2004/02/17 by Tom Cecil, Cecil, Tom, Quo-Shin Chi +4 · 4 citations
Mathematics · #53C40 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #math.DG #msc:53C40
paper · pdf · doi:10.48550/arxiv.math/0402272
79 pages, no figures, improved version of a pre-existing preprint with the same title
arxiv created 2004/02/17 · openalex publication_date 2004/02/17 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let M be an isoparametric hypersurface in the sphere Sn with four distinct principal curvatures. Münzner showed that the four principal curvatures can have at most two distinct multiplicities m1, m2, and Stolz showed that the pair (m1,m2) must either be (2,2), (4,5), or be equal to the multiplicities of an isoparametric hypersurface of FKM-type, constructed by Ferus, Karcher and Münzner from orthogonal representations of Clifford algebras. In this paper, we prove that if the multiplicities satisfy m2 ≥ 3m1 - 1, then the isoparametric hypersurface M must be of FKM-type. Together with known results of Takagi for the case m1 = 1, and Ozeki and Takeuchi for m1 = 2, this handles all possible pairs of multiplicities except for 10 cases, for which the classification problem remains open. The paper improves the result of a pre-existing preprint with the same title, in which 14 cases remained open.