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Some of the most worrisome potential singularity models for the meancurvature flow of $3$-dimensional hypersurfaces in $\mathbb{R}^4$ arenoncollapsed wing-like flows, i.e. noncollapsed flows that are asymptotic to awedge. In this paper, we rule out this potential scenario, not just amongself-similarly translating singularity models, but in fact among all ancientnoncollapsed flows in $\mathbb{R}^4$. Specifically, we prove that for anyancient noncollapsed mean curvature flow $M_t=\partial K_t$ in $\mathbb{R}^4$the blowdown $\lim_{\lambda\to 0} \lambda\cdot {K_{t_0}}$ is always a point,halfline, line, halfplane, plane or hyperplane, but never a wedge. In our proofwe introduce a fine bubble-sheet analysis, which generalizes the fine neckanalysis that has played a major role in many recent papers. Our result is alsoa key first step towards the classification of ancient noncollapsed flows in$\mathbb{R}^4$, which we will address in a series of subsequent papers.

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