Abstract
We study the flow M~t~ of a smooth, strictly convex hypersurface by its mean curvature in β^n + 1^. The surface remains smooth and convex, shrinking monotonically until it disappears at a critical time T and point x^*^ (which is due to Huisken). This is equivalent to saying that the corresponding rescaled mean curvature flow converges to a sphere S^n^ of radius βn. In this paper we will study the rate of exponential convergence of a rescaled flow. We will present here a method that tells us that the rate of the exponential decay is at least 2/n. We can define the βarrival timeβ u of a smooth, strictly convex, nβdimensional hypersurface as it moves with normal velocity equal to its mean curvature via u(x) = t if x β M~t~ for x β Int(M~0~). Huisken proved that, for n β₯ 2, u(x) is C^2^ near x^*^. The case n = 1 has been treated by Kohn and Serfaty [11]; they proved C^3^βregularity of u. As a consequence of the obtained rate of convergence of the mean curvature flow, we prove that u is not necessarily C^3^ near x^*^ for n β₯ 2. We also show that the obtained rate of convergence 2/n, which arises from linearizing a mean curvature flow, is the optimal one, at least for n β₯ 2. Β© 2007 Wiley Periodicals, Inc.