A note on spacelike weighted translating solitons in weighted Lorentzian product spaces

We apply some maximum principles to establish rigidity results concerning complete spacelike weighted translating solitons immersed in a weighted Lorentzian product space R1xPfn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}_1\times {\mathbb {P}}<^>n_f$$\end{document} endowed with a potential function f and whose Riemannian base Pn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {P}}<^>n$$\end{document} is supposed to be complete and with nonnegative Bakry-& Eacute;mery-Ricci tensor. As applications, we derive new Calabi-Bernstein type results for entire spacelike weighted translating graphs constructed over Pn.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {P}}<^>n.$$\end{document}