Isoparametric hypersurfaces with four principal curvatures, III

Abstract

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.

Isoparametric functions and nodal solutions of the Yamabe equation

“…The classification of isoparametric hypersurfaces in the sphere was an outstanding problem that remained open for a long time. Since the article by Cartan [7] lots of works have been carried out in order to achieve the total classification of isoparametric hypersurfaces in the sphere, see for instance the articles by Munzner [40] and [41], Abresch [1], Hsiang and Lawson [29], Ferus et al [19], Stolz [49], Cecil et al [8], Immervoll [30], Chi [10] and [11], Miyaoka [37] and [39]. In a recent paper Chi [12], classifies the isoparametric hypersurfaces in the sphere with four distinct principal curvatures and multiplicity pair (7,8).…”

Submanifold geometry in symmetric spaces of noncompact type

“…In round spheres S n there are inhomogeneous isoparametric hypersurfaces (with constant principal curvatures) [51]. In fact, the classification problem in spheres is much more involved; for more information, we refer the reader to some of the latest advances in the topic, such as [25,29,60,78,83]. In spaces of nonconstant curvature, the problem becomes very complicated.…”

Extrinsic geometry of the Gromoll-Meyer sphere

“…The isoparametric theory initiated from the study of E. Cartan in real space forms in 1940s, which caught much caution recently as the accomplishment of the classifications of isoparametric hypersurfaces in the unit sphere ( [CCJ07], [Chi13], [Miy13], [Miy16], [Chi16]). The applications of isoparametric theory are more and more abundant, see for example [TY13], [QTY13], [TXY14] and [GTY18].…”