Rate of convergence of the mean curvature flow

We study the flow M t of a smooth, strictly convex hypersurface by its mean curvature in ℝ 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 √ n . 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 “arrival time” u of a smooth, strictly convex, n ‐dimensional hypersurface as it moves with normal velocity equal to its mean curvature via u ( x ) = t if x ∈ M t for x ∈ Int( M 0 ). Huisken proved that, for n ≥ 2, u ( x ) is C 2 near x * . The case n = 1 has been treated by Kohn and Serfaty [11]; they proved C 3 ‐regularity 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 ≥ 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 ≥ 2. © 2007 Wiley Periodicals, Inc.