Classification of integrable Weingarten surfaces possessing an sl(2)-valued zero curvature representation

摘要

In this paper we classify Weingarten surfaces integrable in the sense ofsoliton theory. The criterion is that the associated Gauss equation possessesan sl(2)-valued zero curvature representation with a nonremovable parameter.Under certain restrictions on the jet order, the answer is given by a thirdorder ordinary differential equation to govern the functional dependence of theprincipal curvatures. Employing the scaling and translation (offsetting)symmetry, we give a general solution of the governing equation in terms ofelliptic integrals. We show that the instances when the elliptic integralsdegenerate to elementary functions were known to nineteenth century geometers.Finally, we characterize the associated normal congruences.