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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 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.
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 7 (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
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 4 (0 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
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 3 (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 1 (0 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.
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 1 (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.
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 1 (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.
EQUIPARTITION OF ENERGY IN GEOMETRIC SCATTERING THEORY
"... Abstract. In this note, we use an elementary argument to show that the existence and unitarity of radiation fields implies asymptotic partition of energy for the corresponding wave equation. This argument establishes the equipartition of energy for the wave equation on scattering manifolds, asymptot ..."
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Abstract. In this note, we use an elementary argument to show that the existence and unitarity of radiation fields implies asymptotic partition of energy for the corresponding wave equation. This argument establishes the equipartition of energy for the wave equation on scattering manifolds, asymptotically hyperbolic manifolds, asymptotically complex hyperbolic manifolds, and the Schwarzschild spacetime. It also establishes equipartition of energy for the energycritical semilinear wave equation on R3. 1.