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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.
FORWARD AND INVERSE SCATTERING ON MANIFOLDS WITH ASYMPTOTICALLY CYLINDRICAL ENDS
, 905
"... Abstract. We study an inverse problem for a noncompact Riemannian manifold whose ends have the following properties: On each end, the Riemannian metric is assumed to be a shortrange perturbation of the metric of the form (dy) 2 + h(x, dx), h(x, dx) being the metric of some compact manifold of codi ..."
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Cited by 13 (4 self)
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Abstract. We study an inverse problem for a noncompact Riemannian manifold whose ends have the following properties: On each end, the Riemannian metric is assumed to be a shortrange perturbation of the metric of the form (dy) 2 + h(x, dx), h(x, dx) being the metric of some compact manifold of codimension 1. Moreover one end is exactly cylindrical, i.e. the metric is equal to (dy) 2 + h(x, dx). Given two such manifolds having the same scattering matrix on that exactly cylindrical end for all energy, we show that these two manifolds are isometric. 1.
A Conformally Invariant Holographic Two–Point Function on the Berger Sphere
, 2004
"... We apply our previous work on Green’s functions for the four–dimensional quaternionic Taub–NUT manifold to obtain a scalar two–point function on the homogeneously squashed three–sphere (otherwise known as the Berger sphere), which lies at its conformal infinity. Using basic notions from conformal ge ..."
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Cited by 11 (0 self)
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We apply our previous work on Green’s functions for the four–dimensional quaternionic Taub–NUT manifold to obtain a scalar two–point function on the homogeneously squashed three–sphere (otherwise known as the Berger sphere), which lies at its conformal infinity. Using basic notions from conformal geometry and the theory of boundary value problems, in particular the Dirichlet–to–Robin operator, we establish that our two–point correlation function is conformally invariant and corresponds to a boundary operator of conformal dimension one. It is plausible that the methods we use could have more general
The radiation field is a Fourier integral operator
 Ann. Inst. Fourier (Grenoble
, 2005
"... In this note, we exhibit explicitly the form of the “radiation field ” of F. G. Friedlander on two different types of manifolds: scattering manifolds, and asymptotically hyperbolic manifolds. The former class consists of manifolds with ends that look asymptotically like the large ends of cones, and ..."
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Cited by 10 (1 self)
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In this note, we exhibit explicitly the form of the “radiation field ” of F. G. Friedlander on two different types of manifolds: scattering manifolds, and asymptotically hyperbolic manifolds. The former class consists of manifolds with ends that look asymptotically like the large ends of cones, and
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 9 (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
SCATTERING AND INVERSE SCATTERING ON ACH MANIFOLDS
, 2006
"... We study scattering and inverse scattering theories for asymptotically complex hyperbolic manifolds. We show the existence of the scattering operator as a meromorphic family of operators in the Heisenberg calculus on the boundary, which is a contact manifold with a pseudohermitian structure. Then ..."
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Cited by 7 (1 self)
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We study scattering and inverse scattering theories for asymptotically complex hyperbolic manifolds. We show the existence of the scattering operator as a meromorphic family of operators in the Heisenberg calculus on the boundary, which is a contact manifold with a pseudohermitian structure. Then we define radiation fields as in the real asymptotically hyperbolic case, and reconstruct the scattering operator from those fields. As an application we show that the manifold, including its topology and the metric, are determined up to invariants by the scattering matrix at all energies.
Introduction to spectral theory and inverse problems on asymptotically hyperbolic manifolds, preprint
, 2011
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A SUPPORT THEOREM FOR THE RADIATION FIELDS ON ASYMPTOTICALLY EUCLIDEAN MANIFOLDS
, 709
"... We prove a support theorem for the radiation fields on asymptotically Euclidean manifolds with metrics which are warped products near infinity. It generalizes to this setting the well known support theorem for the Radon transform in R n. The main reason we are interested in proving such a theorem is ..."
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Cited by 6 (1 self)
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We prove a support theorem for the radiation fields on asymptotically Euclidean manifolds with metrics which are warped products near infinity. It generalizes to this setting the well known support theorem for the Radon transform in R n. The main reason we are interested in proving such a theorem is the possible application to the problem of reconstructing an asymptotically Euclidean manifold from the scattering
RADIATION FIELDS ON SCHWARZSCHILD SPACETIME
"... Abstract. In this paper we define the radiation field for the wave equation on the Schwarzschild black hole spacetime. In this context it has two components: the rescaled restriction of the time derivative of a solution to null infinity and to the event horizon. In the process, we establish some reg ..."
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Cited by 3 (1 self)
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Abstract. In this paper we define the radiation field for the wave equation on the Schwarzschild black hole spacetime. In this context it has two components: the rescaled restriction of the time derivative of a solution to null infinity and to the event horizon. In the process, we establish some regularity properties of solutions of the wave equation on the spacetime. In particular, we prove that the regularity of the solution across the event horizon and across null infinity is determined by the regularity and decay rate of the initial data at the event horizon and at infinity. We also show that the radiation field is unitary with respect to the conserved energy and prove support theorems for each piece of the radiation field. 1.
Radiation fields for semilinear wave equations
 In preparation
, 2012
"... Abstract. We define the radiation fields of solutions to critical semilinear wave equations in R 3 and use them to define the scattering operator. We also prove a support theorem for the radiation fields with radial initial data. This extends the well known support theorem for the Radon transform to ..."
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Cited by 2 (1 self)
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Abstract. We define the radiation fields of solutions to critical semilinear wave equations in R 3 and use them to define the scattering operator. We also prove a support theorem for the radiation fields with radial initial data. This extends the well known support theorem for the Radon transform to this setting and can also be interpreted as a PaleyWiener theorem for the distorted nonlinear Fourier transform of radial functions. 1.