Electron. J. Differential Equations, Vol. 2020 (2020), No. 83, pp. 1-19.

Solutions to mean curvature equations in weighted standard static spacetimes

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.
DOI: 10.58997/ejde.2020.83

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Henrique F. de Lima
Departamento de Matemática
Universidade Federal de Campina Grande
58.429-970 Campina Grande, Paraíba, Brazil
email: henrique@mat.ufcg.edu.br
André F. A. Ramalho
Departamento de Matemática
Universidade Federal de Campina Grande
58.429-970 Campina Grande, Paraíba, Brazil
email: andre@mat.ufcg.edu.br
Marco Antonio L. Velásquez
Departamento de Matemática
Universidade Federal de Campina Grande
58.429-970 Campina Grande, Paraíba, Brazil
email: marco.velasquez@mat.ufcg.edu.br

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