The resolution of the bounded L 2 curvature conjecture in general relativity
Séminaire Laurent Schwartz — EDP et applications (2014-2015), Exposé no. 1, 18 p.

This paper reports on the recent proof of the bounded L 2 curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the L 2 -norm of the curvature and a lower bound of the volume radius of the corresponding initial data set.

DOI : 10.5802/slsedp.65
Klainerman, Sergiu 1 ; Rodnianski, Igor 1 ; Szeftel, Jérémie 2

1 Department of Mathematics Princeton University Princeton NJ 08544
2 Laboratoire Jacques-Louis Lions Université Pierre et Marie Curie Paris 75005
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Klainerman, Sergiu; Rodnianski, Igor; Szeftel, Jérémie. The resolution of the bounded $L^2$ curvature conjecture in general relativity. Séminaire Laurent Schwartz — EDP et applications (2014-2015), Exposé no. 1, 18 p. doi : 10.5802/slsedp.65. http://archive.numdam.org/articles/10.5802/slsedp.65/

[1] H. Bahouri, J.-Y. Chemin, Équations d’ondes quasilinéaires et estimation de Strichartz, Amer. J. Math., 121, 1337–1777, 1999. | MR | Zbl

[2] H. Bahouri, J.-Y. Chemin, Équations d’ondes quasilinéaires et effet dispersif, IMRN, 21, 1141–1178, 1999. | MR | Zbl

[3] Y. C. Bruhat, Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires, Acta Math. 88, 141–225, 1952. | MR | Zbl

[4] E. Cartan, Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion, C. R. Acad. Sci. (Paris), 174, 593–595, 1922.

[5] D. Christodoulou, Bounded variation solutions of the spherically symmetric Einstein-scalar field equations, Comm. Pure and Appl. Math, 46, 1131–1220, 1993. | MR | Zbl

[6] D. Christodoulou, The instability of naked singularities in the gravitational collapse of a scalar field, Ann. of Math., 149, 183–217,1999. | EuDML | MR | Zbl

[7] A. Fischer, J. Marsden, The Einstein evolution equations as a first-order quasi-linear symmetric hyperbolic system. I, Comm. Math. Phys. 28, 1–38, 1972. | MR | Zbl

[8] T. J. R. Hughes, T. Kato, J. E. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rational Mech. Anal. 63, 273–394, 1977. | MR | Zbl

[9] S. Klainerman, M. Machedon, Space-time estimates for null forms and the local existence theorem, Communications on Pure and Applied Mathematics, 46, 1221–1268, 1993. | MR | Zbl

[10] S. Klainerman, M. Machedon, Finite energy solutions of the Maxwell-Klein-Gordon equations, Duke Math. J. 74, 19–44, 1994. | MR | Zbl

[11] S. Klainerman, M. Machedon, Finite Energy Solutions for the Yang-Mills Equations in 1+3 , Annals of Math. 142, 39–119, 1995. | MR | Zbl

[12] S. Klainerman, PDE as a unified subject, Proceeding of Visions in Mathematics, GAFA 2000 (Tel Aviv 1999). Geom Funct. Anal. 2000, Special Volume , Part 1, 279–315. | MR | Zbl

[13] S. Klainerman, I. Rodnianski, Improved local well-posedness for quasi-linear wave equations in dimension three, Duke Math. J. 117 (1), 1–124, 2003. | MR | Zbl

[14] S. Klainerman, I. Rodnianski, Rough solutions to the Einstein vacuum equations, Annals of Math. 161, 1143–1193, 2005. | MR | Zbl

[15] S. Klainerman, I. Rodnianski, Bilinear estimates on curved space-times, J. Hyperbolic Differ. Equ. 2 (2), 279–291, 2005. | MR | Zbl

[16] S. Klainerman, I. Rodnianski, Casual geometry of Einstein vacuum space-times with finite curvature flux, Inventiones 159, 437–529, 2005. | MR | Zbl

[17] S. Klainerman, I. Rodnianski, Sharp trace theorems on null hypersurfaces, GAFA 16 (1), 164–229, 2006. | MR | Zbl

[18] S. Klainerman, I. Rodnianski, A geometric version of Littlewood-Paley theory, GAFA 16 (1), 126–163, 2006. | MR | Zbl

[19] S. Klainerman, I. Rodnianski, On a break-down criterion in General Relativity, J. Amer. Math. Soc. 23, 345–382, 2010. | MR | Zbl

[20] S. Klainerman, I. Rodnianski, J. Szeftel, The Bounded L 2 Curvature Conjecture, , 91 p, 2012. | arXiv

[21] J. Krieger, W. Schlag, Concentration compactness for critical wave maps, Monographs of the European Mathematical Society, 2012. | MR

[22] H. Lindblad, Counterexamples to local existence for quasilinear wave equations, Amer. J. Math. 118 (1), 1–16, 1996. | MR | Zbl

[23] H. Lindblad, I. Rodnianski, The weak null condition for the Einstein vacuum equations, C. R. Acad. Sci. 336, 901–906, 2003. | MR | Zbl

[24] D. Parlongue, An integral breakdown criterion for Einstein vacuum equations in the case of asymptotically flat spacetimes, eprint1004.4309, 88 p, 2010.

[25] F. Planchon, I. Rodnianski, Uniqueness in general relativity, preprint.

[26] G. Ponce, T. Sideris, Local regularity of nonlinear wave equations in three space dimensions, Comm. PDE 17, 169–177, 1993. | MR | Zbl

[27] H. F. Smith, A parametrix construction for wave equations with C 1,1 coefficients, Ann. Inst. Fourier (Grenoble) 48, 797–835, 1998. | Numdam | MR | Zbl

[28] H.F. Smith, D. Tataru, Sharp local well-posedness results for the nonlinear wave equation, Ann. of Math. 162, 291–366, 2005. | MR | Zbl

[29] S. Sobolev, Méthodes nouvelles pour résoudre le problème de Cauchy pour les équations linéaires hyperboliques normales, Matematicheskii Sbornik, 1 (43), 31–79, 1936. | Zbl

[30] E. Stein, Harmonic Analysis, Princeton University Press, 1993. | MR | Zbl

[31] J. Sterbenz, D. Tataru, Regularity of Wave-Maps in dimension 2+1, Comm. Math. Phys. 298 (1), 231–264, 2010. | MR | Zbl

[32] J. Sterbenz, D. Tataru, Energy dispersed large data wave maps in 2+1 dimensions, Comm. Math. Phys. 298 (1), 139–230, 2010. | MR | Zbl

[33] J. Szeftel, Parametrix for wave equations on a rough background I: Regularity of the phase at initial time, , 145 p, 2012. | arXiv

[34] J. Szeftel, Parametrix for wave equations on a rough background II: Construction of the parametrix and control at initial time, , 84 p, 2012. | arXiv

[35] J. Szeftel, Parametrix for wave equations on a rough background III: Space-time regularity of the phase, , 276 p, 2012. | arXiv

[36] J. Szeftel, Parametrix for wave equations on a rough background IV: Control of the error term, , 284 p, 2012. | arXiv

[37] J. Szeftel, Sharp Strichartz estimates for the wave equation on a rough background, , 30 p, 2013. | arXiv

[38] T. Tao, Global regularity of wave maps I–VII, preprints.

[39] D. Tataru, Local and global results for Wave Maps I, Comm. PDE 23, 1781–1793, 1998. | MR | Zbl

[40] D. Tataru. Strichartz estimates for operators with non smooth coefficients and the nonlinear wave equation, Amer. J. Math. 122, 349–376, 2000. | MR | Zbl

[41] D. Tataru, Strichartz estimates for second order hyperbolic operators with non smooth coefficients, J.A.M.S. 15 (2), 419–442, 2002. | MR | Zbl

[42] Q. Wang, Improved breakdown criterion for Einstein vacuum equation in CMC gauge, Comm. Pure Appl. Math. 65 (1), 21–76, 2012. | MR | Zbl

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