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Classification of integrable Weingarten surfaces possessing an \(\mathfrak{sl}(2)\)-valued zero curvature representation. (English) Zbl 1204.53004

Summary: We classify Weingarten surfaces integrable in the sense of soliton theory. The criterion is that the associated Gauss equation possesses an \(\mathfrak{sl}(2)\)-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.

MSC:

53A05 Surfaces in Euclidean and related spaces
35Q53 KdV equations (Korteweg-de Vries equations)
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