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67
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.
Pseudodifferential Operators on Manifolds with A LIE STRUCTURE AT INFINITY
, 2003
"... Several interesting examples of noncompact manifolds M0 whose geometry at infinity is described by Lie algebras of vector fields V ⊂ Γ(M; T M) (on a compactification of M0 to a manifold with corners M) were studied for instance in [28, 31, 46]. In [1], the geometry of manifolds described by Lie alg ..."
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Cited by 28 (13 self)
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Several interesting examples of noncompact manifolds M0 whose geometry at infinity is described by Lie algebras of vector fields V ⊂ Γ(M; T M) (on a compactification of M0 to a manifold with corners M) were studied for instance in [28, 31, 46]. In [1], the geometry of manifolds described by Lie algebras of vector fields – baptised “manifolds with a Lie structure at infinity ” there – was studied from an axiomatic point of view. In this paper, we define and study the algebra Ψ ∞ 1,0,V (M0), which is an algebra of pseudodifferential operators canonically associated to a manifold M0 with the Lie structure at infinity V ⊂ Γ(M; T M). We show that many of the properties of the usual algebra of pseudodifferential operators on a compact manifold extend to Ψ ∞ 1,0,V (M0). We also consider the algebra Diff ∗ V (M0) of differential operators on M0 generated by V and C ∞ (M), and show that Ψ ∞ 1,0,V (M0) is a “microlocalization” of Diff ∗ V (M0). We also define and study semiclassical and “suspended ” versions of the algebra Ψ ∞ 1,0,V (M0). Thus, our constructions solves a conjecture of Melrose [28].
Gluing and Wormholes for the Einstein Constraint Equations
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2002
"... We establish a general gluing theorem for constant mean curvature solutions of the vacuum Einstein constraint equations. This allows one to take connected sums of solutions or to glue a handle (wormhole) onto any given solution. Away from this handle region, the initial data sets we produce can be m ..."
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Cited by 25 (8 self)
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We establish a general gluing theorem for constant mean curvature solutions of the vacuum Einstein constraint equations. This allows one to take connected sums of solutions or to glue a handle (wormhole) onto any given solution. Away from this handle region, the initial data sets we produce can be made as close as desired to the original initial data sets. These constructions can be made either when the initial manifold is compact or asymptotically Euclidean or asymptotically hyperbolic, with suitable corresponding conditions on the extrinsic curvature. In the compact setting a mild nondegeneracy condition is required. In the final section of the paper, we list a number ways this construction may be used to produce new types of vacuum spacetimes.
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 18 (14 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
Propagation of singularities for the wave equation on manifolds with corners
 In Séminaire: Équations aux Dérivées Partielles, 2004–2005, Sémin. Équ. Dériv. Partielles
"... Abstract. In this paper we describe the propagation of C ∞ and Sobolev singularities for the wave equation on C ∞ manifolds with corners M equipped with a Riemannian metric g. That is, for X = M ×Rt, P = D2 t −∆M, and u ∈ H1 loc (X) solving Pu = 0 with homogeneous Dirichlet or Neumann boundary condi ..."
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Cited by 18 (11 self)
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Abstract. In this paper we describe the propagation of C ∞ and Sobolev singularities for the wave equation on C ∞ manifolds with corners M equipped with a Riemannian metric g. That is, for X = M ×Rt, P = D2 t −∆M, and u ∈ H1 loc (X) solving Pu = 0 with homogeneous Dirichlet or Neumann boundary conditions, we show that WFb(u) is a union of maximally extended generalized broken bicharacteristics. This result is a C ∞ counterpart of Lebeau’s results for the propagation of analytic singularities on real analytic manifolds with appropriately stratified boundary, [11]. Our methods rely on bmicrolocal positive commutator estimates, thus providing a new proof for the propagation of singularities at hyperbolic points even if M has a smooth boundary (and no corners). 1.
Microlocal analysis of asymptotically hyperbolic and Kerr–de Sitter spaces, preprint
"... Abstract. In this paper we develop a general, systematic, microlocal framework for the Fredholm analysis of nonelliptic problems, including high energy (or semiclassical) estimates, which is stable under perturbations. This framework, described in Section 2, is relatively simple given modern microl ..."
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Cited by 18 (7 self)
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Abstract. In this paper we develop a general, systematic, microlocal framework for the Fredholm analysis of nonelliptic problems, including high energy (or semiclassical) estimates, which is stable under perturbations. This framework, described in Section 2, is relatively simple given modern microlocal analysis, and only takes a bit over a dozen pages after the statement of notation. It resides on a compact manifold without boundary, hence in the standard setting of microlocal analysis, including semiclassical analysis. The rest of the paper is devoted to applications. Many natural applications arise in the setting of nonRiemannian bmetrics in the context of Melrose’s bstructures. These include asymptotically Minkowski metrics, asymptotically de Sittertype metrics on a blowup of the natural compactification and Kerrde Sittertype metrics. The simplest application, however, is to provide a new approach to analysis on Riemannian or Lorentzian (or indeed, possibly of other signature) conformally compact spaces (such as asymptotically hyperbolic or de Sitter spaces). The results include, in particular, a new construction of the meromorphic extension of the resolvent of the Laplacian in the Riemannian case, as well as high energy estimates for the spectral parameter in strips of the complex plane. For these results, only Section 2 and Section 4.44.9, starting with the paragraph of (4.8), are strictly needed. The appendix written by Dyatlov relates his analysis of resonances on exact Kerrde Sitter space (which then was used to analyze the wave equation in that setting) to the more general method described here. 1.
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 14 (2 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 14 (7 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.
Propagation of singularities for the wave equation on conic manifolds, Inventiones Mathematicae 156
, 2004
"... Abstract. For the wave equation associated to the Laplacian on a compact manifold with boundary with a conic metric (with respect to which the boundary is metrically a point) the propagation of singularities through the boundary is analyzed. Under appropriate regularity assumptions the diffracted, n ..."
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Cited by 13 (6 self)
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Abstract. For the wave equation associated to the Laplacian on a compact manifold with boundary with a conic metric (with respect to which the boundary is metrically a point) the propagation of singularities through the boundary is analyzed. Under appropriate regularity assumptions the diffracted, nondirect, wave produced by the boundary is shown to have Sobolev regularity greater than the incoming wave.
Continuity of Edge and Corner Pseudodifferential Operators
, 1998
"... . A theorem of Calder'onVaillancourt type is obtained for a class of pseudodifferential operators with operatorvalued symbols, and strongly continuous (in general nonsmooth) groups of isomorphisms involved in the symbol estimates. The theory of pseudodifferential operators on singular manifolds, ..."
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Cited by 12 (4 self)
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. A theorem of Calder'onVaillancourt type is obtained for a class of pseudodifferential operators with operatorvalued symbols, and strongly continuous (in general nonsmooth) groups of isomorphisms involved in the symbol estimates. The theory of pseudodifferential operators on singular manifolds, i.e. manifolds with singular geometries in the sense of piecewise smooth Riemannian metrics, has seen a fruitful developement throughout the past decades. In particular, pseudodifferential operators on compact manifolds with conical singularities, edges, and corners were intensively studied. In this connection we want to mention the works of Melrose [10], Plamenevskij [13], Schulze [17], [19], [20], and their coworkers. Less attention is paid to the case of noncompact singular manifolds. Indeed, the noncompactness of an underlying configuration may be viewed as a further kind of singularity, whose treatment requires a precise control of the operators `at infinity'. The analysis of noncomp...