Improved breakdown criterion for Einstein vacuum equations in CMC gauge

Let $\M_*=\cup_{t\in [t_0, t_*)} \Sigma_t$ be a part of vacuum globally hyperbolic space-time $(\bM, \bg)$, foliated by constant mean curvature hypersurfaces $\Sigma_t$ with $t_0<t_*<0$. We show that the foliation can be extended beyond $t_*$ if the second fundamental form $k$ and the lapse function $n$ satisfy $$\int_{t_0}^{t_*}(\|k\|_{L^\infty(\Sigma_t)}+\|\nab \log n\|_{L^\infty(\Sigma_t)}) dt <\infty.$$ This improves the existing breakdown criteria for Einstein vacuum equations.