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 $\kappa$-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.

revised version, to appear in Invent. Math

References in corpus (1)

• Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions

Cited by in corpus (5)

• Ancient solutions to the Ricci flow in dimension 3
• Ancient mean curvature flows and their spacetime tracks
• Convexity of 2-convex translating solitons to the mean curvature flow in ${\mathbb{R}}^{n+1}$
• Uniqueness of ancient compact non-collapsed solutions to the 3-dimensional Ricci flow
• Unique asymptotics of ancient compact non-collapsed solutions to the 3-dimensional Ricci flow