Hypersurfaces with constant mth mean curvature in the spheres

摘要

In this paper, we prove that, for 1 <= m <= n 1, n >= 3, k >= 2, given a constant c between (cot pi/k)(m) and k(2)-2 (k(2)+m-2-m)(m-2/2), there exists at least one compact non-isoparametric embedded hypersurface with mth mean curvature H-m = c in a unit sphere Sn+1. As corollaries, we also prove that for m = n-1, n-2 and m >= 2, given any positive constant c, there exists at least one compact non-isoparametric embedded hypersurface with H-m = c in Sn+1. Moreover, our results are the generalization of the results of Perdomo [9] (when m = 1), Wei et al. [7] (when m = 2, 4). (C) 2016 Elsevier B.V. All rights reserved.