Zhou Gang

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 $l\ge 2$, 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.

In the present paper we study a type of generic singularity of mean curvature flow modelled on the bubble-sheet ${\mathbb{S}}^1\times {\mathbb{R}}^3$ , and we derive an asymptotic profile for a neighborhood of singularity.

We consider one of the generic regimes of formation of singularities. We obtain a detailed description of a possibly small, but fixed, neighborhood of the blowup point, up to (and including) the blowup time, and find that it is mean convex. This confirms a conjecture by Ilmanen. And we find that the singularity is isolated from the other ones.

We study the formation of generic singularities of mean curvature flow by combining the different approaches, specifically the methods in studying blowup of nonlinear heat equations, the techniques used by the author and the collaborators for mean curvature flow, and these invented by Colding and Minicozzi. We study the solution in a neighborhood of the blowup point, in some generic regimes, we find the key parameters take favorable signs and have sharp decay rates. We provide the remainder estimates in different norms.