2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/34501We prove global completeness in the expanding direction of spacetimes satisfying the vacuum Einstein equations on a manifold of the form $Σ\times S^{1}\times R$ where $Σ$ 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.Conference in honor of J. Leray, Tokyo 2001, Kajitani and Vaillant ed. to appear BirkhauserGeneral Relativity and Quantum Cosmologytext