Around the bounded L 2 curvature conjecture in general relativity
Sergiu Klainerman1; Igor Rodnianski1; Jeremie Szeftel2
1 Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton NJ 08544-1000 USA
2 Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton NJ 08544-1000 USA and Mathématiques Appliquées de Bordeaux, UMR CNRS 5466, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence cedex FRANCE
Journées équations aux dérivées partielles (2008), article no. 9, 15 p.

We report on recent progress obtained on the construction and control of a parametrix to the homogeneous wave equation □ g φ=0, where ≫ is a rough metric satisfying the Einstein vacuum equations. Controlling such a parametrix as well as its error term when one only assumes L 2 bounds on the curvature tensor R of ≫ is a major step towards the proof of the bounded L 2 curvature conjecture.

DOI: 10.5802/jedp.53
Sergiu Klainerman 1; Igor Rodnianski 1; Jeremie Szeftel 2

1 Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton NJ 08544-1000 USA
2 Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton NJ 08544-1000 USA and Mathématiques Appliquées de Bordeaux, UMR CNRS 5466, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence cedex FRANCE 
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Sergiu Klainerman; Igor Rodnianski; Jeremie Szeftel. Around the bounded $L^2$ curvature conjecture in general relativity. Journées équations aux dérivées partielles (2008), article  no. 9, 15 p. doi : 10.5802/jedp.53. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.53/

[1] Hajer Bahouri; Jean-Yves Chemin Équations d’ondes quasilinéaires et effet dispersif, Internat. Math. Res. Notices (1999) no. 21, pp. 1141-1178 | MR | Zbl

[2] Hajer Bahouri; Jean-Yves Chemin Équations d’ondes quasilinéaires et estimations de Strichartz, Amer. J. Math., Volume 121 (1999) no. 6, pp. 1337-1377 | MR | Zbl

[3] Robert Bartnik Existence of maximal surfaces in asymptotically flat spacetimes, Comm. Math. Phys., Volume 94 (1984) no. 2, pp. 155-175 | MR | Zbl

[4] Demetrios Christodoulou; Sergiu Klainerman The global nonlinear stability of the Minkowski space, Princeton Mathematical Series, 41, Princeton University Press, Princeton, NJ, 1993 | MR | Zbl

[5] Y. Fourès-Bruhat Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires, Acta Math., Volume 88 (1952), pp. 141-225 | MR | Zbl

[6] Thomas J. R. Hughes; Tosio Kato; Jerrold E. Marsden Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rational Mech. Anal., Volume 63 (1976) no. 3, p. 273-294 (1977) | MR | Zbl

[7] S. Klainerman; M. Machedon Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math., Volume 46 (1993) no. 9, pp. 1221-1268 | MR | Zbl

[8] S. Klainerman; M. Machedon Estimates for null forms and the spaces H s,δ , Internat. Math. Res. Notices (1996) no. 17, pp. 853-865 | MR | Zbl

[9] S. Klainerman; I. Rodnianski Improved local well-posedness for quasilinear wave equations in dimension three, Duke Math. J., Volume 117 (2003) no. 1, pp. 1-124 | MR | Zbl

[10] S. Klainerman; I. Rodnianski A geometric approach to the Littlewood-Paley theory, Geom. Funct. Anal., Volume 16 (2006) no. 1, pp. 126-163 | MR

[11] S. Klainerman; I. Rodnianski Sharp trace theorems for null hypersurfaces on Einstein metrics with finite curvature flux, Geom. Funct. Anal., Volume 16 (2006) no. 1, pp. 164-229 | MR

[12] Sergiu Klainerman PDE as a unified subject, Geom. Funct. Anal. (2000) no. Special Volume, Part I, pp. 279-315 GAFA 2000 (Tel Aviv, 1999) | MR | Zbl

[13] Sergiu Klainerman; Igor Rodnianski Ricci defects of microlocalized Einstein metrics, J. Hyperbolic Differ. Equ., Volume 1 (2004) no. 1, pp. 85-113 | MR | Zbl

[14] Sergiu Klainerman; Igor Rodnianski Bilinear estimates on curved space-times, J. Hyperbolic Differ. Equ., Volume 2 (2005) no. 2, pp. 279-291 | MR

[15] Sergiu Klainerman; Igor Rodnianski Causal geometry of Einstein-vacuum spacetimes with finite curvature flux, Invent. Math., Volume 159 (2005) no. 3, pp. 437-529 | MR | Zbl

[16] Sergiu Klainerman; Igor Rodnianski The causal structure of microlocalized rough Einstein metrics, Ann. of Math. (2), Volume 161 (2005) no. 3, pp. 1195-1243 | MR | Zbl

[17] Sergiu Klainerman; Igor Rodnianski Rough solutions of the Einstein-vacuum equations, Ann. of Math. (2), Volume 161 (2005) no. 3, pp. 1143-1193 | MR | Zbl

[18] Hans Lindblad Counterexamples to local existence for semi-linear wave equations, Amer. J. Math., Volume 118 (1996) no. 1, pp. 1-16 | MR | Zbl

[19] Gustavo Ponce; Thomas C. Sideris Local regularity of nonlinear wave equations in three space dimensions, Comm. Partial Differential Equations, Volume 18 (1993) no. 1-2, pp. 169-177 | MR | Zbl

[20] Hart F. Smith A parametrix construction for wave equations with C 1,1 coefficients, Ann. Inst. Fourier (Grenoble), Volume 48 (1998) no. 3, pp. 797-835 | Numdam | MR | Zbl

[21] Hart F. Smith; Christopher D. Sogge On Strichartz and eigenfunction estimates for low regularity metrics, Math. Res. Lett., Volume 1 (1994) no. 6, pp. 729-737 | MR | Zbl

[22] Hart F. Smith; Daniel Tataru Sharp local well-posedness results for the nonlinear wave equation, Ann. of Math. (2), Volume 162 (2005) no. 1, pp. 291-366 | MR | Zbl

[23] Elias M. Stein Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, 43, Princeton University Press, Princeton, NJ, 1993 (With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III) | MR | Zbl

[24] Daniel Tataru Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation, Amer. J. Math., Volume 122 (2000) no. 2, pp. 349-376 | MR | Zbl

[25] Daniel Tataru Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. III, J. Amer. Math. Soc., Volume 15 (2002) no. 2, p. 419-442 (electronic) | MR | Zbl

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