  We define the class of weakly regular spacetimes with T2 symmetry, and
investigate their global geometry structure. We formulate the initial value
problem for the Einstein vacuum equations with weak regularity, and establish
the existence of a global foliation by the level sets of the area R of the
orbits of symmetry, so that each leaf can be regarded as an initial
hypersurface. Except for the flat Kasner spacetimes which are known explicitly,
R takes all positive values. Our weak regularity assumptions only require that
the gradient of R is continuous while the metric coefficients belong to the
Sobolev space H1 (or have even less regularity).
