Isoparametric hypersurfaces with four principal curvatures

Let M be an isoparametric hypersurface in the sphere S n with four distinct principal curvatures. Mnzner 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 Mnzner from orthogonal representations of Clifford algebras. In this paper, we prove that if the multiplicities satisfy m 2 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.