Rigidity of CMC Spacelike Hypersurfaces in a Lorentzian Product Space Under a Dirichlet Integral Condition
Year: 2026
Type: article
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 $$\mathbb {R}_1\times M^n$$ R 1 × M n , whose Riemannian base $$M^n$$ M n has nonnegative sectional curvature, must be a slice $$\{t\}\times M^n$$ { t } × M n . In particular, we conclude that the spacelike hyperplanes of the Lorentz–Minkowski space $$\mathbb {L}_1^{n+1}$$ L 1 n + 1 are the only complete CMC spacelike hypersurfaces immersed in $$\mathbb {L}_1^{n+1}$$ 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.
Source: Milan Journal of Mathematics
Author: Henrique Fernandes de Lima
Institution: Universidade Federal de Campina Grande
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W7168954111