Future complete Einsteinian space times with U(1) isometry group

We prove that spacetimes satisfying the vacuum Einstein equations on a manifold of the form Sigma x U(1) x R, where Sigma is a compact surface of genus G > 1 and where the Cauchy data is invariant with respect to U(1) and sufficiently small exist for an infinite proper time in the expanding direction. (C) 2001 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.