Around the bounded L 2 curvature conjecture in general relativity
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.

@article{JEDP_2008____A9_0,
     author = {Klainerman, Sergiu and Rodnianski, Igor and Szeftel, Jeremie},
     title = {Around the bounded $L^2$ curvature conjecture in general relativity},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2008},
     doi = {10.5802/jedp.53},
     language = {en},
     url = {http://www.numdam.org/item/JEDP_2008____A9_0}
}
Klainerman, Sergiu; Rodnianski, Igor; Szeftel, Jeremie. 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. http://www.numdam.org/item/JEDP_2008____A9_0/

[1] Bahouri, Hajer; Chemin, Jean-Yves Équations d’ondes quasilinéaires et effet dispersif, Internat. Math. Res. Notices (1999) no. 21, pp. 1141-1178 | MR 1728676 | Zbl 0938.35106

[2] Bahouri, Hajer; Chemin, Jean-Yves Équations d’ondes quasilinéaires et estimations de Strichartz, Amer. J. Math., Tome 121 (1999) no. 6, pp. 1337-1377 | MR 1719798 | Zbl 0952.35073

[3] Bartnik, Robert Existence of maximal surfaces in asymptotically flat spacetimes, Comm. Math. Phys., Tome 94 (1984) no. 2, pp. 155-175 | MR 761792 | Zbl 0548.53054

[4] Christodoulou, Demetrios; Klainerman, Sergiu The global nonlinear stability of the Minkowski space, Princeton University Press, Princeton, NJ, Princeton Mathematical Series, Tome 41 (1993) | MR 1316662 | Zbl 0827.53055

[5] Fourès-Bruhat, Y. Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires, Acta Math., Tome 88 (1952), pp. 141-225 | MR 53338 | Zbl 0049.19201

[6] Hughes, Thomas J. R.; Kato, Tosio; Marsden, Jerrold E. Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rational Mech. Anal., Tome 63 (1976) no. 3, p. 273-294 (1977) | MR 420024 | Zbl 0361.35046

[7] Klainerman, S.; Machedon, M. Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math., Tome 46 (1993) no. 9, pp. 1221-1268 | MR 1231427 | Zbl 0803.35095

[8] Klainerman, S.; Machedon, M. Estimates for null forms and the spaces H s,δ , Internat. Math. Res. Notices (1996) no. 17, pp. 853-865 | MR 1420552 | Zbl 0909.35095

[9] Klainerman, S.; Rodnianski, I. Improved local well-posedness for quasilinear wave equations in dimension three, Duke Math. J., Tome 117 (2003) no. 1, pp. 1-124 | MR 1962783 | Zbl 1031.35091

[10] Klainerman, S.; Rodnianski, I. A geometric approach to the Littlewood-Paley theory, Geom. Funct. Anal., Tome 16 (2006) no. 1, pp. 126-163 | MR 2221254 | Zbl pre05029441

[11] Klainerman, S.; Rodnianski, I. Sharp trace theorems for null hypersurfaces on Einstein metrics with finite curvature flux, Geom. Funct. Anal., Tome 16 (2006) no. 1, pp. 164-229 | MR 2221255 | Zbl pre05029445

[12] Klainerman, Sergiu PDE as a unified subject, Geom. Funct. Anal. (2000) no. Special Volume, Part I, pp. 279-315 (GAFA 2000 (Tel Aviv, 1999)) | MR 1826256 | Zbl 1002.35002

[13] Klainerman, Sergiu; Rodnianski, Igor Ricci defects of microlocalized Einstein metrics, J. Hyperbolic Differ. Equ., Tome 1 (2004) no. 1, pp. 85-113 | MR 2052472 | Zbl 1063.53051

[14] Klainerman, Sergiu; Rodnianski, Igor Bilinear estimates on curved space-times, J. Hyperbolic Differ. Equ., Tome 2 (2005) no. 2, pp. 279-291 | MR 2151111 | Zbl pre02202321

[15] Klainerman, Sergiu; Rodnianski, Igor Causal geometry of Einstein-vacuum spacetimes with finite curvature flux, Invent. Math., Tome 159 (2005) no. 3, pp. 437-529 | MR 2125732 | Zbl 1136.58018

[16] Klainerman, Sergiu; Rodnianski, Igor The causal structure of microlocalized rough Einstein metrics, Ann. of Math. (2), Tome 161 (2005) no. 3, pp. 1195-1243 | MR 2180401 | Zbl 1089.83007

[17] Klainerman, Sergiu; Rodnianski, Igor Rough solutions of the Einstein-vacuum equations, Ann. of Math. (2), Tome 161 (2005) no. 3, pp. 1143-1193 | MR 2180400 | Zbl 1089.83006

[18] Lindblad, Hans Counterexamples to local existence for semi-linear wave equations, Amer. J. Math., Tome 118 (1996) no. 1, pp. 1-16 | MR 1375301 | Zbl 0855.35080

[19] Ponce, Gustavo; Sideris, Thomas C. Local regularity of nonlinear wave equations in three space dimensions, Comm. Partial Differential Equations, Tome 18 (1993) no. 1-2, pp. 169-177 | MR 1211729 | Zbl 0803.35096

[20] Smith, Hart F. A parametrix construction for wave equations with C 1,1 coefficients, Ann. Inst. Fourier (Grenoble), Tome 48 (1998) no. 3, pp. 797-835 | Numdam | MR 1644105 | Zbl 0974.35068

[21] Smith, Hart F.; Sogge, Christopher D. On Strichartz and eigenfunction estimates for low regularity metrics, Math. Res. Lett., Tome 1 (1994) no. 6, pp. 729-737 | MR 1306017 | Zbl 0832.35018

[22] Smith, Hart F.; Tataru, Daniel Sharp local well-posedness results for the nonlinear wave equation, Ann. of Math. (2), Tome 162 (2005) no. 1, pp. 291-366 | MR 2178963 | Zbl 1098.35113

[23] Stein, Elias M. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, Princeton, NJ, Princeton Mathematical Series, Tome 43 (1993) (With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III) | MR 1232192 | Zbl 0821.42001

[24] Tataru, Daniel Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation, Amer. J. Math., Tome 122 (2000) no. 2, pp. 349-376 | MR 1749052 | Zbl 0959.35125

[25] Tataru, Daniel Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. III, J. Amer. Math. Soc., Tome 15 (2002) no. 2, p. 419-442 (electronic) | MR 1887639 | Zbl 0990.35027