Edited by Foxit PDF Editor Copyright (c) by Foxit Software Company, 2004 For Evaluation Only. 1 Timelike Minimal Surfaces via Loop Groups The first named author would like to dedicate this paper to the memory of his father Yukio J. INOGUCHI1, and M. TODA2 1Department of Mathematics Education, Faculty of Education, Utsunomiya University, Utsunomiya, 321-8505, Japan. e-mail:
[email protected] 2Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79407, U.S.A. e-mail:
[email protected] (Received: 30 July 2003; in final form: 24 May 2004) Abstract. This work consists of two parts. In Part I, we shall give a systematic study of Lorentz conformal structure from structural viewpoints. We study manifolds with split-complex structure. We apply general results on split-complex structure for the study of Lorentz surfaces. In Part II, we study the conformal realization of Lorentz surfaces in the Minkowski 3-space via conformal minimal immersions. We apply loop group theoretic Weierstrass-type representation of timelike constant mean curvature for timelike minimal surfaces. Classical integral representation formula for timelike minimal surfaces will be recovered from loop group theoretic viewpoint. Mathematics Subject Classifications (2000): 53A10, 58E20, 22E67. Key words: Lorentz surfaces, harmonic maps, loop groups, timelike minimal surfaces. Introduction The classical wave equation: −utt + uxx = 0 over a plane R2(t, x) can be understood as the harmonicity equation with respect to the Lorentz metric −dt2 +dx2. It is easy to see that Lorenz harmonicity is invariant under conformal transformation of (R2(t, x), −dt2 + dx2). This observation tells us that for the study of second order PDE of hyperbolic type, it is convenient to introduce the notion of Lorentz conformal structure.