Injectivity radius and optimal regularity of Lorentzian manifolds with bounded curvature
Philippe G. LeFloch1
1 Université Pierre et Marie Curie (Paris VI) Laboratoire Jacques-Louis Lions (UMR CNRS 7598) 4 place Jussieu 75252 Paris (France)
Séminaire de théorie spectrale et géométrie, Tome 26 (2007-2008), pp. 77-90.

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.

DOI : 10.5802/tsg.261
Classification : 83C05, 53C50, 53C12
Keywords: Lorentzian geometry, injectivity radius, constant mean curvature foliation, harmonic coordinates
Philippe G. LeFloch 1

1 Université Pierre et Marie Curie (Paris VI) Laboratoire Jacques-Louis Lions (UMR CNRS 7598) 4 place Jussieu 75252 Paris (France) 
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Philippe G. LeFloch. 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. https://proceedings.centre-mersenne.org/articles/10.5802/tsg.261/

[1] M.T. Anderson, Convergence and rigidity of metrics under Ricci curvature bounds, Invent. Math. 102 (1990), 429–445. | MR | Zbl

[2] M.T. Anderson, Regularity for Lorentz metrics under curvature bounds, Jour. Math. Phys. 44 (2003), 2994–3012. | MR | Zbl

[3] L. Andersson and V. Moncrief, Elliptic-hyperbolic systems and the Einstein equations, Ann. Inst. Henri Poincaré 4 (2003), 1–34. | MR | Zbl

[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. | MR | Zbl

[5] R. Bartnik and L. Simon, Spacelike hypersurfaces with prescribed boundary values and mean curvature, Commun. Math. Phys. 87 (1982), 131–152. | MR | Zbl

[6] A. Besse, Einstein manifolds, Ergebenisse Math. Series 3, Springer Verlag, 1987. | MR | Zbl

[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. | MR | Zbl

[8] B.-L. Chen and P.G. LeFloch, Injectivity radius estimates for Lorentzian manifolds, Commun. Math. Phys. 278 (2008), 679–713. | MR | Zbl

[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. | Numdam | MR | Zbl

[11] C. Gerhardt, H-surfaces in Lorentzian manifolds, Commun. Math. Phys. 89 (1983), 523–533. | MR | Zbl

[12] S. Hawking and G.F. Ellis, The large scale structure of space-time, Cambridge Univ. Press, 1973. | MR | Zbl

[13] J. Jost and H. Karcher, Geometrische Methoden zur Gewinnung von a-priori-Schranken für harmonische Abbildungen, Manuscripta Math. 40 (1982), 27–77. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

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