Voir la notice de l'acte provenant de la source Numdam
We review recent work on the local geometry and optimal regularity of Lorentzian manifolds with bounded curvature. Our main results provide an estimate of the injectivity radius of an observer, and a local canonical foliations by CMC (Constant Mean Curvature) hypersurfaces, together with spatially harmonic coordinates. In contrast with earlier results based on a global bound for derivatives of the curvature, our method requires only a sup-norm bound on the curvature near the given observer.
LeFloch, Philippe G. 1
@article{TSG_2007-2008__26__77_0,
author = {LeFloch, Philippe G.},
title = {Injectivity radius and optimal regularity of {Lorentzian} manifolds with bounded curvature},
journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie},
pages = {77--90},
publisher = {Institut Fourier},
address = {Grenoble},
volume = {26},
year = {2007-2008},
doi = {10.5802/tsg.261},
zbl = {1191.53052},
mrnumber = {2654598},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.5802/tsg.261/}
}
TY - JOUR AU - LeFloch, Philippe G. TI - Injectivity radius and optimal regularity of Lorentzian manifolds with bounded curvature JO - Séminaire de théorie spectrale et géométrie PY - 2007-2008 SP - 77 EP - 90 VL - 26 PB - Institut Fourier PP - Grenoble UR - http://geodesic.mathdoc.fr/articles/10.5802/tsg.261/ DO - 10.5802/tsg.261 LA - en ID - TSG_2007-2008__26__77_0 ER -
%0 Journal Article %A LeFloch, Philippe G. %T Injectivity radius and optimal regularity of Lorentzian manifolds with bounded curvature %J Séminaire de théorie spectrale et géométrie %D 2007-2008 %P 77-90 %V 26 %I Institut Fourier %C Grenoble %U http://geodesic.mathdoc.fr/articles/10.5802/tsg.261/ %R 10.5802/tsg.261 %G en %F TSG_2007-2008__26__77_0
LeFloch, Philippe G. Injectivity radius and optimal regularity of Lorentzian manifolds with bounded curvature. Séminaire de théorie spectrale et géométrie, Tome 26 (2007-2008), pp. 77-90. doi : 10.5802/tsg.261. http://geodesic.mathdoc.fr/articles/10.5802/tsg.261/
[1] M.T. Anderson, Convergence and rigidity of metrics under Ricci curvature bounds, Invent. Math. 102 (1990), 429–445. | Zbl | MR
[2] M.T. Anderson, Regularity for Lorentz metrics under curvature bounds, Jour. Math. Phys. 44 (2003), 2994–3012. | Zbl | MR
[3] L. Andersson and V. Moncrief, Elliptic-hyperbolic systems and the Einstein equations, Ann. Inst. Henri Poincaré 4 (2003), 1–34. | Zbl | MR
[4] L. Andersson and V. Moncrief, Future complete vacuum spacetimes, in “The Einstein equations and the large scale behavior of gravitational fields”, Birkhäuser, Basel, 2004, pp. 299–330. | Zbl | MR
[5] R. Bartnik and L. Simon, Spacelike hypersurfaces with prescribed boundary values and mean curvature, Commun. Math. Phys. 87 (1982), 131–152. | Zbl | MR
[6] A. Besse, Einstein manifolds, Ergebenisse Math. Series 3, Springer Verlag, 1987. | Zbl | MR
[7] J. Cheeger, M. Gromov, and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Diff. Geom. 17 (1982), 15–53. | Zbl | MR
[8] B.-L. Chen and P.G. LeFloch, Injectivity radius estimates for Lorentzian manifolds, Commun. Math. Phys. 278 (2008), 679–713. | Zbl | MR
[9] B.-L. Chen and P.G. LeFloch, Local foliations and optimal regularity of Einstein spacetimes, submitted.
[10] D.M. DeTurck and J.L. Kazdan, Some regularity theorems in Riemannian geometry. Ann. Sci. École Norm. Sup. 14 (1981), 249–260. | Zbl | MR | mathdoc-id
[11] C. Gerhardt, H-surfaces in Lorentzian manifolds, Commun. Math. Phys. 89 (1983), 523–533. | Zbl | MR
[12] S. Hawking and G.F. Ellis, The large scale structure of space-time, Cambridge Univ. Press, 1973. | Zbl | MR
[13] J. Jost and H. Karcher, Geometrische Methoden zur Gewinnung von a-priori-Schranken für harmonische Abbildungen, Manuscripta Math. 40 (1982), 27–77. | Zbl | MR
[14] S. Klainerman and I. Rodnianski, On the radius of injectivity of null hypersurfaces, J. Amer. Math. Soc. 21 (2008), 775–795. | MR
[15] S. Klainerman and I. Rodnianski, On the breakdown criterion in general relativity, preprint, 2008.
[16] R. Penrose, Techniques of differential topology in relativity, CBMS-NSF Region. Conf. Series Appli. Math., Vol. 7, 1972. | Zbl | MR
[17] P. Petersen, Convergence theorems in Riemannian geometry, in “Comparison Geometry” (Berkeley, CA, 1992–93), MSRI Publ. 30, Cambridge Univ. Press, 1997, pp. 167–202. | Zbl | MR
Cité par Sources :