Rigidity of CMC Spacelike Hypersurfaces in a Lorentzian Product Space Under a Dirichlet Integral Condition

We apply Bochner’s technique combined with a mild Dirichlet integral condition to show that a complete constant mean curvature (CMC) spacelike hypersurface immersed with positive semi-definite modified second fundamental form in a Lorentzian product space R₁× Mⁿ R 1 × M n , whose Riemannian base Mⁿ M n has nonnegative sectional curvature, must be a slice \× Mⁿ { t } × M n . In particular, we conclude that the spacelike hyperplanes of the Lorentz–Minkowski space L₁ⁿ⁺¹ L 1 n + 1 are the only complete CMC spacelike hypersurfaces immersed in L₁ⁿ⁺¹ L 1 n + 1 , whose modified second fundamental form is positive semi-definite and the height function with respect to a spacelike hyperplane satisfies such a Dirichlet integral condition. A version of the classical Calabi–Bernstein Theorem to the context of Lorentzian product spaces is also given.