In this paper we show that a Dupin hypersurface with constant M\"{o}bius
curvatures is M\"{o}bius equivalent to either an isoparametric hypersurface in
the sphere or a cone over an isoparametric hypersurface in a sphere. We also
show that a Dupin hypersurface with constant Laguerre curvatures is Laguerre
equivalent to a flat Laguerre isoparametric hypersurface. These results solve
the major issues related to the conjectures of Cecil et al on the
classification of Dupin hypersurfaces.Comment: 45 pages. arXiv admin note: text overlap with arXiv:math/0512090 by
other author
