A nonexistence result for wing-like mean curvature flows in $\mathbb{R}^4$
| Autor: | Choi, Kyeongsu, Haslhofer, Robert, Hershkovits, Or |
|---|---|
| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | Geom. Topol. 28 (2024) 3095-3134 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.2140/gt.2024.28.3095 |
| Popis: | 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. Comment: 40 pages |
| Databáze: | arXiv |
| Externí odkaz: |
|