Complete Hypersurfaces With Constant Mean Curvature In Sⁿ⁺¹(1)

Let Mⁿ be an n-dimensional complete hypersurface in the (n + 1)-dimensional unitsphere Sⁿ⁺¹ with constant mean curvature H. Let {e_i}_i=1ⁿ be a local orthonormal basisof Mⁿ with respect to the induced metric and {w_i}_i=1ⁿ be its dual basis of 1-forms. LetII=∑_i,jh_ijw_i(?)w_j be the second fundamental form and H=(?)∑_ih_ii be the meancurvature. Denote S=|II|²=∑_i,j(h_ij)² and |Φ|=(?).In this paper, we study complete hypersurfaces of the unit sphere Sⁿ⁺¹ with constantmean curvature H. As the main result, we will prove the followingMain Theorem. For each constant H, we define two positive constants B⁺(H) andB^-(H) byThen we haveFor any B∈[B^-(H),B⁺(H)], there is a complete hypersurface Mⁿâ†'Sⁿ⁺¹ withconstant mean curvature H, such that the corresponding function |Φ| satisfies the relationsup |Φ| = B(H).