Causal geometry of rough Einstein CMCSH spacetime

Abstract

This is the second (and last) part of a series in which we consider very rough solutions to Cauchy problem for the Einstein vacuum equations in constant mean curvature and spatial harmonic (CMCSH) gauge, and we obtain a local well-posedness result in Hs with s > 2. The novelty of our approach lies in that, without resorting to the standard paradifferential regularization over the rough Einstein metric g, we manage to implement the commuting vector field approach and prove a Strichartz estimate for the geometric wave equation □g ϕ = 0 in a direct manner. This direct treatment would not work without gaining sufficient regularity on the background geometry. In this paper, we analyze the geometry of null hypersurfaces in rough Einstein spacetimes in terms of Hs data. We provide an integral control on the spatial supremum of the connection coefficients [Formula: see text], ζ, which is crucially tied to the Strichartz estimates established in the first part.