Isoparametric foliation and Yau conjecture on the first eigenvalue

2012/01/03 by Zizhou Tang, ZiZhou Tang, Tang, ZiZhou +2
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #math.DG

paper · pdf · doi:10.48550/arxiv.1201.0666

to appear in J.Diff.Geom

openalex publication_date 2012/01/03 · arxiv created 2012/07/17 · arxiv updated 2012/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A well known conjecture of Yau states that the first eigenvalue of every closed minimal hypersurface Mⁿ in the unit sphere Sⁿ⁺¹(1) is just its dimension n. The present paper shows that Yau conjecture is true for minimal isoparametric hypersurfaces. Moreover, the more fascinating result of this paper is that the first eigenvalues of the focal submanifolds are equal to their dimensions in the non-stable range.

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