Isoparametric hypersurfaces with four principal curvatures

Abstract

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.

On the classification of isoparametric hypersurfaces with four distinct curvatures in spheres

“…For the possible pairs (m 1 , m 2 ) with m 1 < m 2 we have (m 1 , m 2 ) = (4, 5) or 2 φ(m1−1) divides m 1 + m 2 + 1, where φ : N → N is given by φ(m) = {i | 1 ≤ i ≤ m and i ≡ 0, 1, 2, 4 (mod 8)} ; see [15]. These results imply that the inequality m 2 ≥ 2m 1 − 1 in Theorem 1.1 is satisfied for all possible pairs (m 1 , m 2 ) with m 1 ≤ m 2 except for the five pairs (2,2), (3,4), (4,5), (6,9), and (7,8).…”

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AbstractIn this paper we give a new proof for the classification result in [3]. We show that isoparametric hypersurfaces with four distinct principal curvatures in spheres are of Clifford type provided that the multiplicities m 1 , m 2 of the principal curvatures satisfy m 2 ≥ 2m 1 − 1.

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Willmore submanifolds in the unit sphere via isoparametric functions

“…By a further discussion (for details, one can find it in the previous version (arXiv0402272v1, Page 19) of [CCJ07]), we have…”

“…In virtue of Münzner's work [Mün80] [Mün81], the number g of distinct principal curvatures must be 1, 2, 3, 4 or 6, and there are at most two multiplicities {m 1 , m 2 } of principal curvatures (m 1 = m 2 , if g is odd). The classification problem has been completed except for one case (see [Tho00] and [Cec08] for excellent surveys and [CCJ07], [Imm08], [Chi13], [Miy13], [TY13a], [TXY14] for recent progresses).…”

Algebraic nature of singular Riemannian foliations in spheres

“…He proved a finiteness result controlling the topology of the hypersurfaces and an algebraicity result building a bridge between geometry and algebra: any isoparametric hypersurface is given as the zero set of a polynomial equation. Starting from these results essentially all isoparametric hypersurfaces have been classified by combining deep topological, geometric and algebraic insights [1], [16], [24], [8], [11].…”