Hypersurface In Unit Sphere Research Articles

Isoparametric hypersurfaces and complex structures

The main purpose of this note is to construct almost complex or complex structures on certain isoparametric hypersurfaces in unit spheres. As a consequence, complex structures on S1 × S7 × S6, and on S1 × S3 × S2 with vanishing first Chern class, are built.

Isoparametric foliations, a problem of Eells–Lemaire and conjectures of Leung

In this paper, two sequences of minimal isoparametric hypersurfaces are constructed via representations of Clifford algebras. Based on these, we give estimates on eigenvalues of the Laplacian of the focal submanifolds of isoparametric hypersurfaces in unit spheres. This improves results of [TY13] and [TXY14]. Eells and Lemaire [EL83] posed a problem to characterize the compact Riemannian manifold M for which there is an eigenmap from M to S^n. As another application of our constructions, the focal maps give rise to many examples of eigenmaps from minimal isoparametric hypersurfaces to unit spheres. Most importantly, by investigating the second fundamental forms of focal submanifolds of isoparametric hypersurfaces in unit spheres, we provide infinitely many counterexamples to two conjectures of Leung [Le91] (posed in 1991) on minimal submanifolds in unit spheres. Notice that these conjectures of Leung have been proved in the case that the normal connection is flat [HV01].

Gap phenomenon of an abstract Willmore type functional of hypersurface in unit sphere.

For an n-dimensional hypersurface in unit sphere, we introduce an abstract Willmore type called W (n,F)-Willmore functional, which generalizes the well-known classic Willmore functional. Its critical point is called the W (n,F)-Willmore hypersurface, for which the variational equation and Simons' type integral equalities are obtained. Moreover, we construct a few examples of W (n,F)-Willmore hypersurface and give a gap phenomenon characterization by use of our integral formula.