Mean curvature flow with generic initial data

Abstract

We show that the mean curvature flow of generic closed surfaces in ${\mathbb{R}}^3$ avoids asymptotically conical and non-spherical compact singularities. We also show that the mean curvature flow of generic closed low-entropy hypersurfaces in ${\mathbb{R}}^4$ is smooth until it disappears in a round point. The main technical ingredient is a long-time existence and uniqueness result for ancient mean curvature flows that lie on one side of asymptotically conical or compact shrinking solitons.

arXiv commentFinal version, to appear in Invent. Math

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Cited by in corpus (4)

• Stability of neckpinch singularities
• Ancient finite entropy flows by powers of curvature in ${\mathbb{R}}^2$
• Classification of ancient flows by sub-affine-critical powers of curvature in ${\mathbb{R}}^2$
• Mass Drop and Multiplicity in Mean Curvature Flow