Convex solutions to the mean curvature flow

Xu-Jia Wang¹

¹ Centre for Mathematics and its Applications, The Australian National University, ACT 0200, Australia

Annals of mathematics, Tome 173 (2011) no. 3, pp. 1185-1239.

Résumé

In this paper we study the classification of ancient convex solutions to the mean curvature flow in $\Bbb{R}^{n+1}$. An open problem related to the classification of type II singularities is whether a convex translating solution is $k$-rotationally symmetric for some integer $2\le k\le n$, namely whether its level set is a sphere or cylinder $S^{k-1}\times \Bbb{R}^{n-k}$. In this paper we give an affirmative answer for entire solutions in dimension $2$. In high dimensions we prove that there exist nonrotationally symmetric, entire convex translating solutions, but the blow-down in space of any entire convex translating solution is $k$-rotationally symmetric. We also prove that the blow-down in space-time of an ancient convex solution which sweeps the whole space $\mathbb{R}^{n+1}$ is a shrinking sphere or cylinder.

Détail

DOI : 10.4007/annals.2011.173.3.1