Hypersurfaces with constant mean curvature in a hyperbolic space

Let Mn be an n-dimensional complete connected and oriented hypersurface in a hyperbolic space Hn+1(c) with non-zero constant mean curvature H and two distinct principal curvatures. In this paper, we show that (1) if the multiplicities of the two distinct principal curvatures are greater than 1, then Mn is isometric to the Riemannian product Sk(r) × Hn−k (−1/(r2 + ρ2)), where r > 0 and 1 < k < n − 1; (2) if H2 > −c and one of the two distinct principal curvatures is simple, then Mn is isometric to the Riemannian product Sn−1(r) × H1 (−1/(r2 + ρ2)) or S1(r) × Hn−1 (−1/(r2 + ρ2)) r > 0, if one of the following conditions is satisfied (i) on Mn or (ii) on Mn or (iii) on Mn, where t1 and t2 are the positive real roots of (1.5).