On the Chern conjecture for isoparametric hypersurfaces

Abstract

For a closed hypersurface M n ⊂ S n+1 (1) with constant mean curvature and constant non-negative scalar curvature, the present paper shows that if tr(A k ) are constants for k = 3, . . . , n − 1 for shape operator A, then M is isoparametric. The result generalizes the theorem of de Almeida and Brito [dB90] for n = 3 to any dimension n, strongly supporting Chern's conjecture.In particular, for S ≤ n, one has either S ≡ 0 or S ≡ n on M n .

Biharmonic hypersurfaces with constant scalar curvature in space forms

“…For n ≥ 4, Chern's conjecture remains open. Some important progress had been made recently, see for examples [27,28] and references therein.…”

A note on the Chern Conjecture in dimension four

“…A recent great progress of Tang-Wei-Yan [33] and Tang-Yan [34] generalized the theorem of de Almeida and Brito [11] for n = 3 to any dimension n, strongly supporting Chern Conjecture 1.2. Note that the scalar curvature R M ≥ 0 for all isoparametric hypersurfaces and it can be found in [34].…”

“…Theorem 1.4. (Tang and Yan [34]) Let M n (n ≥ 4) be a closed immersed hypersurface in S n+1 . If the following conditions are satisfied:…”

A lower bound for $L_2$ length of second fundamental form on minimal hypersurfaces

“…Strongly supporting Chern Conjecture 1.2, a recent remarkable progress by Tang-Wei-Yan [55] and Tang-Yan [58] generalized the theorem of de Almeida and Brito [22] for n = 3 to arbitrary dimension n: A closed immersed hypersurface M n in S n+1 having constant 1, 2, • • • , (n − 1)-th mean curvatures and nonnegative scalar curvature R M ≥ 0 is isoparametric. de Almeida-Brito-Scherfner-Weiss [23] showed: A closed immersed hypersurface M n in S n+1 having constant Gauss-Kronecker curvature K M and 3 distinct principal curvatures everywhere is isoparametric.…”

“…For the four dimensional case, Deng-Gu-Wei [24] proved that if M 4 is a closed Willmore minimal hypersurface with constant scalar curvature in S 5 , then it is isoparametric. In other words, in dimension four [24] dropped the nonnegativity assumption R M ≥ 0 of [58], under the new condition Willmore which is equivalent to that the third mean curvature vanishes other than being only a constant as in [58]. In fact, given some pinching restrictions other than identities to the third mean curvature and the Gauss-Kronecker curvature, one can also remove the nonnegativity assumption R M ≥ 0 of [58] in dimension four (cf.…”