ON THE RADIUS OF INJECTIVITY OF NULL HYPERSURFACES.

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 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 of the causal past (future) sets . 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 . 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

[1]  S. Klainerman,et al.  Sharp trace theorems for null hypersurfaces on Einstein metrics with finite curvature flux , 2003, math/0309459.

[2]  S. Klainerman,et al.  A geometric approach to the Littlewood–Paley theory , 2003, math/0309463.

[3]  S. Klainerman,et al.  Causal geometry of Einstein-Vacuum spacetimes with finite curvature flux , 2003, math/0308123.

[4]  Michael T. Anderson.  Regularity for Lorentz metrics under curvature bounds , 2002, gr-qc/0209073.

[5]  Michael T. Anderson.  Cheeger-Gromov Theory and Applications to General Relativity , 2002, gr-qc/0208079.

[6]  K. Roberts,et al.  Thesis , 2002 .

[7]  Michael T. Anderson.  On Long-Time Evolution in General Relativity¶and Geometrization of 3-Manifolds , 2000, gr-qc/0006042.

[8]  G. Galloway.  Maximum Principles for Null Hypersurfaces and Null Splitting Theorems , 1999, math/9909158.

[9]  G. Wei,et al.  Comparison Geometry with Integral Curvature Bounds , 1997 .

[10]  Y. Okada.  Solvability of Convolution Operators , 1994 .

[11]  S. Klainerman,et al.  The Global Nonlinear Stability of the Minkowski Space. , 1994 .

[12]  J. Cheeger,et al.  Diffeomorphism finiteness for manifolds with ricci curvature andLn/2-norm of curvature bounded , 1991 .

[13]  Jerrold E. Marsden,et al.  Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity , 1977 .

[14]  D. S. Jones,et al.  THE WAVE EQUATION ON A CURVED SPACE‐TIME , 1977 .

[15]  George F. R. Ellis,et al.  The Large Scale Structure of Space-Time , 2023 .

[16]  Y. Fourès-Bruhat,et al.  Théorème d'existence pour certains systèmes d'équations aux dérivées partielles non linéaires , 1952 .

[17]  P. Petersen.  Convergence Theorems in Riemannian Geometry , 1997 .

[18]  J. Cheeger.  FINITENESS THEOREMS FOR RIEMANNIAN MANIFOLDS. , 1970 .