2022 Theses Doctoral

Structure and symmetry of singularity models of mean curvature flow

Zhu, Jingze

In this thesis, we study the structure and symmetry of singularity models of mean curvature flow.

In chapter 1, we prove the quantitative long range curvature estimate and related results. The famous structure theorem of White asserts that in convex 𝛼-noncollapsed ancient solutions to the mean curvature flow, rescaled curvature is bounded in terms of rescaled distance. We improve this result and show that rescaled curvature is bounded by a quadratic function of rescaled distance using Ecker-Huisken's interior estimate. This method together with an induction on scale argument similar to the work of Brendle-Huisken can push the result to high curvature regions. We show that for a mean convex flow and any 𝑅 > 0, the rescaled curvature is bounded by 𝑪(𝑅+1)² in a parabolic neighborhood of rescaled size 𝑅 in the high curvature regions.

We will then describe how this can be applied to give an alternative proof to a simplified version of White's structure theorem.

In chapter 2, we discuss the symmetry structure of translators. We show that with mild assumptions, every convex, noncollapsed translator in ℝ⁴ has 𝑆𝑂(2) symmetry. In higher dimensions, we can prove an analogous result with a curvature assumption. With mild assumptions, we show that every convex, uniformly 3-convex, noncollapsed translator in ℝⁿ+¹ has 𝑆𝑂(n-1) symmetry.