Isoparametric foliation and Yau conjecture on the first eigenvalue, II

Abstract

This is a continuation of Tang and Yan, which investigated the first eigenvalues of minimal isoparametric hypersurfaces with $g=4$ distinct principal curvatures and focal submanifolds in unit spheres. For the focal submanifolds with $g=6$, the present paper obtains estimates on all the eigenvalues, among others, giving an affirmative answer in one case to the problem posed in Tang and Yan, which may be regarded as a generalization of Yau's conjecture. In two of the four unsettled cases in Tang and Yan for focal submanifolds $M_1$ of OT-FKM-type, we prove the first eigenvalues to be their dimensions, respectively.