An integral breakdown criterion for Einstein vacuum equations in the case of asymptotically flat spacetimes

Abstract

We will give in this paper the proof of an integral breakdown criterion for Einstein vacuum equations. In a recent article of S.Klainerman and I.Rodnianski a new breakdown criterion was proved as a result of a sequence of articles involving new techniques. However, in this article, the authors mentioned that it was likely possible to prove a sharper result involving an integral condition instead of a pointwise one. This paper is concerned with giving the proof of this improvement. Moreover the proof of this breakdown criterion was written in the original article for a foliation of constant mean curvature, we will present it here for a maximal foliation which leads to some difficulties due to the non-compacity of the leaves of such a foliation and the use of weighted Sobolev norms.