Generic Dynamics of Mean Curvature Flows with Asymptotically Conical Singularities

TL;DR

This paper investigates the generic behavior of mean curvature flows near asymptotically conical singularities, introducing new analytical tools to understand their dynamics and showing that generic perturbations avoid such singularities.

Contribution

It develops the Feynman-Kac formula and invariant cone method for noncompact settings, advancing the understanding of mean curvature flow dynamics with conical singularities.

Findings

Perturbed flows typically avoid conical singularities.

New methods handle noncompact asymptotically conical shrinkers.

Asymptotic behavior of linearized equations is precisely characterized.

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.