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41
Microlocal analysis of asymptotically hyperbolic and Kerr–de Sitter spaces
, 2010
"... 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 ana ..."
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Cited by 62 (13 self)
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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.
Theorems on existence and global dynamics for the Einstein equations
 Living Rev. Relativ
"... This article is a guide to theorems on existence and global dynamics of solutions of the Einstein equations. It draws attention to open questions in the field. The local in time Cauchy problem, which is relatively well understood, is surveyed. Global results for solutions with various types of symme ..."
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Cited by 33 (3 self)
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This article is a guide to theorems on existence and global dynamics of solutions of the Einstein equations. It draws attention to open questions in the field. The local in time Cauchy problem, which is relatively well understood, is surveyed. Global results for solutions with various types of symmetry are discussed. A selection of results from Newtonian theory and special relativity which offer useful comparisons is presented. Treatments of global results in the case of small data and results on constructing spacetimes with prescribed singularity structure are given. A conjectural picture of the asymptotic behaviour of general cosmological solutions of the Einstein equations is built up. Some miscellaneous topics connected with the main theme are collected in a separate section. 1 1
Extensions of the Stability Theorem of the Minkowski Space
 in General Relativity. Solutions of the EinsteinMaxwell Equations. AMS/IP. Studies in Advanced Mathematics
, 2009
"... In this paper, we sketch the proof of the extension of the stability theorem of the Minkowski space in General Relativity done explicitly in [6], [7]. We discuss solutions of the Einstein vacuum (EV) equations (obtained in the author’s Ph.D. thesis [6] in 2007). We solve the Cauchy problem for more ..."
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Cited by 24 (3 self)
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In this paper, we sketch the proof of the extension of the stability theorem of the Minkowski space in General Relativity done explicitly in [6], [7]. We discuss solutions of the Einstein vacuum (EV) equations (obtained in the author’s Ph.D. thesis [6] in 2007). We solve the Cauchy problem for more general, asymptotically flat initial data than in the pioneering work [21] of D. Christodoulou and S. Klainerman or than in any other work. Moreover, we describe precisely the asymptotic behaviour. Our relaxed assumptions on the initial data yield a spacetime curvature which is not bounded in L ∞ (M). As a major result, we encounter in our work borderline cases, which we discuss in this paper as well. The fact that certain of our estimates are borderline in view of decay indicates that the conditions in our main theorem are sharp in so far as the assumptions on the decay at infinity on the initial data are concerned. Thus, the borderline cases are a consequence of our relaxed assumptions on the data, [6], [7]. They are not present in the other works, as all of them place stronger assumptions on their data. We work with an invariant formulation of the EV equations. Our main proof is based on a bootstrap argument. To close the argument, we have to show that the
MATHEMATICAL GENERAL RELATIVITY: A SAMPLER
, 2010
"... We provide an introduction to selected recent advances in the mathematical understanding of Einstein’s theory of gravitation. ..."
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Cited by 23 (2 self)
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We provide an introduction to selected recent advances in the mathematical understanding of Einstein’s theory of gravitation.
Topological Censorship for Kaluza–Klein SpaceTimes
, 1424
"... Abstract. The standard topological censorship theorems require asymptotic hypotheses which are too restrictive for several situations of interest. In this paper we prove a version of topological censorship under significantly weaker conditions, compatible, e.g., with solutions with Kaluza–Klein asym ..."
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Cited by 21 (9 self)
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Abstract. The standard topological censorship theorems require asymptotic hypotheses which are too restrictive for several situations of interest. In this paper we prove a version of topological censorship under significantly weaker conditions, compatible, e.g., with solutions with Kaluza–Klein asymptotic behavior. In particular we prove simple connectedness of the quotient of the domain of outer communications by the group of symmetries for models which are asymptotically flat, or asymptotically antide Sitter, in a Kaluza–Klein sense. This allows one, e.g., to define the twist potentials needed for the reduction of the field equations in uniqueness theorems. Finally, the methods used to prove the above are used to show that weakly trapped compact surfaces cannot be seen from Scri. 1.
Global solutions of the Einstein– Maxwell equations in higher dimension
 Class. Quantum Grav
"... We consider the EinsteinMaxwell equations in spacedimension n. We point out that the LindbladRodnianski stability proof applies to those equations whatever the spacedimension n ≥ 3. In even spacetime dimension n + 1 ≥ 6 we use the standard conformal method on a Minkowski background to give a si ..."
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Cited by 14 (5 self)
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We consider the EinsteinMaxwell equations in spacedimension n. We point out that the LindbladRodnianski stability proof applies to those equations whatever the spacedimension n ≥ 3. In even spacetime dimension n + 1 ≥ 6 we use the standard conformal method on a Minkowski background to give a simple proof that the maximal globally hyperbolic development of initial data sets which are sufficiently close to the data for Minkowski spacetime and which are Schwarzschildian outside of a compact set lead to geodesically complete spacetimes, with a complete Scri, with smooth conformal structure, and with the gravitational field approaching the Minkowski metric along null directions at least as fast as r −(n−1)/2. 1
The wave equation on extreme ReissnerNordström black hole spacetimes: stability and instability results, arXiv:1006.0283
"... We consider solutions to the linear wave equation gψ = 0 on a suitable globally hyperbolic subset of an extreme ReissnerNordström spacetime, arising from regular initial data prescribed on a Cauchy hypersurface Σ0 crossing the future event horizon H+. We obtain boundedness, decay, nondecay and ..."
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Cited by 9 (4 self)
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We consider solutions to the linear wave equation gψ = 0 on a suitable globally hyperbolic subset of an extreme ReissnerNordström spacetime, arising from regular initial data prescribed on a Cauchy hypersurface Σ0 crossing the future event horizon H+. We obtain boundedness, decay, nondecay and blowup results. Our estimates hold up to and including H+. The fundamental new aspect of this problem is the degeneracy of the redshift on the event horizon H+. Several new analytical features of
ON THE UNIQUENESS AND GLOBAL DYNAMICS OF ADS SPACETIMES
"... Abstract. We study global aspects of complete, nonsingular asymptotically locally AdS spacetimes solving the vacuum Einstein equations whose conformal infinity is an arbitrary globally stationary spacetime. It is proved that any such solution which is asymptotically stationary to the past and futur ..."
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Cited by 8 (2 self)
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Abstract. We study global aspects of complete, nonsingular asymptotically locally AdS spacetimes solving the vacuum Einstein equations whose conformal infinity is an arbitrary globally stationary spacetime. It is proved that any such solution which is asymptotically stationary to the past and future is itself globally stationary. This gives certain rigidity or uniqueness results for exact AdS and related spacetimes. 1.
The nature of spacetime singularities
 In 100 Years of Relativity – SpaceTime Structure: Einstein and
, 2005
"... Present knowledge about the nature of spacetime singularities in the context of classical general relativity is surveyed. The status of the BKL picture of cosmological singularities and its relevance to the cosmic censorship hypothesis are discussed. It is shown how insights on cosmic censorship als ..."
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Cited by 6 (0 self)
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Present knowledge about the nature of spacetime singularities in the context of classical general relativity is surveyed. The status of the BKL picture of cosmological singularities and its relevance to the cosmic censorship hypothesis are discussed. It is shown how insights on cosmic censorship also arise in connection with the idea of weak null singularities inside black holes. Other topics covered include matter singularities and critical collapse. Remarks are made on possible future directions in research on spacetime singularities. 1
SEMILINEAR WAVE EQUATIONS ON ASYMPTOTICALLY DE SITTER, KERRDE SITTER AND MINKOWSKI SPACETIMES
"... Abstract. In this paper we show the small data solvability of suitable semilinear wave and KleinGordon equations on geometric classes of spaces, which include socalled asymptotically de Sitter and Kerrde Sitter spaces, as well as asymptotically Minkowski spaces. These spaces allow general infini ..."
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Cited by 5 (2 self)
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Abstract. In this paper we show the small data solvability of suitable semilinear wave and KleinGordon equations on geometric classes of spaces, which include socalled asymptotically de Sitter and Kerrde Sitter spaces, as well as asymptotically Minkowski spaces. These spaces allow general infinities, called conformal infinity in the asymptotically de Sitter setting; the Minkowski type setting is that of nontrapping Lorentzian scattering metrics introduced by Baskin, Vasy and Wunsch. Our results are obtained by showing the global Fredholm property, and indeed invertibility, of the underlying linear operator on suitable L2based function spaces, which also possess appropriate algebra or more complicated multiplicative properties. The linear framework is based on the banalysis, in the sense of Melrose, introduced in this context by Vasy to describe the asymptotic behavior of solutions of linear equations. An interesting feature of the analysis is that resonances, namely poles of the inverse of the Mellin transformed bnormal operator, which are ‘quantum ’ (not purely symbolic) objects, play an important role. 1.