  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.
