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Title: Willmore canal surfaces in Euclidean space
Authors: Musso, Emilio
Nicolodi, Lorenzo
Keywords: Willmore surfacesWillmore canal surfacesIsothermic sur facesConformai geometry
Issue Date: 1999
Publisher: Università degli Studi di Trieste. Dipartimento di Scienze Matematiche
Source: Emilio Musso and Lorenzo Nicolodi, "Willmore canal surfaces in Euclidean space", in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 31 (1999), pp. 177-202.
Series/Report no.: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics
31 (1999)
We study envelopes of 1-parameter families of spheres (including planes) in Euclidean space which are critical points of the Willmore functional (Willmore canal surfaces). We prove that Willmore canal surfaces are isothermic surfaces and hence conformally equivalent to surfaces of revolution, cones or cylinders. We provide explicit formulae for all solution surfaces. In the generic case the formulae involve Weierstrass's elliptic functions. There are two exceptional cases which can be integrated by using elementary functions only, namely the catenoid and the stereographic projection of the minimal Clifford torus in S3. To obtain the solution surfaces we explicitly integrate the linear differential system defining the Willmore canal surfaces.
Type: Article
ISSN: 0049-4704
Appears in Collections:Rendiconti dell'Istituto di Matematica dell'Università di Trieste: an International Journal of Mathematics vol.31 (1999)

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