73 research outputs found

    Isoparametric foliation and a problem of Besse on generalizations of Einstein condition

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    The focal sets of isoparametric hypersurfaces in spheres with g = 4 are all Willmore submanifolds, being minimal but mostly non-Einstein ([TY1], [QTY]). Inspired by A.Gray's view, the present paper shows that, these focal sets are all A- manifolds but rarely Ricci parallel, except possibly for the only unclassified case. As a byproduct, it gives infinitely many simply-connected examples to the problem 16.56 (i) of Besse concerning generalizations of the Einstein condition.Comment: To appear in Advances in Mathematic

    Isoparametric foliation and Yau conjecture on the first eigenvalue

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    A well known conjecture of Yau states that the first eigenvalue of every closed minimal hypersurface MnM^n in the unit sphere Sn+1(1)S^{n+1}(1) is just its dimension nn. The present paper shows that Yau conjecture is true for minimal isoparametric hypersurfaces. Moreover, the more fascinating result of this paper is that the first eigenvalues of the focal submanifolds are equal to their dimensions in the non-stable range.Comment: to appear in J.Diff.Geo

    On the Chern conjecture for isoparametric hypersurfaces

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    For a closed hypersurface MnβŠ‚Sn+1(1)M^n\subset S^{n+1}(1) with constant mean curvature and constant non-negative scalar curvature, the present paper shows that if tr(Ak)\mathrm{tr}(\mathcal{A}^k) are constants for k=3,…,nβˆ’1k=3,\ldots, n-1 for shape operator A\mathcal{A}, then MM is isoparametric. The result generalizes the theorem of de Almeida and Brito \cite{dB90} for n=3n=3 to any dimension nn, strongly supporting Chern's conjecture.Comment: 27 page

    Schoen-Yau-Gromov-Lawson theory and isoparametric foliations

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    Motivated by the celebrated Schoen-Yau-Gromov-Lawson surgery theory on metrics of positive scalar curvature, we construct a double manifold associated with a minimal isoparametric hypersurface in the unit sphere. The resulting double manifold carries a metric of positive scalar curvature and an isoparametric foliation as well. To investigate the topology of the double manifolds, we use K-theory and the representation of the Clifford algebra for the FKM-type, and determine completely the isotropy subgroups of singular orbits for homogeneous case.Comment: 24 pages, to appear in Communications in Analysis and Geometr
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