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**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.

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.

Reviewer: Weihuan Chen (Beijing)

### MSC:

53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |

53C40 | Global submanifolds |