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

Choi, Kyeongsu¹ ; Haslhofer, Robert² ; Hershkovits, Or³

¹ School of Mathematics, Korea Institute for Advanced Study, Seoul, South Korea
² Department of Mathematics, University of Toronto, Toronto, ON, Canada
³ Institute of Mathematics, Hebrew University, Jerusalem, Israel

Geometry & topology, Tome 28 (2024) no. 7, pp. 3095-3134.

Résumé

Some of the most worrisome potential singularity models for the mean curvature flow of three-dimensional hypersurfaces in ${ℝ}^4$ 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 ${ℝ}^4$. Specifically, we prove that for any ancient noncollapsed mean curvature flow $M_t=\partial K_t$ in ${ℝ}^4$ the blowdown $\underset{\lambda \to 0}{\lim }\lambda \cdot K_{t_0}$ 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 ${ℝ}^4$, which we will address in a series of subsequent papers.

Détail

DOI : 10.2140/gt.2024.28.3095

Keywords: mean curvature flow, singularities, ancient solutions

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