Darboux transformations on timelike constant mean curvature surfaces
One of the central topics of classical differential geometry in the 19th century was the transformation theory of surfaces, the best known example being the Bäcklund transformation of constant negative curvature surfaces in Euclidean space, developed by Bianchi and Bäcklund. Nowadays transformations for solutions to a soliton equation which add solitons to a given solution are generally called Bäcklund transformations. The construction of multisoliton solutions from the vacuum solution by using Bäcklund transformations are called the direct method. Recently \textit{I. Sterling} and \textit{H. Wente} [Indiana J. Math. 42, 1239-1266 (1993; Zbl 0803.53009)] studied Bäcklund transformations of constant mean curvature (CMC) surfaces. Further, \textit{H. Muto} [``A dressing up procedure for a constant mean curvature surface'', in Ohnita (Ed.) Workshop on Harmonic Maps and Related Topics, Tokyo Metropolitan University, 41-48 (1995)] obtained another formulation of such Bäcklund transformations, by reformulating the direct method as a transformation on framings and obtaining what is refered to as the Darboux transformation of CMC surfaces. In the paper under review, the author studies Bäcklund and Darboux transformations for timelike CMC surfaces in Minkowski 3-space with real distinct principal curvatures. Since the induced metric of a timelike surface is indefinite, Bäcklund and Darboux transformations are more complicated than in the case of CMC surfaces in Euclidean space. In particular, there exist four kinds of Bäcklund transformations. The paper is organized as follows. In Section 1 the author deals with extended framings in order to find special framings for timelike CMC surfaces, which will be useful to study such immersions via the Darboux transformation theory. This is done by using the \(2\times 2\) matrix formalism of the Gauss-Codazzi equations introduced by the author in his previous paper [Tokyo J. Math. 21, 141-152 (1998; Zbl 0930.53042)]. Section 2 is devoted to Bäcklund transformations. In her thesis [One-parameter families of surfaces with constant curvature in Lorentz 3-space, Ph.D. Thesis, Brown University (1980)], \textit{L. McNertney} developed the theory of Bäcklund transformations on timelike surfaces of constant positive curvature. Here the author recalls and rewrites McNertney's results, and shows that there exist four kinds of Bäcklund transformations. Observe that the parallel surface procedure induces transformations on timelike CMC surfaces. In the next section, the author treats induced transformations on timelike CMC surfaces and gives a reformulation via extended framings. Section 4 is devoted to Darboux transformations. In particular, the author considers the following problem: Let \(\Phi_\lambda\) be the appropriate extended framing found in Section 1. Find polynominal maps \(R_\lambda\) into the twisted loop group of \(SL_2\mathbb{R}\) (or \(GL_2\mathbb{R}\)) such that \(\Phi_\lambda R_\lambda\) is also a solution to the extended framing equation determined by \(\Phi_\lambda\). Such polynomial maps \(R_\lambda\) are traditionally called Darboux matrices, and their existence (and uniqueness) is given in the main result of the paper (Theorem 4.5). As an application, one-soliton solutions are explicitly presented, which are obtained from the vacuum solution. Finally, in Section 5 the author makes some additional comments and remarks to the case of Bäcklund transformations on spacelike surfaces.
scientific article; zbMATH DE number 1369698
scientific article