Möbius isoparametric hypersurfaces with three distinct principal curvatures. II (Q627399)

The authors study connected smooth and umbilic-point-free hypersurfaces \(x: M^n \to S^{n+1}\) in the \((n+1)\)-sphere \(S^{n+1}\) whose Möbius invariant 1-form vanishes and whose principal curvatures are all constant. Such hypersurfaces of \(S^{n+1}\) are called Möbius isoparametric hypersurfaces. They prove a classification theorem for Möbius isoparametric hypersurfaces which have only three distinct Möbius principal curvatures, thus extending their result in Part I [ibid. 232, No.~2, 289--311 (2007; Zbl 1154.53011)]. It is shown that there are exactly three types of such surfaces: The first type belongs to parallel Möbius second fundamental form and has three distinct Möbius principal curvatures with arbitrary multiplicities \(p\), \(q\), \(n-p-q\). The second type are the Euclidean isoparametric surfaces contained in \(S^{ n+1}\). The third type consists of certain minimal hypersurfaces where the multiplicities of the three Möbius principal curvatures are \(m\), \(m\) and \(n-2m > m\).