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Upper and lower bounds on resonances for manifolds hyperbolic near infinity
 Comm. Part. Dif. Eq
"... Abstract. For a conformally compact manifold that is hyperbolic near infinity and of dimension n + 1, we complete the proof of the optimal O(rn+1) upper bound on the resonance counting function, correcting a mistake in the existing literature. In the case of a compactly supported perturbation of a h ..."
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Cited by 18 (10 self)
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Abstract. For a conformally compact manifold that is hyperbolic near infinity and of dimension n + 1, we complete the proof of the optimal O(rn+1) upper bound on the resonance counting function, correcting a mistake in the existing literature. In the case of a compactly supported perturbation of a hyperbolic manifold, we establish a Poisson formula expressing the regularized wave trace as a sum over scattering resonances. This leads to an rn+1 lower bound on the counting function for scattering poles. Contents
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 16 (6 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.
Wave decay on convex cocompact hyperbolic manifolds
 Comm. Math. Phys
, 2009
"... Abstract. For convex cocompact hyperbolic quotients X = Γ\H n+1, we analyze the longtime asymptotic of the solution of the wave equation u(t) with smooth compactly supported initial data f = (f0, f1). We show that, if the Hausdorff dimension δ of the limit set is less than n/2, then u(t) = Cδ(f)e ..."
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Cited by 11 (2 self)
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Abstract. For convex cocompact hyperbolic quotients X = Γ\H n+1, we analyze the longtime asymptotic of the solution of the wave equation u(t) with smooth compactly supported initial data f = (f0, f1). We show that, if the Hausdorff dimension δ of the limit set is less than n/2, then u(t) = Cδ(f)e (δ − n 2)t /Γ(δ − n/2 + 1) + e (δ − n 2)t R(t) where Cδ(f) ∈ C ∞ (X) and R(t)  = O(t − ∞). We explain, in terms of conformal theory of the conformal infinity of X, the special cases δ ∈ n/2 − N where the leading asymptotic term vanishes. In a second part, we show for all ǫ> 0 the existence of an infinite number of resonances (and thus zeros of Selberg zeta function) in the strip {−nδ − ǫ < Re(λ) < δ}. As a byproduct we obtain a lower bound on the remainder R(t) for generic initial data f. 1.
INVERSE SCATTERING RESULTS FOR MANIFOLDS HYPERBOLIC NEAR INFINITY
"... Abstract. We study the inverse resonance problem for conformally compact manifolds which are hyperbolic outside a compact set. Our results include compactness of isoresonant metrics in dimension two and of isophasal negatively curved metrics in dimension three. In dimensions four or higher we prove ..."
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Cited by 8 (0 self)
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Abstract. We study the inverse resonance problem for conformally compact manifolds which are hyperbolic outside a compact set. Our results include compactness of isoresonant metrics in dimension two and of isophasal negatively curved metrics in dimension three. In dimensions four or higher we prove topological finiteness theorems under the negative curvature assumption. Contents
On the local Nirenberg problem for the Qcurvatures
 Pacific J. Math
"... Abstract. The local image of each conformal Qcurvature operator on the sphere admits no scalar constraint although identities of Kazdan–Warner type hold for its graph. 1. ..."
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Cited by 8 (1 self)
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Abstract. The local image of each conformal Qcurvature operator on the sphere admits no scalar constraint although identities of Kazdan–Warner type hold for its graph. 1.
RESOLVENT OF THE LAPLACIAN ON GEOMETRICALLY FINITE HYPERBOLIC MANIFOLDS
"... Abstract. For geometrically finite hyperbolic manifolds Γ\Hn+1, we prove the meromorphic extension of the resolvent of Laplacian, Poincare ́ series, Einsenstein series and scattering operator to the whole complex plane. We also deduce the asymptotics of lattice points of Γ in large balls of Hn+1 in ..."
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Cited by 6 (1 self)
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Abstract. For geometrically finite hyperbolic manifolds Γ\Hn+1, we prove the meromorphic extension of the resolvent of Laplacian, Poincare ́ series, Einsenstein series and scattering operator to the whole complex plane. We also deduce the asymptotics of lattice points of Γ in large balls of Hn+1 in terms of the Hausdorff dimension of the limit set of Γ. 1.
Generalized Krein formula and determinants for PoincaréEinstein manifolds
 Amer. J. Math
"... Abstract. We define for a class of even dimensional asymptotically hyperbolic manifold (X, g) a natural generalized Krein spectral function ξ (on R) with derivative the renormalized trace of the spectral measure of the Laplacian. It is related to the scattering operator S(λ) of ∆g by −2πi∂zξ(z) = T ..."
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Cited by 4 (0 self)
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Abstract. We define for a class of even dimensional asymptotically hyperbolic manifold (X, g) a natural generalized Krein spectral function ξ (on R) with derivative the renormalized trace of the spectral measure of the Laplacian. It is related to the scattering operator S(λ) of ∆g by −2πi∂zξ(z) = TR(∂zS ( n 2 + iz)S−1 ( n − iz)) where TR is the KontsevichVishik 2 trace. For even PoincaréEinstein metrics, we define the determinant of S(λ) using methods of KontsevichVishik and show that it is a conformal invariant of the conformal boundary (M,[h0]) depending meromorphically on λ, with divisors given by the resonances multiplicity and the dimensions of kernels of the conformal Laplacians (Pk)k∈N of [h0]. We finally prove that ξ is the phase of det S(λ) on the essential spectrum, we compute the determinant of Pk with respect to ξ and, as an application, det Pk is expressed explicitly in term of the Selberg zeta function for convex cocompact hyperbolic manifolds. 1.
Generalized Krein formula and determinants for even dimensional Poincaré–Einstein manifolds
"... Abstract. We define for a class of even dimensional asymptotically hyperbolic manifold (X, g) a natural generalized Krein spectral function ξ (on R) with derivative the renormalized trace of the spectral measure of the Laplacian. It is related to the scattering operator S(λ) of ∆g by −2πi∂zξ(z) = T ..."
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Cited by 4 (0 self)
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Abstract. We define for a class of even dimensional asymptotically hyperbolic manifold (X, g) a natural generalized Krein spectral function ξ (on R) with derivative the renormalized trace of the spectral measure of the Laplacian. It is related to the scattering operator S(λ) of ∆g by −2πi∂zξ(z) = TR(∂zS ( n 2 + iz)S−1 ( n + iz)) where TR is the KontsevichVishik 2 trace. For even PoincaréEinstein metrics, we define the determinant of S(λ) using methods of KontsevichVishik and show that it is a conformal invariant of the conformal boundary (M,[h0]) depending meromorphically on λ, with divisors given by the resonances multiplicity and the dimensions of kernels of the conformal Laplacians (Pk)k∈N of [h0]. We finally prove that ξ is the phase of det S(λ) on the essential spectrum, we compute the determinant of Pk with respect to ξ and, as an application, det Pk is expressed explicitly in term of the Selberg zeta function for convex cocompact hyperbolic manifolds. 1.
RESONANCES FOR MANIFOLDS HYPERBOLIC NEAR INFINITY: OPTIMAL LOWER BOUNDS ON ORDER OF GROWTH
"... Abstract. Suppose that (X, g) is a conformally compact (n+1)dimensional manifold that is hyperbolic near infinity in the sense that the sectional curvatures of g are identically equal to minus one outside of a compact set K ⊂ X. We prove that the counting function for the resolvent resonances has m ..."
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Cited by 4 (4 self)
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Abstract. Suppose that (X, g) is a conformally compact (n+1)dimensional manifold that is hyperbolic near infinity in the sense that the sectional curvatures of g are identically equal to minus one outside of a compact set K ⊂ X. We prove that the counting function for the resolvent resonances has maximal order of growth (n + 1) generically for such manifolds. This is achieved by constructing explicit examples of manifolds hyperbolic at infinity for which the resonance counting function obeys optimal lower bounds. Contents
ON THE CRITICAL LINE OF CONVEX COCOMPACT HYPERBOLIC SURFACES
"... Abstract. Let Γ be a convex cocompact Fuchsian group. We formulate a conjecture on the critical line i.e. what is the largest halfplane with finitely many resonances for the Laplace operator on the infinite area hyperbolic surface X = Γ\H2. An upper bound depending on the dimension δ of the limit ..."
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Cited by 3 (2 self)
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Abstract. Let Γ be a convex cocompact Fuchsian group. We formulate a conjecture on the critical line i.e. what is the largest halfplane with finitely many resonances for the Laplace operator on the infinite area hyperbolic surface X = Γ\H2. An upper bound depending on the dimension δ of the limit set is proved which is in favor of the conjecture for small values of δ and in the case when δ> 1 2 and Γ is a subgroup of an arithmetic group. New omega lower bounds for the error term in the hyperbolic lattice point counting problem are derived. 1.