This paper reports on the recent proof of the bounded curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the -norm of the curvature and a lower bound of the volume radius of the corresponding initial data set.
@article{SLSEDP_2014-2015____A1_0,
author = {Sergiu Klainerman and Igor Rodnianski and J\'er\'emie Szeftel},
title = {The resolution of the bounded $L^2$ curvature conjecture in general relativity},
journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
note = {talk:1},
publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2014-2015},
doi = {10.5802/slsedp.65},
language = {en},
url = {https://slsedp.centre-mersenne.org/articles/10.5802/slsedp.65/}
}
TY - JOUR TI - The resolution of the bounded $L^2$ curvature conjecture in general relativity JO - Séminaire Laurent Schwartz — EDP et applications N1 - talk:1 PY - 2014-2015 DA - 2014-2015/// PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique UR - https://slsedp.centre-mersenne.org/articles/10.5802/slsedp.65/ UR - https://doi.org/10.5802/slsedp.65 DO - 10.5802/slsedp.65 LA - en ID - SLSEDP_2014-2015____A1_0 ER -
%0 Journal Article %T The resolution of the bounded $L^2$ curvature conjecture in general relativity %J Séminaire Laurent Schwartz — EDP et applications %Z talk:1 %D 2014-2015 %I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique %U https://doi.org/10.5802/slsedp.65 %R 10.5802/slsedp.65 %G en %F SLSEDP_2014-2015____A1_0
Sergiu Klainerman; Igor Rodnianski; Jérémie Szeftel. The resolution of the bounded $L^2$ curvature conjecture in general relativity. Séminaire Laurent Schwartz — EDP et applications (2014-2015), Talk no. 1, 18 p. doi : 10.5802/slsedp.65. https://slsedp.centre-mersenne.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.
[2] H. Bahouri, J.-Y. Chemin, Équations d’ondes quasilinéaires et effet dispersif, IMRN, 21, 1141–1178, 1999.
[3] Y. C. Bruhat, Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires, Acta Math. 88, 141–225, 1952.
[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.
[6] D. Christodoulou, The instability of naked singularities in the gravitational collapse of a scalar field, Ann. of Math., 149, 183–217,1999.
[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.
[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.
[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.
[10] S. Klainerman, M. Machedon, Finite energy solutions of the Maxwell-Klein-Gordon equations, Duke Math. J. 74, 19–44, 1994.
[11] S. Klainerman, M. Machedon, Finite Energy Solutions for the Yang-Mills Equations in , Annals of Math. 142, 39–119, 1995.
[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.
[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.
[14] S. Klainerman, I. Rodnianski, Rough solutions to the Einstein vacuum equations, Annals of Math. 161, 1143–1193, 2005.
[15] S. Klainerman, I. Rodnianski, Bilinear estimates on curved space-times, J. Hyperbolic Differ. Equ. 2 (2), 279–291, 2005.
[16] S. Klainerman, I. Rodnianski, Casual geometry of Einstein vacuum space-times with finite curvature flux, Inventiones 159, 437–529, 2005.
[17] S. Klainerman, I. Rodnianski, Sharp trace theorems on null hypersurfaces, GAFA 16 (1), 164–229, 2006.
[18] S. Klainerman, I. Rodnianski, A geometric version of Littlewood-Paley theory, GAFA 16 (1), 126–163, 2006.
[19] S. Klainerman, I. Rodnianski, On a break-down criterion in General Relativity, J. Amer. Math. Soc. 23, 345–382, 2010.
[20] S. Klainerman, I. Rodnianski, J. Szeftel, The Bounded Curvature Conjecture, arXiv:1204.1767, 91 p, 2012.
[21] J. Krieger, W. Schlag, Concentration compactness for critical wave maps, Monographs of the European Mathematical Society, 2012.
[22] H. Lindblad, Counterexamples to local existence for quasilinear wave equations, Amer. J. Math. 118 (1), 1–16, 1996.
[23] H. Lindblad, I. Rodnianski, The weak null condition for the Einstein vacuum equations, C. R. Acad. Sci. 336, 901–906, 2003.
[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.
[27] H. F. Smith, A parametrix construction for wave equations with coefficients, Ann. Inst. Fourier (Grenoble) 48, 797–835, 1998.
[28] H.F. Smith, D. Tataru, Sharp local well-posedness results for the nonlinear wave equation, Ann. of Math. 162, 291–366, 2005.
[29] S. Sobolev, Méthodes nouvelles pour résoudre le problème de Cauchy pour les équations linéaires hyperboliques normales, Matematicheskii Sbornik, 1 (43), 31–79, 1936.
[30] E. Stein, Harmonic Analysis, Princeton University Press, 1993.
[31] J. Sterbenz, D. Tataru, Regularity of Wave-Maps in dimension , Comm. Math. Phys. 298 (1), 231–264, 2010.
[32] J. Sterbenz, D. Tataru, Energy dispersed large data wave maps in dimensions, Comm. Math. Phys. 298 (1), 139–230, 2010.
[33] J. Szeftel, Parametrix for wave equations on a rough background I: Regularity of the phase at initial time, arXiv:1204.1768, 145 p, 2012.
[34] J. Szeftel, Parametrix for wave equations on a rough background II: Construction of the parametrix and control at initial time, arXiv:1204.1769, 84 p, 2012.
[35] J. Szeftel, Parametrix for wave equations on a rough background III: Space-time regularity of the phase, arXiv:1204.1770, 276 p, 2012.
[36] J. Szeftel, Parametrix for wave equations on a rough background IV: Control of the error term, arXiv:1204.1771, 284 p, 2012.
[37] J. Szeftel, Sharp Strichartz estimates for the wave equation on a rough background, arXiv:1301.0112, 30 p, 2013.
[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.
[40] D. Tataru. Strichartz estimates for operators with non smooth coefficients and the nonlinear wave equation, Amer. J. Math. 122, 349–376, 2000.
[41] D. Tataru, Strichartz estimates for second order hyperbolic operators with non smooth coefficients, J.A.M.S. 15 (2), 419–442, 2002.
[42] Q. Wang, Improved breakdown criterion for Einstein vacuum equation in CMC gauge, Comm. Pure Appl. Math. 65 (1), 21–76, 2012.
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