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

Abstract

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

$$\kappa $$

κ

-noncollapsed. In this paper, we solve the analogous problem for mean curvature flow in

$${\mathbb {R}}^3$$

R

3

, and prove that the rotationally symmetric bowl soliton is the only noncompact ancient solution of mean curvature flow in

$${\mathbb {R}}^3$$

R

3

which is strictly convex and noncollapsed.