Non linear stability of einsteinian spacetimes with U(1) isometry group

We prove global completeness in the expanding direction of spacetimes satisfying the vacuum Einstein equations on a manifold of the form $\Sigma \times S^{1}\times R$where$\Sigma $isacompactsurfaceofgenus$G>1.$ The Cauchy data are supposed to be invariant with respect to the group $S^1$ and sufficiently small, but we do not impose a restrictive hypothesis made in gr-qc 0112049 on the lowest eigenvalue of a relevant Laplacian. The total energy decay still holds, but its rate depends of the asymptotic value of this eigenvalue.