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}$$isuniformlyspacelikeandasymptoticto$M\times\left{s\right}$forsome$s\in$\mathbb{R}$$ 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 $t\to \mathrm{\infty }$.