The Weierstrass–Enneper system for constant mean curvature surfaces and the completely integrable sigma model

Abstract

The integrability of a system which describes constant mean curvature surfaces by means of the adapted Weierstrass–Enneper inducing formula is studied. This is carried out by using a specific transformation which reduces the initial system to the completely integrable two-dimensional Euclidean nonlinear sigma model. Through the use of the apparatus of differential forms and Cartan theory of systems in involution, it is demonstrated that the general analytic solutions of both systems possess the same degree of freedom. Furthermore, a new linear spectral problem equivalent to the initial Weierstrass–Enneper system is derived via the method of differential constraints. A new procedure for constructing solutions to this system is proposed and illustrated by several elementary examples, including a multi-soliton solution.