Results 1  10
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27
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 21 (2 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.
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.
The decomposition of global conformal invariants: On a conjecture of Deser and
"... We present a proof of a conjecture of Deser and Schwimmer regarding the algebraic structure of “global conformal invariants”; these are defined to be scalar quantities whose integrals over compact manifolds remain invariant under conformal changes of the underlying metric. We prove that any such inv ..."
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We present a proof of a conjecture of Deser and Schwimmer regarding the algebraic structure of “global conformal invariants”; these are defined to be scalar quantities whose integrals over compact manifolds remain invariant under conformal changes of the underlying metric. We prove that any such invariant can be expressed as a linear combination of a local conformal invariant, a divergence, and the ChernGaussBonnet integrand. 1 Introduction. This work is a continuation of [2] and [3], where we confirm a conjecture of Deser and Schwimmer, originally formulated in [5]. The purpose of this introduction is to firstly provide a formulation of the conjecture and discuss some of its implications, and then to give a a very brief synopsis of some of the main ideas
The decomposition of Global Conformal Invariants I
"... This is the first of two papers where we address and partially confirm a conjecture of Deser and Schwimmer, originally postulated in high energy physics. The objects of study are scalar Riemannian quantities constructed out of the curvature and its covariant derivatives, whose integrals over compact ..."
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Cited by 12 (3 self)
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This is the first of two papers where we address and partially confirm a conjecture of Deser and Schwimmer, originally postulated in high energy physics. The objects of study are scalar Riemannian quantities constructed out of the curvature and its covariant derivatives, whose integrals over compact manifolds are invariant under conformal changes of the underlying metric. Our main conclusion is that each such quantity that locally depends only on the curvature tensor (without covariant derivatives) can be written as a linear combination of the ChernGaussBonnet integrand and a scalar conformal invariant. 1
Renormalized area and properly embedded minimal surfaces in hyperbolic 3manifolds
 Commun. Math. Phys
, 2010
"... Abstract We study the renormalized area functional A in the AdS/CFT correspondence, specifically for properly embedded minimal surfaces in convex cocompact hyperbolic 3manifolds (or somewhat more broadly, PoincaréEinstein spaces). Our main results include an explicit formula for the renormalized ..."
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Cited by 10 (2 self)
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Abstract We study the renormalized area functional A in the AdS/CFT correspondence, specifically for properly embedded minimal surfaces in convex cocompact hyperbolic 3manifolds (or somewhat more broadly, PoincaréEinstein spaces). Our main results include an explicit formula for the renormalized area of such a minimal surface Y as an integral of local geometric quantities, as well as formulae for the first and second variations of A which are given by integrals of global quantities over the asymptotic boundary loop γ of Y . All of these formulae are also obtained for a broader class of nonminimal surfaces. The proper setting for the study of this functional (when the ambient space is hyperbolic) requires an understanding of the moduli space of all properly embedded minimal surfaces with smoothly embedded asymptotic boundary. We show that this moduli space is a smooth Banach manifold and develop a Zvalued degree theory for the natural map taking a minimal surface to its boundary curve. We characterize the nondegenerate critical points of A for minimal surfaces in H 3 , and finally, discuss the relationship of A to the Willmore functional.
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.
FAMILIES INDEX FOR MANIFOLDS WITH HYPERBOLIC CUSP SINGULARITIES
, 2008
"... Manifolds with fibered hyperbolic cusp metrics include hyperbolic manifolds with cusps and locally symmetric spaces of Qrank one. We extend Vaillant’s treatment of Diractype operators associated to these metrics by weaking the hypotheses on the boundary families through the use of Fredholm perturb ..."
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Cited by 4 (2 self)
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Manifolds with fibered hyperbolic cusp metrics include hyperbolic manifolds with cusps and locally symmetric spaces of Qrank one. We extend Vaillant’s treatment of Diractype operators associated to these metrics by weaking the hypotheses on the boundary families through the use of Fredholm perturbations as in the family index theorem of Melrose and Piazza and by treating the index of families of such operators. We also extend the index theorem of Moroianu and LeichtnamMazzeoPiazza to families of perturbed Diractype operators associated to fibered cusp metrics (sometimes known as fibered boundary metrics).
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.
CHERNSIMONS LINE BUNDLE ON TEICHMÜLLER SPACE
"... Abstract. Let X be a noncompact geometrically finite hyperbolic 3manifold without cusps of rank 1. The deformation space H of X can be identified with the Teichmüller space T of the conformal boundary of X as the graph of a section in T ∗ T. We construct a Hermitian holomorphic line bundle L on T, ..."
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Cited by 3 (1 self)
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Abstract. Let X be a noncompact geometrically finite hyperbolic 3manifold without cusps of rank 1. The deformation space H of X can be identified with the Teichmüller space T of the conformal boundary of X as the graph of a section in T ∗ T. We construct a Hermitian holomorphic line bundle L on T, with curvature equal to a multiple of the WeilPetersson symplectic form. This bundle has a canonical holomorphic section defined by e 1 π Vol R(X)+2πiCS(X) where VolR(X) is the renormalized volume of X and CS(X) is the ChernSimons invariant of X. This section is parallel on H for the Hermitian connection modified by the (1, 0) component of the Liouville form on T ∗ T. As applications, we deduce that H is Lagrangian in T ∗ T, and that VolR(X) is a Kähler potential for the WeilPetersson metric on T and on its quotient by a certain subgroup of the mapping class group. For the Schottky uniformisation, we use a formula of Zograf to construct an explicit isomorphism of holomorphic Hermitian line bundles between L −1 and the sixth power of the determinant line bundle. 1.
DYNAMICS OF ASYMPTOTICALLY HYPERBOLIC MANIFOLDS
, 809
"... Abstract. We prove a dynamical trace formula for asymptotically hyperbolic (n+1) dimensional manifolds with negative (but not necessarily constant) sectional curvatures which equates the renormalized wave trace to the lengths of closed geodesics. This result generalizes the classical theorem of Duis ..."
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Cited by 2 (1 self)
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Abstract. We prove a dynamical trace formula for asymptotically hyperbolic (n+1) dimensional manifolds with negative (but not necessarily constant) sectional curvatures which equates the renormalized wave trace to the lengths of closed geodesics. This result generalizes the classical theorem of DuistermaatGuillemin for compact manifolds and the results of [37], [38], and [58] for hyperbolic manifolds with infinite volume. A corollary of this dynamical trace formula is a Selberg trace formula for compact perturbations of convex cocompact hyperbolic manifolds which we use to prove a growth estimate for the length spectrum counting function. We then define a dynamical zeta function and prove its analyticity in a half plane. Based on the work of Eberlein [24], [25] and EberleinO’Neill [27] for complete open manifolds with negative sectional curvature known as “visibility manifolds ” we show that the geodesic flow restricted to the nonwandering set satisfies the hypotheses of Parry and Pollicott’s [56] Axiom A flows restricted to a basic set. We then apply their results to prove a prime orbit theorem for the geodesic flow. As a corollary to the prime orbit theorem, for compact perturbations of convex cocompact hyperbolic manifolds we show that the existence of pure point spectrum for the Laplacian is related to the dynamics of the geodesic flow. 1.