Mean curvature flow in asymptotically flat product spacetimes

We consider the long-time behaviour of the mean curvature flow of spacelike hypersurfaces in the Lorentzian product manifold \(M\times\mathbb{R}\), where \(M\) is asymptotically flat. If the initial hypersurface \(F_0\subset M\times\mathbb{R}\) is uniformly spacelike and asymptotic to \(M\times\left\{s\right\}\) for some \(s\in\mathbb{R}\) at infinity, we show that a mean curvature flow starting at \(F_0\) exists for all times and converges uniformly to \(M\times\left\{s\right\}\) as \(t\to \infty\).