  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.
