Degenerate cylindrical singular sets of mean curvature flow are locally contained in C^{2,α} submanifolds, with curvature determined by the flow's asymptotic profile.
Generic Dynamics of Mean Curvature Flows with Asymptotically Conical Singularities
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abstract
This is the second paper in the series to study the generic dynamics of mean curvature flows. We study the initial perturbation of mean curvature flows, whose first singularity is modeled by an asymptotically conical shrinker. The noncompactness of the limiting shrinker creates essential difficulties. We introduce the Feynman-Kac formula to get precise asymptotic behaviour of the linearized rescaled mean curvature equation along an orbit. We also develop the invariant cone method for the noncompact setting for the local dynamics near the shrinker. As a consequence, we prove that after a generic initial perturbation, the perturbed rescaled mean curvature flow avoids the conical singularity.
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Regularity of cylindrical singular sets of mean curvature flow
Degenerate cylindrical singular sets of mean curvature flow are locally contained in C^{2,α} submanifolds, with curvature determined by the flow's asymptotic profile.