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

Abstract

We establish nonexistence results for complete spacelike translating solitons immersed in a Lorentzian product space [Formula: see text], under suitable curvature constraints on the curvatures of the Riemannian base [Formula: see text]. In particular, we obtain Calabi–Bernstein type results for entire translating graphs constructed over [Formula: see text]. 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 [Formula: see text] is non-positively curved, we close our paper constructing new examples of rotationally symmetric spacelike translating solitons embedded into [Formula: see text].