In this chapter we shall describe the spaces of maximal globally hyperbolic (Hausdorff) spaces-times for which the groups $S0(3) \times U(l), S0(3)$ (two-dimensional principal orbits) and $U(l) \times U(l) \times U(l)$ act by isometries on some compact, connected and orientable Cauchy surface. We start by proving the well known result, that symmetries of Cauchy data lead to symmetries of the space-time:

In this Chapter we present the proof of Theorem 1.5.2, namely that the singularity of $(p_1,p_2,p_3) = (\frac{2}{3},\frac{2}{3}, -\frac{1}{3})$ (or permutation thereof) Kasner metrics is stable under $U(1) \times U(1)$ symmetric perturbations. The problem reduces to establishing a priori estimates for a Lorentzian harmonic-type map from two-dimensional Minkowski space-time to the unit two-dimensional hyperboloid of constant negative curvature. We shall start by analyzing the harmonic map equations, the geometric interpretation of the estimates proved below will be given in Section 3.5.