Hypersurfaces with constant mth mean curvature in the spheres

In this paper, we prove that, for 1≤m≤n−1, n≥3, k≥2, given a constant c between (cotπk)m and k2−2n(k2+m−2n−m)m−22, there exists at least one compact non-isoparametric embedded hypersurface with mth mean curvature Hm=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 Hm=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).