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Initial data engineering. (English) Zbl 1080.83002

Summary: We present a local gluing construction for general relativistic initial data sets. The method applies to generic initial data, in a sense which is made precise. In particular the trace of the extrinsic curvature is not assumed to be constant near the gluing points, which was the case for previous such constructions. No global conditions on the initial data sets such as compactness, completeness, or asymptotic conditions are imposed. As an application, we prove existence of spatially compact, maximal globally hyperbolic, vacuum space-times without any closed constant mean curvature spacelike hypersurface.

MSC:

83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
58J45 Hyperbolic equations on manifolds
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[1] Andersson, L., Chruściel, P.T.: On asymptotic behavior of solutions of the constraint equations in general relativity with ”hyperboloidal boundary conditions”. Dissert. Math. 355, 1–100 (1996) · Zbl 0873.35101
[2] Bartnik, R.: Regularity of variational maximal surfaces. Acta Math. 161, 145–181 (1988) · Zbl 0667.53049
[3] Bartnik, R.: Remarks on cosmological spacetimes and constant mean curvature surfaces. Commun. Math. Phys. 117, 615–624 (1988) · Zbl 0647.53044
[4] Bartnik, R.: New definition of quasilocal mass. Phys. Rev. Lett. 62, 2346–2348 (1989)
[5] Bartnik, R.: Energy in general relativity, Tsing Hua Lectures on Geometry and Analysis (S.-T. Yau, ed.), Cambridge, MA: International Press, 1997 · Zbl 0884.53065
[6] Beig, R., Chruściel, P.T.: Killing Initial Data. Class. Quantum. Grav. 14, A83–A92 (1996). A special issue in honour of Andrzej Trautman on the occasion of his 64th Birthday, Tafel, J. (ed.).
[7] Beig, R., Chruściel, P.T., Schoen, R.: KIDs are non-generic. To appear in Ann. Henri Poincaré; http://arxiv.org/list/gr-qc/0403042, 2004
[8] Berger, M., Ebin, D.: Some decompositions of the space of symmetric tensors on a Riemannian manifold. J. Diff. Geom. 3, 379–392 (1969) · Zbl 0194.53103
[9] Bray, H.L.: Proof of the Riemannian Penrose conjecture using the positive mass theorem. J. Diff. Geom. 59, 177–267 (2001). · Zbl 1039.53034
[10] Bray, H.L., Chruściel, P.T.: The Penrose inequality. In: 50 years of the Cauchy problem in general relativity, Chruściel, P.T., Friedrich, H. eds., Basel: Birkhaeuser, 2004 · Zbl 1058.83006
[11] Brill, D.: On spacetimes without maximal surfaces. In: Proceedings of the third Marcel Grossmann meeting (Amsterdam), Hu Ning, ed., Amsterdam: North Holland, 1983, pp. 79–87
[12] Chruściel, P.T., Delay, E.: On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications. Mém. Soc. Math. de France. 94, 1–103 (2003) · Zbl 1058.83007
[13] Corvino, J.: Scalar curvature deformation and a gluing construction for the Einstein constraint equations. Commun. Math. Phys. 214, 137–189 (2000) · Zbl 1031.53064
[14] Corvino, J., Schoen, R.: On the asymptotics for the vacuum Einstein constraint equations. To appear J. Diff. Geom.; http://arxiv.org/list/gr-qc/0301071, 2003 · Zbl 1122.58016
[15] Eardley, D., Witt, D.: Unpublished, 1992
[16] Isenberg, J.: Constant mean curvature solutions of the Einstein constraint equations on closed manifolds. Class. Quantum Grav. 12, 2249–2274 (1995) · Zbl 0840.53056
[17] Isenberg, J., Maxwell, D., Pollack, D.: A gluing construction for non-vacuum solutions of the Einstein constraint equations. http://arxiv.org/list/gr-qc/0501083, 2005 · Zbl 1101.83005
[18] Isenberg, J., Mazzeo, R., Pollack, D.: Gluing and wormholes for the Einstein constraint equations. Commun. Math. Phys. 231, 529–568 (2002) · Zbl 1013.83008
[19] Isenberg, J., Mazzeo, R., Pollack, D.: On the topology of vacuum spacetimes. Annales Henri Poincaré 4, 369–383 (2003) · Zbl 1026.83008
[20] Joyce, D.: Constant scalar curvature metrics on connected sums. Int. J. Math. Sci. no. 7, 405–450 (2003) · Zbl 1026.53019
[21] Moncrief, V.: Spacetime symmetries and linearization stability of the Einstein equations I. J. Math. Phys. 16, 493–498 (1975) · Zbl 0314.53035
[22] Witt, D. M.: Vacuum space-times that admit no maximal slice. Phys. Rev. Lett. 57, 1386–1389 (1986)
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