A sufficient condition for a hypersurface to be isoparametric

2018/03/27 by Zizhou Tang, Dongyi Wei, Tang, Zizhou +3 · 2 citations
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Advanced Differential Geometry Research #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1803.10006

Abstract

Let Mⁿ be a closed Riemannian manifold on which the integral of the scalar curvature is nonnegative. Suppose \mathfraka is a symmetric (0,2) tensor field whose dual (1,1) tensor A has n distinct eigenvalues, and tr(A^k) are constants for k=1,⋯, n-1. We show that all the eigenvalues of A are constants, generalizing a theorem of de Almeida and Brito \citedB90 to higher dimensions. As a consequence, a closed hypersurface Mⁿ in Sⁿ⁺¹ is isoparametric if one takes \mathfraka above to be the second fundamental form, giving affirmative evidence to Chern's conjecture.

Cited by

• Chern Conjecture on Minimal Willmore Hypersurfaces with Constant Scalar Curvature
• Minimal Hypersurfaces with constant scalar curvature in S⁶

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