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VOL. 27 | 1991 $U(1) \times U(1)$ stability of the $(\frac{2}{3},\frac{2}{3}, -\frac{1}{3})$ Kasner metrics


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.


Published: 1 January 1991
First available in Project Euclid: 18 November 2014

Rights: Copyright © 1991, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University. This book is copyright. Apart from any fair dealing for the purpose of private study, research, criticism or review as permitted under the Copyright Act, no part may be reproduced by any process without permission. Inquiries should be made to the publisher.


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