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

Abstract

Some of the most worrisome potential singularity models for the mean curvature flow of $3$-dimensional hypersurfaces in ${\mathbb{R}}^4$ are noncollapsed wing-like flows, i.e. noncollapsed flows that are asymptotic to a wedge. In this paper, we rule out this potential scenario, not just among self-similarly translating singularity models, but in fact among all ancient noncollapsed flows in ${\mathbb{R}}^4$. Specifically, we prove that for any ancient 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 proof we introduce a fine bubble-sheet 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 ${\mathbb{R}}^4$, which we will address in a series of subsequent papers.

40 pages

Cited by in corpus (2)

• Classification of bubble-sheet ovals in ${\mathbb{R}}^4$
• O(2)-symmetry of 3D steady gradient Ricci solitons