EMBEDDED HYPERSURFACES WITH CONSTANT mTH MEAN CURVATURE IN A UNIT SPHERE

In this paper, we study n-dimensional hypersurfaces with constant mth mean curvature in a unit sphere S(n+1)(1) and construct many compact nontrivial embedded hypersurfaces with constant mth mean curvature H(m) > 0 in S(n+1)(1), for 1 <= m <= n - 1. Moreover, if the 2nd mean curvature H(2) takes value between 1/(tan pi/k)(2) and k(2)-2/n for any integer k >= 2 and n >= 3, then there exists an n-dimensional compact nontrivial embedded hypersurface with constant H(2) (i.e. constant scalar curvature) in S(n+1)(1); If the 4th mean curvature H(4) takes value between 1/(tan pi/k)(4) and k(4)-4/n(n-4) for any integer k >= 3 and n >= 5, then there exists an n-dimensional compact nontrivial embedded hypersurface with constant H(4) in S(n+1)(1).