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45
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
Rough solutions of the Einstein constraints on closed manifolds without nearCMC conditions. Submitted for publication. Available as arXiv:0712.0798v1 [grqc
"... ABSTRACT. We consider the conformal decomposition of Einstein’s constraint equations introduced by Lichnerowicz and York, on a closed manifold. We establish existence of nonCMC weak solutions using a combination of a priori estimates for the individual Hamiltonian and momentum constraints, barrier ..."
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Cited by 30 (14 self)
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ABSTRACT. We consider the conformal decomposition of Einstein’s constraint equations introduced by Lichnerowicz and York, on a closed manifold. We establish existence of nonCMC weak solutions using a combination of a priori estimates for the individual Hamiltonian and momentum constraints, barrier constructions and fixedpoint techniques for the Hamiltonian constraint, RieszSchauder theory for the momentum constraint, together with a topological fixedpoint argument for the coupled system. Although we present general existence results for nonCMC weak solutions when the rescaled background metric is in any of the three Yamabe classes, an important new feature of the results we present for the positive Yamabe class is the absence of the nearCMC assumption, if the freely specifiable part of the data given by the tracelesstransverse part of the rescaled extrinsic curvature and the matter fields are sufficiently small, and if the energy density of matter is not identically zero. In this case, the mean extrinsic curvature can be taken to be an arbitrary smooth function without restrictions on the size of its spatial derivatives, so that it can be arbitrarily far from constant, giving what is apparently the first existence results for nonCMC solutions without 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.
Nearconstant mean curvature solutions of the Einstein constraint equations with nonnegative Yamabe metrics. Available as arXiv:0710.0725 [grqc
, 2007
"... Abstract. We show that sets of conformal data on closed manifolds with the metric in the positive or zero Yamabe class, and with the gradient of the mean curvature function sufficiently small, are mapped to solutions of the vacuum Einstein constraint equations. This result extends previous work whic ..."
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Cited by 19 (1 self)
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Abstract. We show that sets of conformal data on closed manifolds with the metric in the positive or zero Yamabe class, and with the gradient of the mean curvature function sufficiently small, are mapped to solutions of the vacuum Einstein constraint equations. This result extends previous work which required the conformal metric to be in the negative Yamabe class, and required the mean curvature function to be nonzero. 1.
The Newtonian limit for perfect fluids
, 810
"... We prove that there exists a class of nonstationary solutions to the EinsteinEuler equations which have a Newtonian limit. The proof of this result is based on a symmetric hyperbolic formulation of the EinsteinEuler equations which contains a singular parameter ǫ = vT /c where vT is a characteris ..."
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Cited by 14 (8 self)
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We prove that there exists a class of nonstationary solutions to the EinsteinEuler equations which have a Newtonian limit. The proof of this result is based on a symmetric hyperbolic formulation of the EinsteinEuler equations which contains a singular parameter ǫ = vT /c where vT is a characteristic velocity scale associated with the fluid and c is the speed of light. The symmetric hyperbolic formulation allows us to derive ǫ independent energy estimates on weighted Sobolev spaces. These estimates are the main tool used to analyze the behavior of solutions in the limit ǫց0. 1
Rough solutions of the Einstein constraint equations on closed manifolds without nearCMC conditions
 36 [ICBM92] [IM96] [IOM04] [Is79] [Is95
"... Abstract. We consider the conformal decomposition of Einstein’s constraint equations introduced by Lichnerowicz and York, on a compact manifold with boundary. We first develop some technical results for the momentum constraint operator under weak assumptions on the problem data, including generalize ..."
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Cited by 13 (2 self)
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Abstract. We consider the conformal decomposition of Einstein’s constraint equations introduced by Lichnerowicz and York, on a compact manifold with boundary. We first develop some technical results for the momentum constraint operator under weak assumptions on the problem data, including generalized Korn inequalities on manifolds with boundary not currently in the literature. We then consider the Hamiltonian constraint, and using order relations on appropriate Banach spaces we derive weak solution generalizations of known sub and supersolutions (barriers). We also establish some related a priori L ∞bounds on any W 1,2solution. The barriers are combined with variational methods to establish existence of solutions to the Hamiltonian constraint in L ∞ ∩ W 1,2. The result is established under weak assumptions on the problem data, and for scalar curvature R having any sign; nonnegative R requires additional positivity assumptions either on the matter energy density or on the tracefree divergencefree part of the extrinsic curvature. Although the formulation is different, the result can be viewed as extending the regularity of the recent result of Maxwell on “rough ” CMC solutions in W k,2 for k> 3/2 down to L ∞ ∩ W 1,2. The results for the individual constraints are then combined to establish existence of nonCMC solutions in W 1,p, p> 3 for the threemetric and in L q, q = 6p/(3 + p) for the extrinsic curvature. The result is obtained using fixedpoint iteration and compactness arguments directly, rather than by building a contraction map. The nonCMC result can be viewed as a type of extension of the regularity of the 1996 nonCMC result of Isenberg and Moncrief down to W 1,p for p> 3, and extending their result to R having any sign. Similarly, the result can also be viewed as type of extension of the recent work of Maxwell on rough solutions from the CMC case to the nonCMC case. Although our presentation is for 3manifolds, the results also hold in higher dimensions with minor adjustments. The results should also extend to other cases such as closed and (fully or partially) open manifolds without substantial difficulty.
The Einstein–scalar field constraints on asymptotically Euclidean manifolds
 ArXiv: grqc/0506101
"... We use the conformal method to obtain solutions of the Einstein– scalar field gravitational constraint equations. Handling scalar fields is a bit more challenging than handling matter fields such as fluids, Maxwell fields or Yang–Mills fields, because the scalar field introduces three extra terms in ..."
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Cited by 12 (6 self)
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We use the conformal method to obtain solutions of the Einstein– scalar field gravitational constraint equations. Handling scalar fields is a bit more challenging than handling matter fields such as fluids, Maxwell fields or Yang–Mills fields, because the scalar field introduces three extra terms into the Lichnerowicz equation, rather than just one. Our proofs are constructive and allow for arbitrary dimension (> 2) as well as low regularity initial data. Dedicated to the memory of S. S. Chern, with admiration for his mathematical discoveries and his character. 1 Introduction. To explain recent observations of far away stars and galaxies, as well as the possible origin of matter elements, it has become more and more relevant in Einsteinian cosmology to admit the existence of a scalar field with a potential which remains to be estimated. On the other hand various considerations, in
On Lorentzian causality with continuous metrics
 Classical Quantum Gravity
"... Abstract. We present a systematic study of causality theory on Lorentzian manifolds with continuous metrics. Examples are given which show that some standard facts in smooth Lorentzian geometry, such as lightcones being hypersurfaces, are wrong when metrics which are merely continuous are considere ..."
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Cited by 9 (0 self)
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Abstract. We present a systematic study of causality theory on Lorentzian manifolds with continuous metrics. Examples are given which show that some standard facts in smooth Lorentzian geometry, such as lightcones being hypersurfaces, are wrong when metrics which are merely continuous are considered. We show that existence of time functions remains true on domains of dependence with continuous metrics, and that C0,1 differentiability of the metric suffices for many key results of the smooth causality theory.
A large class of non constant mean curvature solutions of the Einstein constraint equations on an asymptotically hyperbolic manifold
 Comm. Math. Phys
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Axisymmetric evolution of Einstein equations and mass conservation
, 2009
"... For axisymmetric evolution of isolated systems, we prove that there exists a gauge such that the total mass can be written as a positive definite integral on the spacelike hypersurfaces of the foliation and the integral is constant along the evolution. The conserved mass integral controls the square ..."
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Cited by 7 (6 self)
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For axisymmetric evolution of isolated systems, we prove that there exists a gauge such that the total mass can be written as a positive definite integral on the spacelike hypersurfaces of the foliation and the integral is constant along the evolution. The conserved mass integral controls the square of the extrinsic curvature and the square of first derivatives of the intrinsic metric. We also discuss applications of this result for the global existence problem in axial symmetry. 1