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In this paper, we investigate the large time behavior of interfaces movingwith 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 becharacterized by its head and tail speed, which depends continuously on itsoverall direction of propagation $\nu$. If head speed equals tail speed at agiven direction $\nu$, the interface has a unique large-scale speed in thatdirection. In general the interface develops linearly growing "long fingers" inthe direction where the equality breaks down. We discuss these results in bothgeneral setting and in laminar setting, where further results are obtained dueto regularity properties of the flow.

Head and tail speeds of mean curvature flow with forcing