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Isoparametric hypersurfaces with four principal curvatures. II. (English) Zbl 1243.53094

The author shows that an isoparametric hypersurface with four principal curvatures and multiplicities \((3,4)\) in \(S^{15}\) is one constructed by Ozeki and Takeuchi and Ferus, Karcher, and Munzner, referred to collectively as of OT-FKM type. This new approach also gives a considerably simpler proof, that an isoparametric hypersurface with four principal curvatures in spheres with the multiplicity constraint \(m_{2}\geq 2m_{1}-1\) is of OT-FKM type, which left unsettled exactly the four anomalous multiplicity pairs \((4,5), (3,4), (7,8)\) and \((6,9)\), where the last three are closely tied, respectively, with the quaternion algebra, the octonion algebra, and the complexified octonion algebra, whereas the first stands alone in that it cannot be of OT-FKM type. The cases for the multiplicity pairs \((4,5), (6,9)\), and \((7,8)\) remain open now.
For part I, cf. [Th. E. Cecil, Sh.-Q. Chi, G. R. Jensen, Ann. Math. (2) 166, No. 1, 1–76 (2007; Zbl 1143.53058)]; see also [the author, Nagoya Math. J. 193, 129–154 (2009; Zbl 1165.53032)].
Reviewer: Jan Kurek (Lublin)

MSC:

53C40 Global submanifolds
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