Half polarized U(1)U(1) symmetric vacuum spacetimes with AVTD behavior

In a previous work, we used a polarization condition to show that there is a family of U(1)U(1) symmetric solutions of the vacuum Einstein equations on Σ×S1×RΣ×S1×R (ΣΣ any two-dimensional manifold) such that each exhibits AVTD1 behavior in the neighbourhood of its singularity. Here we consider the general case of S1S1 bundles over the base Σ×RΣ×R and determine a condition, called the half polarization condition, necessary and sufficient in our context, for AVTD behavior near the singularity.