Electronic Journal of Differential Equations, 
Vol. 2020 (2020), No. 83, pp. 1-19.

Title: Solutions to mean curvature equations in weighted standard static spacetimes
       
Authors: Henrique F. de Lima, Andre F. A. Ramalho, Marco Antonio L. Velasquez

Abstract:
 In this article, we study the solutions for the mean curvature
 equation in a weighted standard static spacetime,
 $\mathbb{P}_f^n\times_\rho\mathbb{R}_1$, having a warping function $\rho$
 whose  weight function f does not depend on the parameter $t\in\mathbb{R}$.
 We establish a f-parabolicity criterion to study the rigidity
 of spacelike hypersurfaces immersed in 
 $\mathbb{P}_f^n\times_\rho\mathbb{R}_1$
 and,  in particular, of entire Killing graphs constructed over the Riemannian base
 $\mathbb{P}^n$.
 Also we give applications to weighted standard static spacetimes of the type
 $\mathbb{G}^n\times_\rho\mathbb{R}_1$, where $\mathbb G^n$ is the Gaussian space.
 
Submitted June 15, 2020. Published July 30, 2020.
Math Subject Classifications: 53C42, 53B30, 53C50.
Key Words: Standard static spacetimes with density; Gaussian space;
           f-parabolic spacelike hypersurface; entire Killing graphs; 
           mean curvature equation.