Convex solutions to the mean curvature flow
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- 1 May 2011
- journal article
- Published by Annals of Mathematics in Annals of Mathematics
- Vol. 173 (3), 1185-1239
- https://doi.org/10.4007/annals.2011.173.3.1
Abstract
In this paper we study the classification of ancient convex solutions to the mean curvature flow in $\R^{n+1}$. 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 $S^{k-1}\times \R^{n-k}$. In this paper we give an affirmative answer for entire solutions in dimension 2. In high dimensions we prove that there exist non-rotationally 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 $\R^{n+1}$ is a shrinking sphere or cylinder.
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