Isoparametric hypersurfaces with four principal curvatures

Let M be an isoparametric hypersurface in the sphere S-n with four distinct principal curvatures. Munzner 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 Ferns, Karcher and Munzner 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.