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

TL;DR

This paper classifies integrable Weingarten surfaces using soliton theory, identifying conditions under which their Gauss equations admit an sl(2)-valued zero curvature representation, and provides explicit solutions involving elliptic integrals.

Contribution

It introduces a classification criterion for integrable Weingarten surfaces based on zero curvature representations and solves the governing equations explicitly.

Findings

Classification of integrable Weingarten surfaces

Explicit solutions via elliptic integrals

Connection to classical geometry results

Abstract

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