Isoparametric hypersurfaces with four principal curvatures, III

The classification work Isoparametric hypersurfaces with four principal curvatures, and Isoparametric hypersurfaces with four principal curvatures, II, 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. The quaternion and the octonion algebras play a fundamental role in their geometric structures. A unifying theme in Isoparametric hypersurfaces with four principal curvatures, and Isoparametric hypersurfaces with four principal curvatures, II, and the present sequel to them is Serre’s criterion of normal varieties. Its technical side pertinent to our situation that we developed in Isoparametric hypersurfaces with four principal curvatures, and Isoparametric hypersurfaces with four principal curvatures, II and extend in this sequel is instrumental. The classification leaves only the case of multiplicity pair $\{7, 8\}$ open.