Quaternionic representation of the moving frame for surfaces in Euclidean three-space and Lax pair

The study of surfaces in threeand higher-dimensional spaces has seen a resurgence of interest recently due to various applications of these surfaces to various areas of mathematical physics, especially to the area of integrable systems [1, 6, 8]. The particular class of surfaces known as minimal surfaces with constant mean curvature has many applications to various physical problems. It is the intention here to review and establish the Gauss-Codazzi equations for surfaces in Euclidean three-space. Next, a quaternionic representation is introduced for the moving frame of the conformally parametrized surface. It will be shown how the frame equations can be written using quaternions by means of an SU(2) matrix. The main new element here is a straightforward derivation of a Lax pair based on the use of quaternions, and an application of this result to the generalized Weierstrass representation [2]. Some specific examples of solutions for the resulting equations are given, and a particular application to the case of constant mean curvature surfaces under Konopelchenko’s generalization of the Weierstrass representation is presented [7]. We begin by establishing some general notions with regard to orientable surfaces in three-dimensional Euclidean space. Under such a parametrization, which is called conformal, the surface S can be given by a vector-valued function