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42
Microlocal analysis of asymptotically hyperbolic and Kerr–de Sitter spaces, preprint
"... Abstract. In this paper we develop a general, systematic, microlocal framework for the Fredholm analysis of nonelliptic problems, including high energy (or semiclassical) estimates, which is stable under perturbations. This framework, described in Section 2, is relatively simple given modern microl ..."
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Cited by 19 (7 self)
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Abstract. In this paper we develop a general, systematic, microlocal framework for the Fredholm analysis of nonelliptic problems, including high energy (or semiclassical) estimates, which is stable under perturbations. This framework, described in Section 2, is relatively simple given modern microlocal analysis, and only takes a bit over a dozen pages after the statement of notation. It resides on a compact manifold without boundary, hence in the standard setting of microlocal analysis, including semiclassical analysis. The rest of the paper is devoted to applications. Many natural applications arise in the setting of nonRiemannian bmetrics in the context of Melrose’s bstructures. These include asymptotically Minkowski metrics, asymptotically de Sittertype metrics on a blowup of the natural compactification and Kerrde Sittertype metrics. The simplest application, however, is to provide a new approach to analysis on Riemannian or Lorentzian (or indeed, possibly of other signature) conformally compact spaces (such as asymptotically hyperbolic or de Sitter spaces). The results include, in particular, a new construction of the meromorphic extension of the resolvent of the Laplacian in the Riemannian case, as well as high energy estimates for the spectral parameter in strips of the complex plane. For these results, only Section 2 and Section 4.44.9, starting with the paragraph of (4.8), are strictly needed. The appendix written by Dyatlov relates his analysis of resonances on exact Kerrde Sitter space (which then was used to analyze the wave equation in that setting) to the more general method described here. 1.
Resonances and scattering poles on asymptotically hyperbolic manifolds
 Math. Res. Lett
"... Abstract. On an asymptotically hyperbolic manifold (X, g), we show that the poles (called resonances) of the meromorphic extension of the resolvent (∆g − λ(n − λ)) −1 coincide, with multiplicities, with the poles (called scattering poles) of the renormalized scattering operator, except for the point ..."
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Cited by 16 (8 self)
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Abstract. On an asymptotically hyperbolic manifold (X, g), we show that the poles (called resonances) of the meromorphic extension of the resolvent (∆g − λ(n − λ)) −1 coincide, with multiplicities, with the poles (called scattering poles) of the renormalized scattering operator, except for the points of n 2 − N. At each λk: = n − k with k ∈ N, the resonance multiplicity 2 m(λk) and the scattering pole multiplicity ν(λk) do not always coincide: ν(λk) − m(λk) is the dimension of the kernel of a differential operator on the boundary ∂ ¯ X introduced by Graham and Zworski; in the asymptotically Einstein case, this operator is the kth conformal Laplacian. 1.
Renormalizing curvature integrals on PoincaréEinstein manifolds
, 2005
"... After analyzing renormalization schemes on a PoincaréEinstein manifold, we study the renormalized integrals of scalar Riemannian invariants. The behavior of the renormalized volume is wellknown, and we show any scalar Riemannian invariant renormalizes similarly. We consider characteristic forms ..."
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Cited by 15 (2 self)
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After analyzing renormalization schemes on a PoincaréEinstein manifold, we study the renormalized integrals of scalar Riemannian invariants. The behavior of the renormalized volume is wellknown, and we show any scalar Riemannian invariant renormalizes similarly. We consider characteristic forms and their behavior under a variation of the PoincaréEinstein structure, and obtain, from the renormalized integral of the Pfaffian, an extension of the GaussBonnet theorem.
Generalized Krein formula, Determinants and Selberg zeta function in even dimension
 Amer. J. Math
"... Abstract. For a class of even dimensional asymptotically hyperbolic (AH) manifolds, we develop a generalized BirmanKrein theory to study scattering asymptotics and, when the curvature is constant, to analyze Selberg zeta function. The main objects we construct for an AH manifold (X, g) are, on the ..."
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Cited by 13 (5 self)
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Abstract. For a class of even dimensional asymptotically hyperbolic (AH) manifolds, we develop a generalized BirmanKrein theory to study scattering asymptotics and, when the curvature is constant, to analyze Selberg zeta function. The main objects we construct for an AH manifold (X, g) are, on the first hand, a natural spectral function ξ for the Laplacian ∆g, which replaces the counting function of the eigenvalues in this infinite volume case, and on the other hand the determinant of the scattering operator SX(λ) of ∆g on X. Both need to be defined through regularized functional: renormalized trace on the bulk X and regularized determinant on the conformal infinity ( ∂ ¯ X,[h0]). We show that det SX(λ) is meromorphic in λ, with divisors given by resonance multiplicities and dimensions of kernels of GJMS conformal Laplacians (Pk)k∈N of ( ∂ ¯ X,[h0]), moreover ξ(z) is proved to be the phase of det SX ( n 2 + iz) on the essential spectrum {z ∈ R+}. Applying this theory to convex cocompact quotients X = Γ\Hn+1 of hyperbolic space Hn+1, we obtain the functional equation Z(λ)/Z(n − λ) = (det SHn+1(λ)) χ(X) /det SX(λ) for Selberg zeta function Z(λ) of X, where χ(X) is the Euler characteristic of X. This describes the poles and zeros of Z(λ), computes det Pk in term of Z ( n n − k)/Z ( + k) and implies a sharp Weyl asymptotic for ξ(z). 2 2 1.
The wave equation on asymptotically de Sitterlike spaces
, 2007
"... Abstract. In this paper we obtain the asymptotic behavior of solutions of the KleinGordon equation on Lorentzian manifolds (X ◦ , g) which are de Sitterlike at infinity. Such manifolds are Lorentzian analogues of the socalled Riemannian conformally compact (or asymptotically hyperbolic) spaces. Un ..."
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Cited by 11 (6 self)
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Abstract. In this paper we obtain the asymptotic behavior of solutions of the KleinGordon equation on Lorentzian manifolds (X ◦ , g) which are de Sitterlike at infinity. Such manifolds are Lorentzian analogues of the socalled Riemannian conformally compact (or asymptotically hyperbolic) spaces. Under global assumptions on the (null)bicharacteristic flow, namely that the boundary of the compactification X is a union of two disjoint manifolds, Y±, and each bicharacteristic converges to one of these two manifolds as the parameter along the bicharacteristic goes to +∞, and to the other manifold as the parameter goes to −∞, we also define the scattering operator, and show that it is a Fourier integral operator associated to the bicharacteristic flow from Y+ to Y−. 1.
GLUING SEMICLASSICAL RESOLVENT ESTIMATES VIA PROPAGATION OF SINGULARITIES
"... Abstract. We use semiclassical propagation of singularities to give a general method for gluing together resolvent estimates. As an application we prove estimates for the analytic continuation of the resolvent of a Schrödinger operator for certain asymptotically hyperbolic manifolds in the presence ..."
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Cited by 10 (5 self)
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Abstract. We use semiclassical propagation of singularities to give a general method for gluing together resolvent estimates. As an application we prove estimates for the analytic continuation of the resolvent of a Schrödinger operator for certain asymptotically hyperbolic manifolds in the presence of trapping which is sufficiently mild in one of several senses. As a corollary we obtain local exponential decay for the wave propagator and local smoothing for the Schrödinger propagator. 1.
A renormalized index theorem for some complete asymptotically regular metrics: the GaussBonnet theorem
, 2005
"... The GaussBonnet Theorem is studied for edge metrics as a renormalized index theorem. These metrics include the PoincaréEinstein metrics of the AdS/CFT correspondence. Renormalization is used to make sense of the curvature integral and the dimensions of the L²cohomology spaces as well as to carry ..."
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Cited by 8 (1 self)
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The GaussBonnet Theorem is studied for edge metrics as a renormalized index theorem. These metrics include the PoincaréEinstein metrics of the AdS/CFT correspondence. Renormalization is used to make sense of the curvature integral and the dimensions of the L²cohomology spaces as well as to carry out the heat equation proof of the index theorem. For conformally compact metrics even mod x m, the finite time supertrace of the heat kernel on conformally compact manifolds is shown to renormalize independently of the choice of special boundary defining function.
Radiation fields, scattering and inverse scattering on asymptotically hyperbolic manifolds
, 2004
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PROPAGATION THROUGH TRAPPED SETS AND SEMICLASSICAL RESOLVENT ESTIMATES
"... Abstract. Motivated by the study of resolvent estimates in the presence of trapping, we prove a semiclassical propagation theorem in a neighborhood of a compact invariant subset of the bicharacteristic flow which is isolated in a suitable sense. Examples include a global trapped set and a single iso ..."
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Cited by 8 (3 self)
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Abstract. Motivated by the study of resolvent estimates in the presence of trapping, we prove a semiclassical propagation theorem in a neighborhood of a compact invariant subset of the bicharacteristic flow which is isolated in a suitable sense. Examples include a global trapped set and a single isolated periodic trajectory. This is applied to obtain microlocal resolvent estimates with no loss compared to the nontrapping setting. 1.
The determinant of the DirichlettoNeumann map for surfaces with boundary
, 2007
"... For any orientable compact surface with boundary, we compute the regularized determinant of the DirichlettoNeumann (DN) map in terms of particular values of dynamical zeta functions by using natural uniformizations, one due to MazzeoTaylor, the other to OsgoodPhillipsSarnak. We also relate in ..."
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Cited by 6 (2 self)
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For any orientable compact surface with boundary, we compute the regularized determinant of the DirichlettoNeumann (DN) map in terms of particular values of dynamical zeta functions by using natural uniformizations, one due to MazzeoTaylor, the other to OsgoodPhillipsSarnak. We also relate in any dimension the DN map for the Yamabe operator to the scattering operator for a conformally compact related problem by using uniformization.