Abstract
Isothermic surfaces are surfaces which allow a conformal curvature line parametrisation. They form an integrable system, and Darboux transforms of isothermic surfaces obey Bianchi permutability: for two distinct spectral parameters, the corresponding Darboux transforms have a common Darboux transform which can be computed algebraically. In this paper, we discuss two-step Darboux transforms with the same spectral parameter, and show that these are obtained by a Sym-type construction: All two-step Darboux transforms of an isothermic surface are given, without further integration, by parallel sections of the associated family of the isothermic surface, either algebraically or by differentiation against the spectral parameter.
Highlights
First defined by Bour in [3] as surfaces which admit conformal curvature lines, isothermic surfaces have enjoyed massive interest in the late 19th and early 20th century
We obtain the parallel sections φ1 = rλ(φ) of f1 by applying the gauge matrix to parallel sections φ given by the isothermic surface f, for spectral parameter away from the pole of rλ
Given an isothermic surface f with associated family dλ and a Darboux transform f1 given by spectral parameter 1 ∈ R and d 1 -parallel section φ1 = φ11, Bianchi permutability allows to compute Darboux transforms of f1 for all spectral parameter 2 = 1 by solely knowing the parallel sections of the family of flat connections of f and performing an algebraic operation
Summary
First defined by Bour in [3] as surfaces which admit conformal curvature lines, isothermic surfaces have enjoyed massive interest in the late 19th and early 20th century. The gauge has a pole, the family dλ1 = rλ · dλ extends into , and we give an explicit form of the associated family With this at hand, we obtain the parallel sections φ1 = rλ(φ) of f1 by applying the gauge matrix to parallel sections φ given by the isothermic surface f , for spectral parameter away from the pole of rλ. We obtain all non-trivial two-step Darboux transforms with same spectral parameter without need for a second integration, a principle we call generalised Bianchi permutability. Since the main ingredients for our construction are the associated family and the simple factor dressing, we expect our results to be templates for similar results for other surface classes allowing simple factor dressing, such as CMC surfaces in space forms, and completely integrable differential equations. This should allow to construct new surfaces and, more generally, new solutions to differential equations given by complete integrability
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