Calabi-Bernstein theorems for constant mean curvature spacelike hypersurfaces

In this paper, we establish new Calabi-Bernstein theorems for spacelike hypersurfaces with constant mean curvature in Lorentzian product spaces of the form M n × R 1 , where ( M n , 〈 , 〉 M ) is a complete, noncompact Riemannian manifold with either nonnegative Ricci curvature or sectional curvature bounded from below, and the product M n × R 1 is endowed with the Lorentzian metric 〈 , 〉 = 〈 , 〉 M − d t 2 . More precisely, we prove that the only entire spacelike graphs of constant mean curvature, given by t = u ( x ) are the slices u ( x ) = constant , under the following conditions: (i) M n has nonnegative Ricci curvature and u ∈ C ∞ ( M n ) satisfies ‖ D u ( x ) ‖ M → 0 as r M ( x ) → ∞ ; or (ii) M n has nonnegative sectional curvature, u is bounded either from above or from below on M and sup M ⁡ ‖ D u ‖ M < 1 . The result under assumption (i) follows from a maximum principle at infinity established in [3, Theorem 2.2] , while the case (ii) is derived using the classical Omori-Yau maximum principle.