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

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 $\mathbb{R}^4$.