Communications in Analysis and Geometry

Volume 22 (2014)

Number 5

Null injectivity estimate under an upper bound on the curvature

Pages: 965 – 996

DOI: http://dx.doi.org/10.4310/CAG.2014.v22.n5.a8

Authors

James D. E. Grant (Department of Mathematics, University of Surrey, Guildford, United Kingdom)

Philippe G. Lefloch (Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique, Université Pierre et Marie Curie (Paris 6), Paris, France)

Abstract

We establish a uniform estimate for the injectivity radius of the past null cone of a point in a general Lorentzian manifold foliated by space-like hypersurfaces and satisfying an upper curvature bound. Precisely, our main assumptions are, on one hand, upper bounds on the null curvature of the spacetime and the lapse function of the foliation and sup-norm bounds on the deformation tensors of the foliation. Our proof is inspired by techniques from Riemannian geometry, and it should be noted that we impose no restriction on the size of the bound satisfied by the curvature or deformation tensors, and allow for metrics that are “far” from the Minkowski one. The relevance of our estimate is illustrated with a class of plane-symmetric spacetimes which satisfy our assumptions but admit no uniform lower bound on the curvature not even in the $L^2$ norm. The conditions we put forward, therefore, lead to a uniform control of the spacetime geometry and should be useful in the context of general relativity.

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