Felix Lubbe

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 $t\to \infty$.

Let $N$ be a complete manifold with bounded geometry, such that ${\sec }_N\le -\sigma <0$ for some positive constant $\sigma$. We investigate the mean curvature flow of the graphs of smooth length-decreasing maps $f:{\mathbb{R}}^m\to N$. In this case, the solution exists for all times and the evolving submanifold stays the graph of a length-decreasing map $f_t$. We further prove uniform decay estimates for all derivatives of order $\ge 2$ of $f_t$ along the flow.

We consider the mean curvature flow of the graph of a smooth map $f:{\mathbb{R}}^2\to {\mathbb{R}}^2$ between two-dimensional Euclidean spaces. If $f$ satisfies an area-decreasing property, the solution exists for all times and the evolving submanifold stays the graph of an area-decreasing map $f_t$. Further, we prove uniform decay estimates for the mean curvature vector of the graph and all higher-order derivatives of the corresponding map $f_t$.