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 and sufficient condition to allow a dual as shown by Ma (Results Math 48:301–309, 2005). 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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Dajczer, M., Vlachos, T. The dual superconformal surface. Ann Glob Anal Geom 48, 1–22 (2015). https://doi.org/10.1007/s10455-015-9453-5
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DOI: https://doi.org/10.1007/s10455-015-9453-5