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

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

On the Chern conjecture for isoparametric hypersurfaces

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

Schoen-Yau-Gromov-Lawson theory and isoparametric foliations

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