73 research outputs found
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
Isoparametric foliation and Yau conjecture on the first eigenvalue
A well known conjecture of Yau states that the first eigenvalue of every
closed minimal hypersurface in the unit sphere is just its
dimension . 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
For a closed hypersurface with constant mean
curvature and constant non-negative scalar curvature, the present paper shows
that if are constants for for
shape operator , then is isoparametric. The result generalizes
the theorem of de Almeida and Brito \cite{dB90} for to any dimension ,
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
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