Rigidity for spacelike submanifolds with parallel Gaussian mean curvature vector under weighted decay and integrability conditions

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.