Spacelike translating solitons in Lorentzian product spaces: Nonexistence, Calabi-Bernstein type results and examples

We establish nonexistence results for complete spacelike translating solitons immersed in a Lorentzian product space R-1 x P-n, under suitable curvature constraints on the curvatures of the Riemannian base P-n. In particular, we obtain Calabi-Bernstein type results for entire translating graphs constructed over P-n. For this, we prove a version of the Omori-Yau's maximum principle for complete spacelike translating solitons. Besides, we also use other two analytical tools related to an appropriate drift Laplacian: a parabolicity criterion and certain integrability properties. Furthermore, under the assumption that the base P-n is non-positively curved, we close our paper constructing new examples of rotationally symmetric spacelike translating solitons embedded into R-1 x P-n.