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

TL;DR

This paper constructs minimal isoparametric hypersurfaces using Clifford algebra representations, improves eigenvalue estimates, and provides counterexamples to longstanding conjectures on minimal submanifolds in spheres.

Contribution

It introduces new minimal isoparametric hypersurfaces, refines eigenvalue bounds, and disproves two conjectures of Leung with explicit counterexamples.

Findings

Constructed two sequences of minimal isoparametric hypersurfaces.

Provided improved eigenvalue estimates for focal submanifolds.

Generated counterexamples to Leung's conjectures on minimal submanifolds.

Abstract

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].