On the classification of isoparametric hypersurfaces with four distinct principal curvatures in spheres

• 成果类型: Article
• 署名作者: Immervoll, Stefan
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2008.168.1011
• 发表日期: 2008
• 页码: 1011-1024
• 关键词: generalized quadrangles; multiplicities

摘要

In this paper we give a new proof for the classification result in [3]. We show that isoparametric hypersurfaces with four distinct principal curvatures in spheres are of Clifford type provided that the multiplicities m(1), m(2) of the principal curvatures satisfy m(2) >= 2m(2) - 1. This inequality is satisfied for all but five possible pairs (m(1), m(2)) with m(1) <= m(2). Our proof implies that for (m(1), m(2)) not equal (1, 1) the Clifford system may be chosen in such a way that the associated quadratic forms vanish on the higher-dimensional of the two focal manifolds. For the remaining five possible pairs (m(1), m(2)) with m(1) <= m(2) (see [13], [1], and [15]) this stronger form of our result is incorrect: for the three pairs (3,4), (6,9), and (7, 8) there are examples of Clifford type such that the associated quadratic forms necessarily vanish on the lower-dimensional of the two focal manifolds, and for the two pairs (2,2) and (4, 5) there exist homogeneous examples that are not of Clifford type; cf. [5, 4.3, 4.4].