A new characterization of the Clifford torus via scalar curvature pinching

摘要

Let M-n be a compact hypersurface with constant mean curvature H in Sn+1. Denote by S the squared norm of the second fundamental form of M. We prove that there exists an explicit positive constant gamma(n) depending only on n such that if vertical bar H vertical bar <= gamma(n) and beta(n, H) <= S <= beta(n, H) + n/23, then S equivalent to beta(n, H) and M is one of the following cases: (i) S-k(root k) x s(n-k)(root n-k), 1 <= k <= n - 1; (ii) S-1(1/root 1 + mu(2)) x Sn-1 (mu/root 1+mu(2)). Here beta(n, H) = n + n(3)/2(n-1) H-2 + n(n-2)/2(n-1)root n(2)H(4)+4(n-1)H-2 and mu = n vertical bar H vertical bar+root n(2)H(2)+4(n-1)/2. For n is an element of {6,12,24}, we give 2 examples to show that the assumption vertical bar H vertical bar <= gamma(n cannot be removed. (C) 2014 Elsevier Inc. All rights reserved.