Future Global in time einsteinian spacetimes with U(1) isometry group

We prove that spacetimes satisfying the vacuum Einstein equations on a manifold of the form $\Sigma \times U(1)\times R$ 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.