2014/01/07 by Marcos Dajczer, Theodoros Vlachos, Dajczer, Marcos +1
Mathematics · #53C42 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53C42
paper · pdf · doi:10.48550/arxiv.1401.1291
arxiv created 2014/01/07 · arxiv updated 2014/01/08
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 \citema. 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).