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On the breakdown criterion in general relativity


Authors: Sergiu Klainerman and Igor Rodnianski
Journal: J. Amer. Math. Soc. 23 (2010), 345-382
MSC (2010): Primary 35J10
DOI: https://doi.org/10.1090/S0894-0347-09-00655-9
Published electronically: December 24, 2009
MathSciNet review: 2601037
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Abstract: We give a geometric criterion for the breakdown of an Einstein-vacuum space-time foliated by a constant mean curvature, or maximal, foliation. More precisely we show that the foliated space-time can be extended as long as the second fundamental form and the first derivatives of the logarithm of the lapse of the foliation remain uniformly bounded. We make no restrictions on the size of the initial data. The paper uses heavily the results of the authors' previous papers.


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  • [And] M. Anderson, On long-time evolution in general relativity and geometrization of $ 3$-manifolds, Comm. Math. Phys. 222 (2001), 533-567. MR 1888088 (2003d:53113)
  • [BKM] J. T. Beale, T. Kato, A. Majda, Remarks on the breakdown of smooth solutions for the $ 3$-D Euler equations, Comm. Math. Phys. 94 (1984), 61-66. MR 763762 (85j:35154)
  • [Br] Y. Choquét-Bruhat, Théorème d'existence pour certains systèmes d'équations aux dérivées partielles non linéaires, Acta Math. 88 (1952), 141-225. MR 0053338 (14:756g)
  • [C-K] D. Christodoulou, S. Klainerman, The global nonlinear stability of the Minkowski space, Princeton Math. Series 41, 1993. MR 1316662 (95k:83006)
  • [Fried] F.G. Friedlander, The Wave Equation on a Curved Space-time, Cambridge University Press, 1976. MR 0460898 (57:889)
  • [HE] S. W. Hawking, G. F. R. Ellis, The Large Scale Structure of Space-time, Cambridge: Cambridge University Press, 1973. MR 0424186 (54:12154)
  • [EM1] D. Eardley, V. Moncrief, The global existence of Yang-Mills-Higgs fields in $ 4$-dimensional Minkowski space. I. Local existence and smoothness properties, Comm. Math. Phys. 83 (1982), no. 2, 171-191. MR 649158 (83e:35106a)
  • [EM2] D. Eardley, V. Moncrief, The global existence of Yang-Mills-Higgs fields in $ 4$-dimensional Minkowski space. II. Completion of proof, Comm. Math. Phys. 83 (1982), no. 2, 193-212. MR 649159 (83e:35106b)
  • [HKM] T. J. R. Hughes, T. Kato and J. E. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rational Mech. Anal. 63 (1977), 273-394. MR 0420024 (54:8041)
  • [Kl] S. Klainerman.
    PDE as a unified subject, Special Volume Geom. Funct. Anal. (2000), 279-315. MR 1826256 (2002e:35001)
  • [Kl-Ma] S. Klainerman, M. Machedon, Finite Energy Solutions for the Yang-Mills Equations in $ {\mathbb{R}}^{3+1}$, Annals of Math. (2) 142 (1995), 39-119. MR 1338675 (96i:58167)
  • [Kl-Ro1] S. Klainerman, I. Rodnianski, Causal geometry of Einstein-vacuum spacetimes with finite curvature flux, Inventiones Math. 159 (2005), 437-529. MR 2125732 (2006e:58042)
  • [Kl-Ro2] S. Klainerman, I. Rodnianski, A geometric approach to Littlewood-Paley theory, Geom. Funct. Anal. 16 (2006), 126-163. MR 2221254 (2007e:58046)
  • [Kl-Ro3] S. Klainerman, I. Rodnianski, Sharp trace theorems for null hypersurfaces on Einstein metrics with finite curvature flux, Geom. Funct. Anal. 16 (2006), 164-229. MR 2221255 (2007e:58047)
  • [Kl-Ro4] S. Klainerman, I. Rodnianski, On the radius of injectivity of null hypersurfaces, J. Amer. Math. Soc. 21 (2008), 775-795. MR 2393426 (2009g:35326)
  • [Kl-Ro5] S. Klainerman, I. Rodnianski, A Kirchhoff-Sobolev parametrix for the wave equation and applications, Journ. of Hyperbolic Equ., 4 (2007), 401-433. MR 2339803 (2008j:35108)
  • [M] V. Moncrief, An integral equation for spacetime curvature in General Relativity, Newton Institute preprint, NI05086-GMR. Surv. Differ. Geom., Vol. 10, Int. Press, Somerville, MA, 2006. MR 2408224 (2009h:53166)
  • [Sob] S. Sobolev, Méthodes nouvelle à resoudre le problème de Cauchy pour les équations linéaires hyperboliques normales, Matematicheskii Sbornik, vol. 1 (43) 1936, 31 -79.
  • [Wang] Q. Wang, Causal geometry of Einstein vacuum space-times. Ph.D. thesis, Princeton University, 2006.

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Additional Information

Sergiu Klainerman
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Email: seri@math.princeton.edu

Igor Rodnianski
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Email: irod@math.princeton.edu

DOI: https://doi.org/10.1090/S0894-0347-09-00655-9
Received by editor(s): February 4, 2008
Published electronically: December 24, 2009
Additional Notes: The first author is partially supported by NSF grant DMS-0070696
The second author is partially supported by NSF grant DMS-0702270
Article copyright: © Copyright 2009 American Mathematical Society

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