On the non-degenerate and degenerate generic singularities formed by mean curvature flow

TL;DR

This paper analyzes generic singularities in mean curvature flow, focusing on those modeled on ^3, providing detailed local descriptions, proving mean convexity and isolation of singularities, and conjecturing about the behavior of remaining cases.

Contribution

It offers a detailed analysis of singularities modeled on ^3 in mean curvature flow, including local descriptions, convexity proofs, and a conjecture on the nature of remaining singularities.

Findings

Neighborhoods of certain singularities are mean convex.

Singularities are shown to be isolated in some cases.

Evidence suggests some singularities cause entire neighborhoods to become singular at blowup.

Abstract

In this paper we study a neighborhood of generic singularities formed by mean curvature flow (MCF). We limit our consideration to the singularities modelled on $\mathbb{S}^3\times\mathbb{R}$ because, compared to the cases $\mathbb{S}^k\times \mathbb{R}^{l}$ with $l\geq 2$, the present case has the fewest possibilities to be considered. For various possibilities, we provide a detailed description for a small, but fixed, neighborhood of singularity, and prove that a small neighborhood of the singularity is mean convex, and the singularity is isolated. For the remaining possibilities, we conjecture that an entire neighborhood of the singularity becomes singular at the time of blowup, and present evidences to support this conjecture. A key technique is that, when looking for a dominating direction for the rescaled MCF, we need a normal form transformation, as a result, the rescaled MCF is parametrized over some chosen curved cylinder, instead over a standard straight one. This is a long paper. The introduction is carefully written to present the key steps and ideas.