  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 $ is a compact surface of genus $G>1.$ 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.
