Isoparametric hypersurfaces with four principal curvatures, II

TL;DR

This paper proves that certain isoparametric hypersurfaces with four principal curvatures and specific multiplicities are of OT-FKM type, introduces a simpler proof method, and discusses unresolved cases related to algebraic structures.

Contribution

It provides a simpler, algebraic approach to classify isoparametric hypersurfaces with four principal curvatures and specific multiplicities, confirming their OT-FKM type in many cases and highlighting open problems.

Findings

Hypersurfaces with multiplicities (3,4) are of OT-FKM type.

New algebraic approach simplifies previous proofs.

Open cases remain for multiplicity pairs (4,5), (6,9), and (7,8).

Abstract

In this sequel, employing more commutative algebra than that explored in \cite{CCJ}, we show that an isoparametric hypersurface with four principal curvatures and multiplicities $(3,4)$ in $S^{15}$ is one constructed by Ozeki-Takeuchi \cite[I]{OT} and Ferus-Karcher-M\"unzner \cite{FKM}, referred to collectively as of OT-FKM type. In fact, this new approach also gives a considerably simpler, both structurally and technically, proof \cite{CCJ} that an isoparametric hypersurface with four principal curvatures in spheres with the multiplicity constraint $m_2\geq 2m_1-1$ 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 by itself in that it cannot be of OT-FKM type. A byproduct of this new approach is that we see that Condition B, introduced by Ozeki and Takeuchi \cite[I]{OT} 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.