## Abstract

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.

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