Weakly regular $T^2$-symmetric spacetimes. The global geometry of future Cauchy developments

Abstract

We provide a geometric well-posedness theory for the Einstein equations within the class of weakly regular vacuum spacetimes with $T^2$-symmetry, as defined in the present paper, and we investigate their global causal structure. Our assumptions allow us to give a meaning to the Einstein equations under weak regularity as well as to solve the initial value problem under the assumed symmetry. First, introducing a frame adapted to the symmetry and identifying certain cancellation properties taking place in the standard expressions of the connection and the curvature, we formulate the initial value problem for the Einstein field equations under the proposed weak regularity assumptions. Second, considering the Cauchy development of any weakly regular initial data set and denoting by $R$ the area of the orbits of symmetry, we establish the existence of a global foliation by the level sets of $R$ such that $R$ grows to infinity in the future direction. Our weak regularity assumptions only require that $R$ is Lipschitz continuous while the metric coefficients describing the initial geometry of the symmetry orbits are in the Sobolev space $H^1$ and the remaining coefficients have even weaker regularity.