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Quasilocal energymomentum and angular momentum in General Relativity: A review article
 Living Rev. Rel
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KIDs are nongeneric
 Poincaré 6 (2005), 155–194, arXiv:grqc/0403042. MR MR2121280 (2005m:83013
"... We prove that generic solutions of the vacuum constraint Einstein equations do not possess any global or local spacetime Killing vectors, on an asymptotically flat Cauchy surface, or on a compact Cauchy surface with mean curvature close to a constant, or for CMC asymptotically hyperbolic initial da ..."
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Cited by 23 (4 self)
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We prove that generic solutions of the vacuum constraint Einstein equations do not possess any global or local spacetime Killing vectors, on an asymptotically flat Cauchy surface, or on a compact Cauchy surface with mean curvature close to a constant, or for CMC asymptotically hyperbolic initial data sets. More generally, we show that nonexistence of global symmetries implies, generically, nonexistence of local ones. As part of the argument, we prove that generic metrics do not possess any local or global conformal Killing vectors. AMS 83C05, PACS 04.20.Cv 1
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.
Asymptotically simple solutions of the vacuum Einstein equations in even dimensions
 Commun. Math. Phys
"... Abstract. We show that a set of conformally invariant equations derived from the FeffermanGraham tensor can be used to construct global solutions of vacuum Einstein equations, in all even dimensions. This gives, in particular, a new, simple proof of Friedrich’s result on the future hyperboloidal st ..."
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Cited by 22 (6 self)
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Abstract. We show that a set of conformally invariant equations derived from the FeffermanGraham tensor can be used to construct global solutions of vacuum Einstein equations, in all even dimensions. This gives, in particular, a new, simple proof of Friedrich’s result on the future hyperboloidal stability of Minkowski spacetime, and extends its validity to even dimensions. 1.
On the center of mass of isolated systems with general asymptotics. Classical Quantum Gravity
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Phase space for the Einstein equations
"... A Hilbert manifold structure is described for the phase space F of asymptotically flat initial data for the Einstein equations. The space of solutions of the constraint equations forms a Hilbert submanifold C ⊂ F. The ADM energymomentum defines a function which is smooth on this submanifold, but wh ..."
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Cited by 15 (2 self)
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A Hilbert manifold structure is described for the phase space F of asymptotically flat initial data for the Einstein equations. The space of solutions of the constraint equations forms a Hilbert submanifold C ⊂ F. The ADM energymomentum defines a function which is smooth on this submanifold, but which is not defined in general on all of F. The ADM Hamiltonian defines a smooth function on F which generates the Einstein evolution equations only if the lapseshift satisfies rapid decay conditions. However a regularised Hamiltonian can be defined on F which agrees with the ReggeTeitelboim Hamiltonian on C and generates the evolution for any lapseshift appropriately asymptotic to a (time) translation at infinity. Finally, critical points for the total (ADM) mass, considered as a function on the Hilbert manifold of constraint solutions, arise precisely at initial data generating stationary vacuum spacetimes.
SPECIFYING ANGULAR MOMENTUM AND CENTER OF MASS FOR VACUUM INITIAL DATA SETS
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THE BOUNDED L2 CURVATURE CONJECTURE
"... Abstract. This is the main paper in a sequence in which we give a complete proof of the bounded L2 curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einsteinvacuum equations depends only on the L2norm of the curvature and a lower bound on the vo ..."
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Cited by 11 (3 self)
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Abstract. This is the main paper in a sequence in which we give a complete proof of the bounded L2 curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einsteinvacuum equations depends only on the L2norm of the curvature and a lower bound on the volume radius of the corresponding initial data set. We note that though the result is not optimal with respect to the standard scaling of the Einstein equations, it is nevertheless critical with respect to another, more subtle, scaling tied to its causal geometry. Indeed, L2 bounds on the curvature is the minimum requirement necessary to obtain lower bounds on the radius of injectivity of causal boundaries. We note also that, while the first nontrivial improvements for well posedness for quasilinear hyperbolic systems in spacetime dimensions greater than 1 + 1 (based on Strichartz estimates) were obtained in [2], [3], [48], [49], [19] and optimized in [20], [36], the result we present here is the first in which the full structure of the quasilinear hyperbolic system, not just its principal part, plays a crucial role. To achieve our goals we recast the Einstein vacuum equations as a quasilinear so(3, 1)valued YangMills theory and introduce a Coulomb type gauge condition in which the equations exhibit a specific new type of null structure compatible with the quasilinear, covariant nature of the equations. To prove the conjecture we formulate and establish bilinear and trilinear estimates on rough backgrounds which allow us to make use of that crucial structure. These require a careful construction and control of parametrices including L2 error bounds which is carried out in [41][44], as well as a proof of sharp Strichartz estimates for the wave equation on a rough background which is carried out in [45]. It is at this level that the null scaling mentioned above makes our problem critical. Indeed, any known notion of a parametrix relies in an essential way on the eikonal equation, and our spacetime possesses, barely, the minimal regularity needed to make sense of its solutions. 1.
LOCALIZED GLUING OF RIEMANNIAN METRICS IN INTERPOLATING THEIR SCALAR CURVATURE
"... Abstract. We show that two smooth nearby Riemannian metrics can be glued interpolating their scalar curvature. The resulting smooth metric is the same as the starting ones outside the gluing region and has scalar curvature interpolating between the original ones. One can then glue metrics while main ..."
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Abstract. We show that two smooth nearby Riemannian metrics can be glued interpolating their scalar curvature. The resulting smooth metric is the same as the starting ones outside the gluing region and has scalar curvature interpolating between the original ones. One can then glue metrics while maintaining inequalities satisfied by the scalar curvature. We also glue asymptotically Euclidean metrics to Schwarzschild ones and the same for asymptotically Delaunay metrics, keeping bounds on the scalar curvature, if any. This extend the Corvino gluing near infinity