Isoparametric hypersurfaces with four principal curvatures

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dc.creatorCecil, Tom
dc.creatorChi, Quo-Shin
dc.creatorJensen, Gary
dc.date2004-02-17
dc.date.accessioned2026-07-07T05:05:31Z
dc.date.available2026-07-07T05:05:31Z
dc.descriptionLet $M$ 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 $m_2 \geq 3m_1 - 1$, 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 10 cases, for which the classification problem remains open. The paper improves the result of a pre-existing preprint with the same title, in which 14 cases remained open.
dc.description79 pages, no figures, improved version of a pre-existing preprint with the same title
dc.identifierhttps://arxiv.org/abs/math/0402272
dc.identifierhttp://arxiv.org/abs/math/0402272
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/70191
dc.subjectDifferential Geometry
dc.subject53C40
dc.titleIsoparametric hypersurfaces with four principal curvatures
dc.typetext

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