Global geometry of T2 symmetric spacetimes with weak regularity

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).