Generic Dynamics of Mean Curvature Flows with Asymptotically Conical Singularities
arXiv:2107.05066 · doi:10.1007/s11425-024-2391-6
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.
48 pages, 1 figure. Accepted by Sci. China Math
References in corpus (9)
- A Topological Property of Asymptotically Conical Self-Shrinkers of Small Entropy
- Mean curvature self-shrinkers of high genus: Non-compact examples
- Construction of Complete Embedded Self-Similar Surfaces under Mean Curvature Flow. Part III
- Uniqueness of asymptotically conical tangent flows
- Asymptotic structure of self-shrinkers
- On the Multiplicity One Conjecture for Mean Curvature Flows of surfaces
- The C^r dependence problem of eigenvalues of the Laplace operator on domains in the plane
- Initial Perturbation of the Mean Curvature Flow for closed limit shrinker
- Closed mean curvature flows with asymptotically conical singularities