Rigidity for spacelike submanifolds with parallel Gaussian mean curvature vector under weighted decay and integrability conditions
Year: 2026
Type: article
Abstract: We establish rigidity results for complete spacelike submanifolds with parallel Gaussian mean curvature vector immersed in the pseudo-Euclidean space $$\mathbb {R}^{n+p}_p$$ R p n + p . Under the assumption $$\Vert \xi \Vert ^2\le 2/p$$ ‖ ξ ‖ 2 ≤ 2 / p , we prove that the submanifold must be an affine spacelike subspace whenever the covariant derivative of the second fundamental form satisfies either an averaged weighted decay condition or a weighted tail integrability condition at infinity. In the critical case $$\Vert \xi \Vert ^2=2/p$$ ‖ ξ ‖ 2 = 2 / p , the same conclusion is obtained under an additional gap assumption on $$\sup _M\Vert A\Vert ^2$$ sup M ‖ A ‖ 2 . The proof is based on a Simons-type differential inequality combined with weighted integral methods with respect to the Gaussian measure. As applications, we derive Bernstein-type consequences for spacelike $$\xi $$ ξ -hypersurfaces in the Lorentz-Minkowski space.
Source: Journal of Geometry
Author: Weiller F. Chaves Barboza
Institution: Universidade Federal de Campina Grande
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
OpenAlex ID: W7225416127