Results 1  10
of
34
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 ..."
Abstract

Cited by 42 (13 self)
 Add to MetaCart
(Show Context)
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.
Inverse Scattering On Asymptotically Hyperbolic Manifolds
 ACTA MATH
, 1998
"... Scattering is defined on compact manifolds with boundary which are equipped with an asymptotically hyperbolic metric, g: A model form is established for such metrics close to the boundary. It is shown ..."
Abstract

Cited by 32 (2 self)
 Add to MetaCart
(Show Context)
Scattering is defined on compact manifolds with boundary which are equipped with an asymptotically hyperbolic metric, g: A model form is established for such metrics close to the boundary. It is shown
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 ..."
Abstract

Cited by 19 (15 self)
 Add to MetaCart
(Show Context)
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
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 ..."
Abstract

Cited by 18 (8 self)
 Add to MetaCart
(Show Context)
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.
Determinants Of Laplacians And Isopolar Metrics On Surfaces Of Infinite Area
 DUKE MATH. J
, 2001
"... We construct a determinant of the Laplacian for infinitearea surfaces which are hyperbolic near infinity and without cusps. In the case of a convex cocompact hyperbolic metric, the determinant can be related to the Selberg zeta function and thus shown to be an entire function of order two with ..."
Abstract

Cited by 17 (5 self)
 Add to MetaCart
(Show Context)
We construct a determinant of the Laplacian for infinitearea surfaces which are hyperbolic near infinity and without cusps. In the case of a convex cocompact hyperbolic metric, the determinant can be related to the Selberg zeta function and thus shown to be an entire function of order two with zeros at the eigenvalues and resonances of the Laplacian. In the hyperbolic near infinity case the determinant is analyzed through the zetaregularized relative determinant for a conformal metric perturbation. We establish that this relative determinant is a ratio of entire functions of order two with divisor corresponding to eigenvalues and resonances of the perturbed and unperturbed metrics. These results are applied to the problem of compactness in the smooth topology for the class of metrics with a given set of eigenvalues and resonances.
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 ..."
Abstract

Cited by 16 (8 self)
 Add to MetaCart
(Show Context)
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.
QCURVATURE AND POINCARÉ METRICS
 MATHEMATICAL RESEARCH LETTERS 9, 139–151 (2002)
, 2002
"... ..."
(Show Context)
Some results on the structure of conformally compact Einstein metrics
, 2005
"... The main result of this paper is that the space of conformally compact Einstein metrics on any given manifold is a smooth, infinite dimensional Banach manifold, provided it is nonempty, generalizing earlier work of GrahamLee and Biquard. We also prove full boundary regularity for such metrics in ..."
Abstract

Cited by 12 (7 self)
 Add to MetaCart
(Show Context)
The main result of this paper is that the space of conformally compact Einstein metrics on any given manifold is a smooth, infinite dimensional Banach manifold, provided it is nonempty, generalizing earlier work of GrahamLee and Biquard. We also prove full boundary regularity for such metrics in dimension 4 and a local existence and uniqueness theorem for such metrics with prescribed metric and stressenergy tensor at conformal infinity, again in dimension 4. This result also holds for LorentzianEinstein metrics with a positive cosmological constant.
Isoscattering Schottky Manifolds
 G. A. F. A
, 1999
"... . We exhibit pairs of infinitevolume, hyperbolic threemanifolds that have the same scattering poles and conformally equivalent boundaries, but which are not isometric. The examples are constructed using Schottky groups and the Sunada construction. In memory of Evsey Dyn'kin 1. Introduction ..."
Abstract

Cited by 9 (8 self)
 Add to MetaCart
. We exhibit pairs of infinitevolume, hyperbolic threemanifolds that have the same scattering poles and conformally equivalent boundaries, but which are not isometric. The examples are constructed using Schottky groups and the Sunada construction. In memory of Evsey Dyn'kin 1. Introduction The purpose of this paper is to explore the spectral geometry of the scattering operator by exhibiting examples of infinite volume hyperbolic threemanifolds that are `isoscattering' in a sense we will make precise, but have distinct geometries. To do so we will work with convex cocompact hyperbolic manifolds associated to Schottky groups of hyperbolic isometries. This class of manifolds, described in greater detail in what follows, may be thought of as interiors X of handlebodies X equipped with a metric that puts the boundary at metric infinity. The interior carries a hyperbolic structure, and the boundary, @ X , carries an induced conformal structure as a Riemann surface. Associated to the ...