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Isoparametric hypersurfaces with four principal curvatures, II

Published online by Cambridge University Press:  11 January 2016

Quo-Shin Chi
Quo-Shin Chi*
Affiliation:
Department of Mathematics, Washington University, St. Louis, Missouri, [email protected]
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Abstract

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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 S15 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 m2≥ 2m1 -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.

Keywords

53C40
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 204 , December 2011 , pp. 1 - 18
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2011

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