A nonexistence result for wing-like mean curvature flows in $\mathbb{R}^4$
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| Format: | Preprint |
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2021
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| _version_ | 1866917854748606464 |
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| author | Choi, Kyeongsu Haslhofer, Robert Hershkovits, Or |
| author_facet | Choi, Kyeongsu Haslhofer, Robert Hershkovits, Or |
| contents | 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 $\lim_{λ\to 0} λ\cdot {K_{t_0}}$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2105_13100 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A nonexistence result for wing-like mean curvature flows in $\mathbb{R}^4$ Choi, Kyeongsu Haslhofer, Robert Hershkovits, Or Differential Geometry Analysis of PDEs 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 $\lim_{λ\to 0} λ\cdot {K_{t_0}}$ 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. |
| title | A nonexistence result for wing-like mean curvature flows in $\mathbb{R}^4$ |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2105.13100 |