Mean curvature flow in asymptotically flat product spacetimes

Abstract

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 $t\to \infty$.

23 pages, final version