Dupin hypersurfaces with four principal Curvatures, II

Abstract

If M is an isoparametric hypersurface in a sphere S n with four distinct principal curvatures, then the principal curvatures κ1, . . . , κ4 can be ordered so that their multiplicities satisfy m 1 = m 2 and m 3 = m 4, and the cross-ratio r of the principal curvatures (the Lie curvature) equals −1. In this paper, we prove that if M is an irreducible connected proper Dupin hypersurface in R n (or S n ) with four distinct principal curvatures with multiplicities m 1 = m 2 ≥ 1 and m 3 = m 4 = 1, and constant Lie curvature r = −1, then M is equivalent by Lie sphere transformation to an isoparametric hypersurface in a sphere. This result remains true if the assumption of irreducibility is replaced by compactness and r is merely assumed to be constant.