A nonexistence result for rotating mean curvature flows in $\mathbb{R}^{4}$

TL;DR

This paper proves that in four-dimensional Euclidean space, there are no rotating ancient mean curvature flows that are noncollapsed and resemble cylinders at ancient times, ruling out certain potential singularity models.

Contribution

It establishes a nonexistence result for rotating ancient flows in , clarifying the structure of possible singularities in mean curvature flow in this setting.

Findings

No rotating ancient noncollapsed flows in  exist.

Rotating ancient flows cannot serve as singularity models in .

The result constrains the types of singularities possible in mean curvature flow in four dimensions.

Abstract

Some worrisome potential singularity models for the mean curvature flow are rotating ancient flows, i.e. ancient flows whose tangent flow at $-\infty$ is a cylinder $\mathbb{R}^k\times S^{n-k}$ and that are rotating within the $\mathbb{R}^k$-factor. We note that while the $\mathbb{R}^k$-factor, i.e. the axis of the cylinder, is unique by the fundamental work of Colding-Minicozzi, the uniqueness of tangent flows by itself does not provide any information about rotations within the $\mathbb{R}^k$-factor. In the present paper, we rule out rotating ancient flows among all ancient noncollapsed flows in $\mathbb{R}^4$.