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arXiv · 1401.1291

The dual superconformal surface

Marcos Dajczer·Theodoros Vlachos
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Abstract

It is shown that a superconformal surface with arbitrary codimension in flat Euclidean space has a (necessarily unique) dual superconformal surface if and only if the surface is S-Willmore, the latter a well-known necessary condition to allow a dual as shown by Ma \cite{ma}. Duality means that both surfaces envelope the same central sphere congruence and are conformal with the induced metric. Our main result is that the dual surface to a superconformal surface can easily be described in parametric form in terms of a parametrization of the latter. Moreover, it is shown that the starting surface is conformally equivalent, up to stereographic projection in the nonflat case, to a minimal surface in a space form (hence, S-Willmore) if and only if either the dual degenerates to a point (flat case) or the two surfaces are conformally equivalent (nonflat case).

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Marcos Dajczer, Theodoros Vlachos. 2014-01-07. The dual superconformal surface. https://arxiv.org/abs/1401.1291

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