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

Abstract 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 $$\mathbb {R}_1\times M^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>R</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>M</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> , whose Riemannian base $$M^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>M</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:math> has nonnegative sectional curvature, must be a slice $$\{t\}\times M^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>t</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>M</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> . In particular, we conclude that the spacelike hyperplanes of the Lorentz–Minkowski space $$\mathbb {L}_1^{n+1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> are the only complete CMC spacelike hypersurfaces immersed in $$\mathbb {L}_1^{n+1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> , 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.