-space coordinates
  14. Spacelike geodesic flow and geodesic distance on anti-de Sitter space
  15. Splittings of the tangent bundle of the unit spacelike tangent bundle
  16. ...and 27 more sections

Key Result

Theorem A

The resolvent $R_{X}(\lambda):C_c^\infty(\mathcal{M})\to \mathcal{D}'(\mathcal{M})$ admits a meromorphic continuation from ${\rm Re}(\lambda)>0$ to $\lambda\in \mathbb{C}$ with poles of finite rank, it has a simple pole at $\lambda_0:=h_{\rm top}-2$ where $h_{\rm top}$ is the topological entropy of with $u$ a measure supported on $\mathcal{K}_+$ and $v$ a measure supported on $\mathcal{K}_-$, whe

Theorems & Definitions (122)

  • Theorem A
  • Theorem B
  • Theorem C
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 112 more