A three-dimensional quasi-Fuchsian Lorentzian manifold M is a globally hyperbolic spacetime diffeomorphic to <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Sigma \times (-1,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Σ</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for a closed orientable surface <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation> of genus <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. It is the quotient <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$M=\Gamma \backslash \Omega _\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi>M</mi> <mo>=</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="true">\</mo> </mrow> <msub> <mi mathvariant="normal">Ω</mi> <mi mathvariant="normal">Γ</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> of an open set <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Omega _\Gamma \subset \textrm{AdS}_3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Ω</mi> <mi mathvariant="normal">Γ</mi> </msub> <mo>⊂</mo> <msub> <mtext>AdS</mtext> <mn>3</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> by a discrete group <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> of isometries of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\textrm{AdS}_3$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>AdS</mtext> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation> which is a particular example of an Anosov representation of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\pi _1(\Sigma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>π</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We first show that the spacelike geodesic flow of M is Axiom A, has a discrete Ruelle resonance spectrum with associated (co-)resonant states, and that the Poincaré series for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> extend meromorphically to <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathbb {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">C</mi> </math></EquationSource> </InlineEquation>. This is then used to prove that there is a natural notion of resolvent of the pseudo-Riemannian Laplacian <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\Box $</EquationSource> <EquationSource Format="MATHML"><math> <mo>□</mo> </math></EquationSource> </InlineEquation> of M, which is meromorphic on <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mathbb {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">C</mi> </math></EquationSource> </InlineEquation> 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

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2026年8月1日

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Benjamin Delarue,
Colin Guillarmou,
Daniel Monclair

摘要

A three-dimensional quasi-Fuchsian Lorentzian manifold M is a globally hyperbolic spacetime diffeomorphic to $\Sigma \times (-1,1)$ $Σ \times (- 1, 1)$ for a closed orientable surface $\Sigma $ $Σ$ of genus $\ge 2$ $\geq 2$ . It is the quotient $M=\Gamma \backslash \Omega _\Gamma $ $M = Γ \ Ω_{Γ}$ of an open set $\Omega _\Gamma \subset \textrm{AdS}_3$ $Ω_{Γ} \subset {AdS}_{3}$ 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.