## Isoparametric hypersurfaces with four principal curvatures.(English)Zbl 1143.53058

This is an important paper on isoparametric hypersurfaces in space forms. The classification problem of isoparametric hypersurfaces in $$\mathbb{R}^n$$ and in the hyperbolic space $$H^n$$ was solved by B. Segre [Atti Accad. Naz. Lincei Rend. (6) 27, 203–207 (1938; Zbl 0019.18403)] and E. Cartan [Ann. Mat. Pura Appl., IV. Ser. 17, 177–191 (1938; Zbl 0020.06505)], respectively. But Cartan’s beautiful theory was relatively unnoticed for thirty years, until it was revived in 1970’s by K. Nomizu’s survey paper [Bull. Am. Math. Soc. 79, 1184–1188 (1974; Zbl 0275.53003)].
After that H. F. Münzner [Math. Ann. 251, 57–71 (1980; Zbl 0417.53030) and Math. Ann. 256, 215–232 (1981; Zbl 0438.53050)] made excellent contributions to the theory of isoparametric hypersurfaces in spheres. He proved that the number $$g$$ of distinct principal curvatures of an isoparametric hypersurfaces in $$S^n$$ must be 1,2,3,4 or 6. Cartan classified isoparametric hypersurfaces with $$g\leq 3$$. The classification of isoparametric hypersurfaces with $$g=4$$ or $$6$$ remained one of the outstanding problems in submanifolds geometry.
Münzner showed that for $$g=6$$, all isoparametric hypersurfaces must have the same multiplicity $$m$$, and then U. Abresch showed that $$m$$ must be 1 or 2. In the case $$m=1$$, J. Dorfmeister and E. Neher proved [Commun. Algebra 13, 2299–2368 (1985; Zbl 0578.53041)] that an isoparametric hypersurface must be homogeneous, but it is not known whether this is true in the case $$m=2$$.
For $$g=4$$, there is a much larger and more diverse collection of known examples. After a series of works of many other geometers, D. Ferus, H. Karcher and H.-F. Münzner [Math. Z. 177, 479–502 (1981; Zbl 0443.53037)] used representation of Clifford algebra to construct for any integer $$m_1>0$$ an infinite series of isoparametric hypersurfaces with four principal curvatures having two different multiplicities $$(m_1, m_2)$$, where $$m_2=k\delta(m_1)-m_1-1$$, $$\delta(m_1)$$ is the integer such that the Clifford algebra $$C_{m_1-1}$$ has an irreducible representation on $$\mathbb{R}^{\delta(m_1)}$$ and $$k$$ is any positive integer for which $$m_2$$ is positive. This kind of isoparametric hypersurfaces are said to be of FKM-type.
The set of FKM-type isoparametric hypersurfaces contains all known examples with $$g=4$$, except two homogeneous examples with multiplicities $$(m_1,m_2)=(2,2)$$ and $$(4,5)$$. S. Stolz [Invent. Math. 138, 253–279 (1999; Zbl 0944.53035)] that the multiplicities of an isoparametric hypersurface with $$g=4$$ must be the same as those in the known examples of FKM-type or the two homogeneous exceptions, this adds weight to the conjecture that the known examples are actually the only isoparametric hypersurfaces with $$g=4$$. In this paper, the authors prove the following
Classification Theorem: Let $$M$$ be an isoparametric hypersurface of $$S^n$$ with $$g=4$$ and multiplicities $$(m_1,m_2)$$ satisfying $$m_2\geq 2m_1-1$$. Then $$M$$ is of FKM-type.
Taken together with the classifications of R. Takagi [J. Differ. Geom. 11, 225–233 (1976; Zbl 0337.53003)] for the case $$m_1=1$$ and H. Ozeki and M. Takeuchi [Tôhoku Math. J., II. Ser. 27, 515–559 (1975; Zbl 0359.53011)] for $$m_1=2$$, this theorem handles all possible pairs $$(m_1,m_2)$$ of multiplicities, with the exception of $$(4,5)$$ and 3 pairs of multiplicities, $$(3,4),(6,9),(7,8)$$ corresponding to isoparametric hypersurfaces of FKM-type. For these 4 pairs, the classification problem for isoparametric hypersurfaces remains open.

### MSC:

 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53C40 Global submanifolds
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