Head and tail speeds of mean curvature flow with forcing

In this paper, we investigate the large time behavior of interfaces moving with motion law $V=-\kappa +g(x)$, where $g$ is positive, Lipschitz and ${\mathbb{Z}}^n$-periodic. It turns out that the behavior of the interface can be characterized by its head and tail speed, which depends continuously on its overall direction of propagation $\nu$. If head speed equals tail speed at a given direction $\nu$, the interface has a unique large-scale speed in that direction. In general the interface develops linearly growing "long fingers" in the direction where the equality breaks down. We discuss these results in both general setting and in laminar setting, where further results are obtained due to regularity properties of the flow.