On the radius of injectivity of null hypersurfaces

Klainerman, Sergiu¹ ; Rodnianski, Igor¹

¹ Department of Mathematics, Princeton University, Princeton, New Jersey 08544

Journal of the American Mathematical Society, Tome 21 (2008) no. 3, pp. 775-795

Résumé

We investigate the regularity of past (future) boundaries of points in regular, Einstein vacuum spacetimes. We provide conditions, expressed in terms of a space-like foliation and which imply, in particular, uniform $L^2$ bounds for the curvature tensor, sufficient to ensure the local nondegeneracy of these boundaries. More precisely we provide a uniform lower bound on the radius of injectivity of the null boundaries $\mathcal {N}^{\pm }(p)$ of the causal past (future) sets $\mathcal {J}^{\pm }(p)$. Such lower bounds are essential in understanding the causal structure and the related propagation properties of solutions to the Einstein equations. They are particularly important in construction of an effective Kirchoff-Sobolev type parametrix for solutions of wave equations on $\mathbf {M}$. Such parametrices are used by the authors in a forthcoming paper to prove a large data break-down criterion for solutions of the Einstein vacuum equations.

Détail

DOI : 10.1090/S0894-0347-08-00592-4

Bibliographie

[1] Anderson, Michael T. Regularity for Lorentz metrics under curvature bounds J. Math. Phys. 2003 2994 3012

[2] Anderson, Michael T. Cheeger-Gromov theory and applications to general relativity 2004 347 377

[3] Anderson, Michael T. On long-time evolution in general relativity and geometrization of 3-manifolds Comm. Math. Phys. 2001 533 567

[4] Anderson, Michael T., Cheeger, Jeff Diffeomorphism finiteness for manifolds with Ricci curvature and 𝐿^{𝑛/2}-norm of curvature bounded Geom. Funct. Anal. 1991 231 252

[5] Fourã¨S-Bruhat, Y. Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires Acta Math. 1952 141 225

[6] Cheeger, Jeff Finiteness theorems for Riemannian manifolds Amer. J. Math. 1970 61 74

[7] Christodoulou, Demetrios, Klainerman, Sergiu The global nonlinear stability of the Minkowski space 1993

[8] Friedlander, F. G. The wave equation on a curved space-time 1975

[9] Galloway, Gregory J. Maximum principles for null hypersurfaces and null splitting theorems Ann. Henri Poincaré 2000 543 567

[10] Hawking, S. W., Ellis, G. F. R. The large scale structure of space-time 1973

[11] 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. 1976

[12] Klainerman, Sergiu, Rodnianski, Igor Causal geometry of Einstein-vacuum spacetimes with finite curvature flux Invent. Math. 2005 437 529

[13] Klainerman, S., Rodnianski, I. A geometric approach to the Littlewood-Paley theory Geom. Funct. Anal. 2006 126 163

[14] Klainerman, S., Rodnianski, I. Sharp trace theorems for null hypersurfaces on Einstein metrics with finite curvature flux Geom. Funct. Anal. 2006 164 229

[15] Petersen, P., Shteingold, S. D., Wei, G. Comparison geometry with integral curvature bounds Geom. Funct. Anal. 1997 1011 1030

[16] Petersen, Peter Convergence theorems in Riemannian geometry 1997 167 202