Isoparametric foliation and Yau conjecture on the first eigenvalue, II

TL;DR

This paper extends the study of eigenvalues of isoparametric hypersurfaces in spheres, providing estimates and confirming parts of Yau's conjecture for specific focal submanifolds with six principal curvatures.

Contribution

It offers new eigenvalue estimates for focal submanifolds with g=6 and confirms the first eigenvalue equals the dimension in two cases, advancing understanding of Yau's conjecture.

Findings

Eigenvalue estimates for g=6 focal submanifolds.

Confirmation that the first eigenvalue equals the dimension in two cases.

Partial affirmative answer to a problem related to Yau's conjecture.

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.