On Dupin hypersurfaces with constant Mubius curvature

摘要

We show that proper Dupin hypersurfaces M-n for n >= 4 in Rn+1 with n distinct principal curvatures and constant Mobius curvature cannot be parametrized by lines of curvature. For n = 3, up to Mobius transformations, there is a unique proper Dupin hypersurface, parametrized by lines of curvature, with three distinct principal curvatures and constant Mobius curvature. Moreover, these hypersurfaces are the only conformally flat proper Dupin hypersurfaces M-3 subset of R-4 with three distinct principal curvatures and constant Mobius curvature.