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Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds
 Gui05c] [Gui06] [GZ95] [GZ97] [GZ99] [His94] [His00] [Jan79] Colin Guillarmou. Resonances and
, 2005
"... Abstract. On an asymptotically hyperbolic manifold (Xn+1, g), Mazzeo and Melrose have constructed the meromorphic extension of the resolvent R(λ): = (∆g − λ(n − λ)) −1 for the Laplacian. However, there are special points on 1 (n − N) that they did not deal with. We 2 show that the points of n − N ar ..."
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Cited by 71 (17 self)
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Abstract. On an asymptotically hyperbolic manifold (Xn+1, g), Mazzeo and Melrose have constructed the meromorphic extension of the resolvent R(λ): = (∆g − λ(n − λ)) −1 for the Laplacian. However, there are special points on 1 (n − N) that they did not deal with. We 2 show that the points of n − N are at most some poles of finite multiplicity, and that the same 2 property holds for the points of n+1 − N if and only if the metric is ‘even’. On the other 2 hand, there exist some metrics for which R(λ) has an essential singularity on n+1 − N and 2 these cases are generic. At last, to illustrate them, we give some examples with a sequence of poles of R(λ) approaching an essential singularity.
Higher symmetries of the laplacian
, 206
"... Abstract. We identify the symmetry algebra of the Laplacian on Euclidean space as an explicit quotient of the universal enveloping algebra of the Lie algebra of conformal motions. We construct analogues of these symmetries on a general conformal manifold. 1. ..."
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Cited by 62 (4 self)
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Abstract. We identify the symmetry algebra of the Laplacian on Euclidean space as an explicit quotient of the universal enveloping algebra of the Lie algebra of conformal motions. We construct analogues of these symmetries on a general conformal manifold. 1.
On the rigidity of conformally compact Einstein manifolds
 Int. Math. Res. Not
"... Abstract. In this paper we prove that a conformally compact Einstein manifold with the round sphere as its conformal infinity has to be the hyperbolic space. We do not assume the manifolds to be spin, but our approach relies on the positive mass theorem for asymptotic flat manifolds. The proof is ba ..."
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Cited by 29 (10 self)
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Abstract. In this paper we prove that a conformally compact Einstein manifold with the round sphere as its conformal infinity has to be the hyperbolic space. We do not assume the manifolds to be spin, but our approach relies on the positive mass theorem for asymptotic flat manifolds. The proof is based on understanding of positive eigenfunctions and compactifications obtained by positive eigenfunctions. In this paper we study the rigidity problem for conformally compact Einstein manifolds with the round sphere as their conformal infinity. Quite recently there has been a great deal of interest in both physics and mathematics community in the socalled AntideSitter/Conformal Field Theory (in short AdS/CFT) correspondence.
Partition functions and doubletrace deformations in AdS/CFT,” [arXiv:hepth/0702163
"... Abstract: We study the effect of a relevant doubletrace deformation on the partition function (and conformal anomaly) of a CFTd at large N and its dual picture in AdSd+1. Three complementary previous results are brought into full agreement with each other: bulk [1] and boundary [2] computations, as ..."
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Cited by 28 (3 self)
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Abstract: We study the effect of a relevant doubletrace deformation on the partition function (and conformal anomaly) of a CFTd at large N and its dual picture in AdSd+1. Three complementary previous results are brought into full agreement with each other: bulk [1] and boundary [2] computations, as well as their formal identity [3]. We show the exact equality between the dimensionally regularized partition functions or, equivalently, fluctuational determinants involved. A series of results then follows: (i) equality between the renormalized partition functions for all d; (ii) for all even d, correction to the conformal anomaly; (iii) for even d, the mapping entails a mixing of UV and IR effects on the same side (bulk) of the duality, with no precedent in the leading order computations; and finally, (iv) a subtle relation between overall coefficients, volume renormalization and IRUV connection. All in all, we get a clean test of the AdS/CFT correspondence beyond the classical SUGRA approximation in the bulk and at subleading O(1) order in the largeN expansion on the boundary. Keywords: AdS/CFT, IRUV Connection, Conformal Anomaly. Contents
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 26 (5 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.
Wave 0trace and length spectrum on convex cocompact hyperbolic manifolds
 Comm. Anal. Geom
"... Abstract. For convex cocompact hyperbolic quotients Γ\H n+1 we obtain a formula relating the 0trace of the wave operator with the resonances and some conformal invariants of the boundary, generalizing a formula of Guillopé and Zworski in dimension 2. Then, by writing this 0trace with the length s ..."
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Cited by 25 (12 self)
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Abstract. For convex cocompact hyperbolic quotients Γ\H n+1 we obtain a formula relating the 0trace of the wave operator with the resonances and some conformal invariants of the boundary, generalizing a formula of Guillopé and Zworski in dimension 2. Then, by writing this 0trace with the length spectrum, we prove precise asymptotics of the number of closed geodesics with an effective, exponentially small error term when the dimension of the limit set of Γ is greater than n 2. 1.
Unique continuation results for Ricci curvature
"... Abstract. Unique continuation results are proved for metrics with prescribed Ricci curvature in the setting of bounded metrics on compact manifolds with boundary, and in the setting of complete conformally compact metrics on such manifolds. In addition, it is shown that the Ricci curvature forms an ..."
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Cited by 24 (16 self)
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Abstract. Unique continuation results are proved for metrics with prescribed Ricci curvature in the setting of bounded metrics on compact manifolds with boundary, and in the setting of complete conformally compact metrics on such manifolds. In addition, it is shown that the Ricci curvature forms an elliptic system in geodesicharmonic coordinates naturally associated with the boundary data. 0. Introduction. In this paper, we study certain issues related to the boundary behavior of metrics with prescribed Ricci curvature. Let M be a compact (n + 1)dimensional manifold with compact nonempty boundary ∂M. We consider two possible classes of Riemannian metrics g on M. First, g may extend smoothly to a Riemannian metric on the closure ¯ M = M ∪∂M, thus inducing a Riemannian
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.