Topologically general U(1) symmetric Einstein spacetimes with AVTD behavior

We use Fuchsian methods to show that, for any two dimensional manifold ${\mathrm{\Sigma }}^2$, there is a large family of U(1) symmetric solutions of the vacuum Einstein equations on the manifold $\mathrm{\Sigma }\times S^1\times \mathbb{R}$, each of which has AVTD behavior in the neighborhood of its singularity.