The Localised Bounded L2-Curvature Theorem

In this paper, we prove a localised version of the bounded L2-curvature theorem of (Klainerman et al. Invent Math 202(1):91-216, 2015). More precisely, we consider initial data for the Einstein vacuum equations posed on a compact spacelike hypersurface Σ with boundary, and show that the time of existence of a classical solution depends only on an L2-bound on the Ricci curvature, an L4-bound on the second fundamental form of ∂Σ⊂Σ, an H1-bound on the second fundamental form, and a lower bound on the volume radius at scale 1 of Σ. Our localisation is achieved by first proving a localised bounded L2-curvature theorem for small data posed on B(0, 1), and then using the scaling of the Einstein equations and a low regularity covering argument on Σ to reduce from large data on Σ to small data on B(0, 1). The proof uses the author's previous works and the bounded L2-curvature theorem as black boxes.