Isoparametric hypersurfaces with four principal curvatures, II

Abstract

In this sequel to an earlier article, employing more commutative algebra than previously, we show that an isoparametric hypersurface with four principal curvatures and multiplicities $(3,4)$ in $S^{15}$ is one constructed by Ozeki and Takeuchi and Ferus, Karcher, and Münzner, referred to collectively as of OT-FKM type. In fact, this new approach also gives a considerably simpler proof, both structurally and technically, that an isoparametric hypersurface with four principal curvatures in spheres with the multiplicity constraint $m_2\ge 2m_1-1$ is of OT-FKM type, which left unsettled exactly the four anomalous multiplicity pairs $(4,5)$, $(3,4)$, $(7,8)$, and $(6,9)$, where the last three are closely tied, respectively, with the quaternion algebra, the octonion algebra, and the complexified octonion algebra, whereas the first stands alone in that it cannot be of OT-FKM type. A by-product of this new approach is that we see that Condition B, introduced by Ozeki and Takeuchi in their construction of inhomogeneous isoparametric hypersurfaces, naturally arises. The cases for the multiplicity pairs $(4,5)$, $(6,9)$, and $(7,8)$ remain open now.