Complete stationary spacelike surfaces in an $n$-dimensional Generalized Robertson-Walker spacetime

TL;DR

This paper establishes uniqueness results for complete stationary spacelike surfaces in higher-dimensional Generalized Robertson-Walker spacetimes, showing under certain curvature conditions that such surfaces are necessarily totally geodesic.

Contribution

It introduces a natural curvature inequality that ensures parabolicity and proves that stationary spacelike surfaces satisfying this are totally geodesic, with explicit examples illustrating the necessity of assumptions.

Findings

Surfaces satisfying the inequality are parabolic.

Such surfaces must be totally geodesic.

Explicit examples in Lorentz-Minkowski spacetime confirm assumptions are necessary.

Abstract

Several uniqueness results for non-compact complete stationary spacelike surfaces in an $n(\geq 3)$-dimensional Generalized Robertson Walker spacetime are obtained. In order to do that, we assume a natural inequality involving the Gauss curvature of the surface, the restrictions of the warping function and the sectional curvature of the fiber to the surface. This inequality gives the parabolicity of the surface. Using this property, a distinguished non-negative superharmonic function on the surface is shown to be constant, which implies that the stationary spacelike surface must be totally geodesic. Moreover, non-trivial examples of stationary spacelike surfaces in the four dimensional Lorentz-Minkowski spacetime are exposed to show that each of our assumptions is needed.