Geodesic
Remarks on the Darboux transform of isothermic surfaces
Documenta mathematica, Tome 2 (1997), pp. 313-333
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We study Darboux and Christoffel transforms of isothermic surfaces in Euclidean space. Using quaternionic calculus we derive a Riccati type equation which characterizes all Darboux transforms of a given isothermic surface. Surfaces of constant mean curvature turn out to be special among all isothermic surfaces: their parallel surfaces of constant mean curvature are Christoffel and Darboux transforms at the same time. We prove – as a generalization of Bianchi's theorem on minimal Darboux transforms of minimal surfaces – that constant mean curvature surfaces in Euclidean space allow ∞3 Darboux transforms into surfaces of constant mean curvature. We indicate the relation between these Darboux transforms and Bäcklund transforms of spherical surfaces.
DOI : 10.4171/dm/32
Classification : 53A10, 53C42
Mots-clés : Riccati equation, isothermic surface, Darboux transformation, Christoffel transformation, constant mean curvature, baecklund transformation 
@article{10_4171_dm_32,
     author = {Udo Hertrich-Jeromin and Franz Pedit},
     title = {Remarks on the {Darboux} transform of isothermic surfaces},
     journal = {Documenta mathematica},
     pages = {313--333},
     year = {1997},
     volume = {2},
     doi = {10.4171/dm/32},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/32/}
}
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Udo Hertrich-Jeromin; Franz Pedit. Remarks on the Darboux transform of isothermic surfaces. Documenta mathematica, Tome 2 (1997), pp. 313-333. doi: 10.4171/dm/32

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