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 [Z. Z. Tang and W. J. Yan, ‘Isoparametric foliation and Yau conjecture on the first eigenvalue’, J. Differential. Geom. 94 (2013) 521–540; Z. Z. Tang, Y. Q. Xie and W. J. Yan, ‘Isoparametric foliation and Yau conjecture on the first eigenvalue, II’, J. Funct. Anal. 266 (2014) 6174–6199]. Eells and Lemaire [Selected topics in harmonic maps, C.B.M.S. Regional Conference Series in Mathematics 50 (American Mathematical Society, Providence, RI, 1983)] posed a problem to characterize the compact Riemannian manifold M for which there is an eigenmap from M to Sn. 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 [‘Minimal submanifolds in a sphere II’, Bull. London Math. Soc. 23 (1991) 387–390] (posed in 1991) on minimal submanifolds in unit spheres. Note that these conjectures of Leung have been proved in the case that the normal connection is flat [T. Hasanis and T. Vlachos, ‘Ricci curvatures and minimal submanifolds’, Pacific J. Math. 197 (2001) 13–24].