In this talk, we will first recall some background and research history of Chern's conjecture, which asserts that a closed, minimally immersed hypersurface of the unit sphere Sn+1(1) with constant scalar  curvature is isoparametric. Next, we introduce our progress in this conjecture. We proved that for a closed hypersurface Mn ⊂ Sn+1(1)  with constant mean curvature and constant non-negative scalar curvature, if tr(Ak) are constants (k = 3,...,n−1)  for shape operator A, then M is isoparametric, which generalizes the theorem of de Almeida and Brito in their 1990's  paper in 《Duke Math. J. 》for n = 3 to any dimension n, strongly supporting Chern’s conjecture. This talk is based on two joint papers with Professor Dongyi Wei and Professor Zizhou Tang.

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