Ancient mean curvature flows that converge rapidly to a compact self-shrinker at time minus infinity must have the same codimension as the shrinker, and if the convergence is super-exponential the flow equals the shrinker.
Regularity of elliptic and parabolic systems
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abstract
We show uniqueness of cylindrical blowups for mean curvature flow in all dimension and all codimension. Cylindrical singularities are known to be the most important; they are the most prevalent in any codimension. Mean curvature flow in higher codimension is a nonlinear parabolic system where many of the methods used for hypersurfaces do not apply and uniqueness of cylindrical blowups remained a major open problem. Our results imply regularity of the singular set for the system.
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math.DG 1years
2019 1verdicts
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Codimension Bounds and Rigidity of Ancient Mean Curvature Flows by the Tangent Flow at $-\infty$
Ancient mean curvature flows that converge rapidly to a compact self-shrinker at time minus infinity must have the same codimension as the shrinker, and if the convergence is super-exponential the flow equals the shrinker.