  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
$\mathbb{R}^4$.
