Isoparametric Hypersurfaces with Four Principal Curvatures

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 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 ≥ 2m₁ - 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 four cases, for which the classification problem remains open.