2005/02/25 by Tom Ilmanen, Ilmanen, Tom, Nataša Šešum +2 · 2 citations
Mathematics · #53C44 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities #math.DG #msc:53C44
paper · pdf · doi:10.48550/arxiv.math/0502530
openalex publication_date 2005/02/25 · arxiv created 2005/05/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the flow Mt of a smooth, strictly convex hypersurface by its mean curvature in Rn+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 \bf Sn 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 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 as u(x) = t, if x∈ Mt for x∈ \Int(M0). Huisken proved that for n≥ 2 u(x) is C2 near x^*. The case n=1 has been treated by Kohn and Serfaty, they proved C3 regularity of u. As a consequence of obtained rate of convergence of the mean curvature flow we prove that u is not C3 near x^* for n≥ 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≥ 2.