The Generalized Weierstrass System for Nonconstant Mean Curvature Surfaces and the Nonlinear Sigma Model
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The Generalized Weierstrass System for Nonconstant Mean Curvature Surfaces and the Nonlinear Sigma Model Paul Bracken
Received: 22 July 2004 / Accepted: 3 July 2006 / Published online: 5 September 2006 © Springer Science + Business Media B.V. 2006
Abstract A study of the generalized Weierstrass system which can be used to induce mean curvature surfaces in three-dimensional Euclidean space is presented. A specific transformation is obtained which reduces the initial system to a twodimensional Euclidean nonlinear sigma model. Some aspects of integrability are discussed, in particular, a connection with a version of the sinh-Gordon equation is established. Finally, some specific solutions are given and a systematic way of calculating multisoliton solutions is presented. Key words Weirstrass system · sinh-Gordon equation · Euclidean space · mean curvature. Mathematics Subject Classifications (2000) 35Q51 · 53A10.
1 Introduction The method which was first formulated by Weierstrass and Enneper [1, 2] for determining minimal surfaces embedded in three-dimensional Euclidean space has recently been generalized by Konopelchenko and Taimanov [3, 4] and it has been the subject of considerable further work [5–8]. One of the reasons for this is that there are many physical applications of minimal surfaces to such areas as integrable systems, statistical mechanics and even string theory [6, 9]. In fact, a direct connection between certain classes of mean curvature surface, namely constant mean curvature, and a particular integrable finite dimensional Hamiltonian system has been established by Konopelchenko and Taimanov [4]. Mean curvature plays a special role among the characteristics of surfaces and their dynamics in many problems that arise in both physics and mathematics [9]. The case in which the mean curvature of the surface
P. Bracken (B) Department of Mathematics, University of Texas, Edinburg, TX 78541-2999, USA e-mail: [email protected]
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Acta Appl Math (2006) 92: 63–76
is constant has been discussed [5, 6] at length. In particular, many properties of such surfaces determined by Konopelchenko’s inducing prescription, such as the relationship to the nonlinear two-dimensional sigma model, integrability and Lax pair have been determined and a Bäcklund transformation has been calculated as well [7, 8]. It is the purpose here to begin an investigation of the case in which the mean curvature of the surface is not constant, but is described by a real-valued function which depends on two independent variables. This is useful for several reasons, in particular for establishing some of the mathematical properties of such surfaces in general, beyond the case of constant mean curvature surfaces. There are many applications of mean curvature surfaces and, of particular interest here, we elaborate on applications to the areas of quantum field theory, two-dimensional gravity and string theory [10, 11]. Classical string theory can be regarded as a study of the geometry of subspaces immersed in higher dimensional spaces or ma
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