Uniqueness of compact tangent flows in Mean Curvature Flow

作者：Schulze Felix^*

来源：Journal fur die Reine und Angewandte Mathematik, 2014, 690(690): 163-172.

DOI：10.1515/crelle-2012-0070

We show, for mean curvature flows in Euclidean space, that if one of the tangent flows at a given space-time point consists of a closed, multiplicity-one, smoothly embedded self-similar shrinker, then it is the unique tangent flow at that point. That is the limit of the parabolic rescalings does not depend on the chosen sequence of rescalings. Furthermore, given such a closed, multiplicity-one, smoothly embedded self-similar shrinker Sigma, we show that any solution of the rescaled flow, which is sufficiently close to Sigma, with Gaussian density ratios greater or equal to that of Sigma, stays for all time close to Sigma and converges to a possibly different self-similarly shrinking solution Sigma%26apos;. The central point in the argument is a direct application of the Lojasiewicz-Simon inequality to Huisken%26apos;s monotone Gaussian integral for Mean Curvature Flow.

• 出版日期2014-5