Isoparametric hypersurfaces with four principal curvatures

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 $m_2 \geq 2m_1 – 1$, 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.

Authors

Thomas E. Cecil

Department of Mathematics
College of the Holy Cross
Worcester, MA 01610
United States

Quo-Shin Chi

Department of Mathematics
Washington University
Saint Louis, MO 63130
United States

Gary R. Jensen

Department of Mathematics
Washington University
Saint Louis, MO 63130
United States