Geodesic
Remarks on the Darboux transform of isothermic surfaces
Documenta mathematica, Tome 2 (1997), pp. 313-333.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 $\infty^3$ Darboux transforms into surfaces of constant mean curvature. We indicate the relation between these Darboux transforms and Bäcklund transforms of spherical surfaces.
Classification : 53A10, 53A50, 53C42
Keywords: isothermic surface, Darboux transformation, Christoffel transformation, Riccati equation, constant mean curvature, baecklund transformation 
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     author = {Hertrich-Jeromin, Udo and Pedit, Franz},
     title = {Remarks on the {Darboux} transform of isothermic surfaces},
     journal = {Documenta mathematica},
     pages = {313--333},
     publisher = {mathdoc},
     volume = {2},
     year = {1997},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_1997__2__a3/}
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Hertrich-Jeromin, Udo; Pedit, Franz. Remarks on the Darboux transform of isothermic surfaces. Documenta mathematica, Tome 2 (1997), pp. 313-333. http://geodesic.mathdoc.fr/item/DOCMA_1997__2__a3/