Improved breakdown criterion for einstein vacuum equations in CMC gauge

Let ${\cal M}_*$ = ∪   t ∈[ t   0 , t   * )Σ t be a part of vacuum globally hyperbolic space‐time ( M , g ), foliated by constant mean curvature hypersurfaces Σ t with t 0 < 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$$ \int_{t_0}^{t_*}(\vert\vert k\vert\vert_{L^\infty(\Sigma_t)}+\vert\vert\nabla \log n\vert\vert_{L^\infty(\Sigma_t)}) dt <\infty. $$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.