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

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 $\hat{I}º$-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.

revised version, to appear in Invent. Math