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69
Homology of pseudodifferential operators on manifolds with corners I. Manifolds with boundary
, 1996
"... Respectfully dedicate to Professor M. Sato on the occasion of his 70th birthday Abstract. Let X be a compact manifold with boundary. Suppose that the boundary is fibred, φ: ∂X − → Y, and let x ∈ C ∞ (X) be a boundary defining function. This data fixes the space of ‘fibred cusp ’ vector fields, consi ..."
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Cited by 98 (21 self)
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Respectfully dedicate to Professor M. Sato on the occasion of his 70th birthday Abstract. Let X be a compact manifold with boundary. Suppose that the boundary is fibred, φ: ∂X − → Y, and let x ∈ C ∞ (X) be a boundary defining function. This data fixes the space of ‘fibred cusp ’ vector fields, consisting of those vector fields V on X satisfying V x = O(x 2) and which are tangent to the fibres of φ; it is a Lie algebra and C ∞ (X) module. This Lie algebra is quantized to the ‘small calculus ’ of pseudodifferential operators Ψ ∗ Φ (X). Mapping properties including boundedness, regularity, Fredholm condition and symbolic maps are discussed for this calculus. The spectrum of the Laplacian of an ‘exact fibred cusp ’ metric is analyzed as is the wavefront set associated to the calculus.
The Divisor of Selberg's Zeta Function for Kleinian Groups
 DUKE MATH. J
, 2000
"... We compute the divisor of Selberg's zeta function for convex cocompact, torsionfree discrete groups ; acting on a real hyperbolic space of dimension n +1. The divisor is determined by the eigenvalues and scattering poles of the Laplacian on X = ;nH n+1 together with the Euler characteristic of X ..."
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Cited by 69 (10 self)
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We compute the divisor of Selberg's zeta function for convex cocompact, torsionfree discrete groups ; acting on a real hyperbolic space of dimension n +1. The divisor is determined by the eigenvalues and scattering poles of the Laplacian on X = ;nH n+1 together with the Euler characteristic of X compactified to a manifold with boundary.Ifn is even, the singularities of the zeta funciton associated to the Euler characteristic of X are identified using work of Bunke and Olbrich.
Fredholm operators and Einstein metrics on conformally compact manifolds
"... Abstract. The main result of this paper is the existence of asymptotically hyperbolic Einstein metrics with prescribed conformal infinity sufficiently close to that of a given asymptotically hyperbolic Einstein metric with nonpositive curvature. If the conformal infinities are sufficiently smooth, t ..."
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Cited by 42 (2 self)
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Abstract. The main result of this paper is the existence of asymptotically hyperbolic Einstein metrics with prescribed conformal infinity sufficiently close to that of a given asymptotically hyperbolic Einstein metric with nonpositive curvature. If the conformal infinities are sufficiently smooth, the resulting Einstein metrics have optimal Hölder regularity at the boundary. The proof is based on sharp Fredholm theorems for selfadjoint geometric linear elliptic operators on asymptotically hyperbolic manifolds. 1.
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 42 (13 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.
Complete Manifolds With Positive Spectrum, II
, 2003
"... In this paper, we continued our investigation of complete manifolds whose spectrum of the Laplacian has an optimal positive lower bound. In particular, we proved a splitting type theorem for ndimensional manifolds that have a finite volume end. This can be viewed as a study of the equality case of ..."
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Cited by 27 (11 self)
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In this paper, we continued our investigation of complete manifolds whose spectrum of the Laplacian has an optimal positive lower bound. In particular, we proved a splitting type theorem for ndimensional manifolds that have a finite volume end. This can be viewed as a study of the equality case of a theorem of Cheng.
the spectrum of an asymptotically hyperbolic Einstein manifold
 Comm. Anal. Geom
, 1995
"... Abstract. This paper relates the spectrum of the scalar Laplacian of an asymptotically hyperbolic Einstein metric to the conformal geometry of its “ideal boundary” at infinity. It follows from work of R. Mazzeo that the essential spectrum of such a metric on an (n + 1)dimensional manifold is the ra ..."
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Cited by 24 (3 self)
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Abstract. This paper relates the spectrum of the scalar Laplacian of an asymptotically hyperbolic Einstein metric to the conformal geometry of its “ideal boundary” at infinity. It follows from work of R. Mazzeo that the essential spectrum of such a metric on an (n + 1)dimensional manifold is the ray [n 2 /4, ∞), with no embedded eigenvalues; however, in general there may be discrete eigenvalues below the continuous spectrum. The main result of this paper is that, if the Yamabe invariant of the conformal structure on the boundary is nonnegative, then there are no such eigenvalues. This generalizes results of R. Schoen, S.T. Yau, and D. Sullivan for the case of hyperbolic manifolds. 1.
Boundary regularity of conformally compact Einstein metrics
 J. DIFFERENTIAL GEOM
, 2005
"... We show that C² conformally compact Riemannian Einstein metrics have conformal compactifications that are smooth up to the boundary in dimension 3 and all even dimensions, and polyhomogeneous in odd dimensions greater than 3. ..."
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Cited by 19 (2 self)
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We show that C² conformally compact Riemannian Einstein metrics have conformal compactifications that are smooth up to the boundary in dimension 3 and all even dimensions, and polyhomogeneous in odd dimensions greater than 3.
Scattering poles for asymptotically hyperbolic manifolds
 Trans. Amer. Math. Soc
"... Abstract. For a class of manifolds X that includes quotients of real hyperbolic (n + 1)dimensional space by a convex cocompact discrete group, we show that the resonances of the meromorphically continued resolvent kernel for the Laplacian on X coincide, with multiplicities, with the poles of the m ..."
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Cited by 18 (8 self)
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Abstract. For a class of manifolds X that includes quotients of real hyperbolic (n + 1)dimensional space by a convex cocompact discrete group, we show that the resonances of the meromorphically continued resolvent kernel for the Laplacian on X coincide, with multiplicities, with the poles of the meromorphically continued scattering operator for X. In order to carry out the proof, we use Shmuel Agmon’s perturbation theory of resonances to show that both resolvent resonances and scattering poles are simple for generic potential perturbations. 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 17 (6 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.