Solutions to mean curvature equations in weighted standard static spacetimes

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.

Keywords

• standard static spacetimes with density
• gaussian space
• f-parabolic spacelike hypersurface
• entire killing graphs
• mean curvature equation