Uniqueness of convex ancient solutions to mean curvature flow in $\mathbb{R}^3$

S. Brendle; K. Choi

Uniqueness of convex ancient solutions to mean curvature flow in $\mathbb{R}^3$

Differential Geometry 2019-01-15 v3 Analysis of PDEs

Authors: S. Brendle , K. Choi

Abstract

A well-known question of Perelman concerns the classification of noncompact ancient solutions to the Ricci flow in dimension $3$ which have positive sectional curvature and are $\kappa$ -noncollapsed. In this paper, we solve the analogous problem for mean curvature flow in $\mathbb{R}^3$ , and prove that the rotationally symmetric bowl soliton is the only noncompact ancient solution of mean curvature flow in $\mathbb{R}^3$ which is strictly convex and noncollapsed.

Keywords

mean curvature flow ancient solution convexity noncollapsed bowl soliton uniqueness

Cite

@article{arxiv.1711.00823,
  title  = {Uniqueness of convex ancient solutions to mean curvature flow in $\mathbb{R}^3$},
  author = {S. Brendle and K. Choi},
  journal= {arXiv preprint arXiv:1711.00823},
  year   = {2019}
}

Comments

revised version, to appear in Invent. Math

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