Improved breakdown criterion for einstein vacuum equations in CMC gauge

Let ${\cal M}_*$ = ∪ Σ_t be a part of vacuum globally hyperbolic space‐time ( M , g ), foliated by constant mean curvature hypersurfaces Σ_t with t₀ < t_* < 0. We improve the existing breakdown criteria for Einstein vacuum equations by showing that the foliation can be extended beyond t_* provided the second fundamental form k and the lapse function n satisfy the weaker condition The proof of this result %in particular relies on the second main result of the paper, which gives a uniform lower bound on the null radius of injectivity. © 2011 Wiley Periodicals, Inc.