Deformations of surfaces associated with integrable Gauss – Mainardi – Codazzi equations

Abstract

Using the formulation of the immersion of a two-dimensional surface into the three-dimensional Euclidean space proposed recently, a mapping from each symmetry of integrable equations to surfaces in ℝ3 can be established. We show that among these surfaces the sphere plays a unique role. Indeed, under the rigid SU(2) rotations all integrable equations are mapped to a sphere. Furthermore we prove that all compact surfaces generated by the infinitely many generalized symmetries of the sine-Gordon equation are homeomorphic to a sphere. We also find some new Weingarten surfaces arising from the deformations of the modified Kurteweg-de Vries and of the nonlinear Schrödinger equations. Surfaces can also be associated with the motion of curves. We study curve motions on a sphere and we identify a new integrable equation characterizing such a motion for a particular choice of the curve velocity. © 2000 American Institute of Physics.