Regularity for Lorentz metrics under curvature bounds

Abstract

Let (M, g) be an (n+1)-dimensional space–time, with bounded curvature, with respect to a bounded framing. If (M, g) is vacuum, or satisfies a weak condition on the stress-energy tensor, then it is shown that (M, g) locally admits coordinate systems in which the Lorentz metric g is well-controlled in the (space–time) Sobolev space L2,p, for any p<∞. This result is essentially optimal. The result allows one to control the regularity of limits of sequences of space–times, with uniformly bounded curvature, and has applications to the structure of boundaries and extensions of space–times.