Geodesic
Isoparametric hypersurfaces with four principal curvatures
Thomas E. Cecil1 ; Quo-Shin Chi2 ; Gary R. Jensen2
1 Department of Mathematics, College of the Holy Cross, Worcester, MA 01610, United States
2 Department of Mathematics, Washington University, Saint Louis, MO 63130, United States
Annals of mathematics, Tome 166 (2007) no. 1, pp. 1-76.

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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 $m_2 \geq 2m_1 – 1$, 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 four cases, for which the classification problem remains open.
DOI : 10.4007/annals.2007.166.1

Thomas E. Cecil 1 ; Quo-Shin Chi 2 ; Gary R. Jensen 2

1 Department of Mathematics, College of the Holy Cross, Worcester, MA 01610, United States
2 Department of Mathematics, Washington University, Saint Louis, MO 63130, United States 
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Thomas E. Cecil; Quo-Shin Chi; Gary R. Jensen. Isoparametric hypersurfaces with four principal curvatures. Annals of mathematics, Tome 166 (2007) no. 1, pp. 1-76. doi : 10.4007/annals.2007.166.1. http://geodesic.mathdoc.fr/articles/10.4007/annals.2007.166.1/

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