Improved breakdown criterion for Einstein vacuum equations in CMC gauge

Abstract

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

Cited by in corpus (1)

• On Ricci coefficients of null hypersurfaces with time foliation in Einstein vacuum space-time