On the Chern conjecture for isoparametric hypersurfaces

Abstract

For a closed hypersurface Mn ⊂ Sn+1(1) with constant mean curvature and constant non-negative scalar curvature, we show that if \({\rm{tr}}\left({{{\cal A}^k}} \right)\) are constants for k = 3, …, n − 1 and the shape operator \({\cal A}\) then M is isoparametric. The result generalizes the theorem of de Almeida and Brito (1990) for n = 3 to any dimension n, strongly supporting the Chern conjecture.