A nonexistence result for wing-like mean curvature flows in R4

Some of the most worrisome potential singularity models for the mean curvature flow of three-dimensional hypersurfaces in R⁴ are noncollapsed wing-like flows, ie noncollapsed flows that are asymptotic to a wedge. We rule out this potential scenario, not just among self-similarly translating singularity models, but in fact among all ancient noncollapsed flows in R⁴. Specifically, we prove that for any ancient noncollapsed mean curvature flow (formula Presented) the blowdown (formula Presented) is always a point, halfline, line, halfplane, plane or hyperplane, but never a wedge. In our proof we introduce a fine bubblesheet analysis, which generalizes the fine neck analysis that has played a major role in many recent papers. Our result is also a key first step towards the classification of ancient noncollapsed flows in R⁴, which we will address in a series of subsequent papers.