Improved Breakdown Criterion for Einstein Vacuum Equations in CMC Gauge

摘要

Let M_*=∪_(t∈[t_0,t_*)) ∑t be a part of vacuum globally hyperbolic space-time(M,g), foliated by constant mean curvature hypersurfaces ∑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 ∫_t_0~t_*(‖K‖L∞(∑_t)+‖▽log n‖L∞(∑_t)dt＜∞. The proof of this result relies on the second main result of the paper, which gives a uniform lower bound on the null radius of injectivity.