On the breakdown criterion in general relativity

Abstract

We give a geometric criterion for the breakdown of an Einstein vacuum space-time foliated by a constant mean curvature, or maximal, foliation. More precisely we show that the foliated space-time can be extended as long as the the second fundamental form and the first derivatives of the logarithm of the lapse of the foliation remain uniformly bounded. No restrictions on the size of the initial data are made.