The rate of convergence of the mean curvature flow

Abstract

We study the flow $M_t$ of a smooth, strictly convex hypersurface by its mean curvature in ${\mathrm{R}}^{n+1}$. The surface remains smooth and convex, shrinking monotonically until it disappears at a critical time $T$ and point $x^{\ast }$ (which is due to Huisken). This is equivalent to saying that the corresponding rescaled mean curvature flow converges to a sphere ${\mathbf{S}}^{\mathbf{n}}$ of radius $\sqrt{n}$. In this paper we will study the rate of exponential convergence of a rescaled flow. We will present here a method that tells us the rate of the exponential decay is at least $\frac{2}{n}$. 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 as $u(x)=t$, if $x\in M_t$ for $x\in \Int(M_0)$. Huisken proved that for $n\ge 2$ $u(x)$ is $C^2$ near $x^{\ast }$. The case $n=1$ has been treated by Kohn and Serfaty, they proved $C^3$ regularity of $u$. As a consequence of obtained rate of convergence of the mean curvature flow we prove that $u$ is not $C^3$ near $x^{\ast }$ for $n\ge 2$. We also show that the obtained rate of convergence $2/n$, that comes out from linearizing a mean curvature flow is the optimal one, at least for $n\ge 2$.