Spacelike translating solitons of the mean curvature flow in Lorentzian product spaces with density

Abstract

By applying suitable Liouville-type results, an appropriate parabolicity criterion, and a version of the Omori-Yau's maximum principle for the drift Laplacian, we infer the uniqueness and nonexistence of complete spacelike translating solitons of the mean curvature flow in a Lorentzian product space $ \mathbb R_1\times\mathbb P^n_f $ endowed with a weight function $ f $ and whose Riemannian base $ \mathbb P^n $ is supposed to be complete and with nonnegative Bakry-Émery-Ricci tensor. When the ambient space is either $ \mathbb R_1\times\mathbb G^n $, where $ \mathbb G^n $ stands for the so-called $ n $-dimensional Gaussian space (which is the Euclidean space $ \mathbb R^n $ endowed with the Gaussian probability measure) or $ \mathbb R_1\times\mathbb H_f^n $, where $ \mathbb H^n $ denotes the standard $ n $-dimensional hyperbolic space and $ f $ is the square of the distance function to a fixed point of $ \mathbb H^n $, we derive some interesting consequences of our uniqueness and nonexistence results. In particular, we obtain nonexistence results concerning entire spacelike translating graphs constructed over $ \mathbb P^n $.