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 ≥ 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.

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The classification work [5], [9] left unsettled only those anomalous isoparametric hypersurfaces with four principal curvatures and multiplicity pair {4, 5}, {6, 9} or {7, 8} in the sphere.By systematically exploring the ideal theory in commutative algebra in conjunction with the geometry of isoparametric hypersurfaces, we show that an isoparametric hypersurface with four principal curvatures and multiplicities {4, 5} in S 19 is homogeneous, and, moreover, an isoparametric hypersurface with four principal curvatures and multiplicities {6, 9} in S 31 is either the inhomogeneous one constructed by Ferus, Karcher and Münzner, or the one that is homogeneous.This classification reveals the striking resemblance between these two rather different types of isoparametric hypersurfaces in the homogeneous category, even though the one with multiplicities {6, 9} is of the type constructed by Ferus, Karcher and Münzner and the one with multiplicities {4, 5} stands alone by itself. The quaternion and the octonion algebras play a fundamental role in their geometric structures.A unifying theme in [5], [9] and the present sequel to them is Serre's criterion of normal varieties. Its technical side pertinent to our situation that we developed in [5], [9] and extend in this sequel is instrumental.The classification leaves only the case of multiplicity pair {7, 8} open.|∇F | 2 (x) = g 2 |x| 2g−2 , (∆F )(x) = (m 2 − m 1 )g 2 |x| g−2 /2 1991 Mathematics Subject Classification. Primary 53C40.

Abstract. The classification of isoparametric hypersurfaces with four principal curvatures in spheres in [2] hinges on a crucial characterization, in terms of four sets of equations of the 2nd fundamental form tensors of a focal submanifold, of an isoparametric hypersurface of the type constructed by Ferus, Karcher and Münzner. The proof of the characterization in [2] is an extremely long calculation by exterior derivatives with remarkable cancellations, which is motivated by the idea that an isoparametric hypersurface is defined by an over-determined system of partial differential equations. Therefore, exterior differentiating sufficiently many times should gather us enough information for the conclusion. In spite of its elementary nature, the magnitude of the calculation and the surprisingly pleasant cancellations make it desirable to understand the underlying geometric principles.In this paper, we give a conceptual, and considerably shorter, proof of the characterization based on Ozeki and Takeuchi's expansion formula for the Cartan-Münzner polynomial. Along the way the geometric meaning of these four sets of equations also becomes clear.

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