Isoparametric hypersurfaces with four principal curvatures

Tom Cecil; Quo-Shin Chi; Gary Jensen

paper

Isoparametric hypersurfaces with four principal curvatures

arXiv:math/0402272

Abstract

Let $M$ be an isoparametric hypersurface in the sphere $S^{n}$ with four distinct principal curvatures. Münzner showed that the four principal curvatures can have at most two distinct multiplicities $m_{1}, m_{2}$ , and Stolz showed that the pair $(m_{1}, m_{2})$ 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 $m_{2} \geq 3 m_{1} - 1$ , then the isoparametric hypersurface $M$ must be of FKM-type. Together with known results of Takagi for the case $m_{1} = 1$ , and Ozeki and Takeuchi for $m_{1} = 2$ , 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.

79 pages, no figures, improved version of a pre-existing preprint with the same title