Isoparametric Hypersurfaces with Four Principal Curvatures

Thomas E. Cecil, College of the Holy Cross
Quo-Shin Chi
Gary R. Jensen

This is an improved version of a pre-existing preprint with the same title. This version was uploaded to on 02/17/2004.

Originally published in Annals of Mathematics, Volume 166, Number 1, July 2007, Pages 1-76.


Let M be an isoparametric hypersurface in the sphere Sn with four distinct principal curvatures. Münzner showed that the four principal curvatures can have at most two distinct multiplicities m1,m2, and Stolz showed that the pair (m1,m2) 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 m2 ≥ 3m1−1, then the isoparametric hypersurface M must be of FKM-type. Together with known results of Takagi for the case m1 = 1, and Ozeki and Takeuchi for m1 = 2, this handles all possible pairs of multiplicities except for 10 cases, for which the classification problem remains open.