On mean curvature flow translators with prescribed ends
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909744743055360 |
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| author | Sun, Ao Wang, Zhihan |
| author_facet | Sun, Ao Wang, Zhihan |
| contents | Given a smooth closed embedded self-shrinker $S$ with index $I$ in $\mathbb{R}^{n}$, we construct an $I$-dimensional family of complete translators polynomially asymptotic to $S\times\mathbb{R}$ at infinity, which answers a long-standing question by Ilmanen. We further prove that $\mathbb{R}^{n+1}$ can be decomposed in many ways into a one-parameter family of closed sets $\coprod_{a\in \mathbb{R}} T_a$, and each closed set $T_a$ contains a complete translator asymptotic to $S\times\mathbb{R}$ at infinity. If the closed set $T_a$ fattens, namely it has nonempty interior, then there are at least two translators asymptotic to each other at an exponential rate, which can be viewed as a kind of nonuniqueness. We show that this fattening phenomenon is non-generic but indeed happens. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_08224 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On mean curvature flow translators with prescribed ends Sun, Ao Wang, Zhihan Differential Geometry 53E10, 53A10 Given a smooth closed embedded self-shrinker $S$ with index $I$ in $\mathbb{R}^{n}$, we construct an $I$-dimensional family of complete translators polynomially asymptotic to $S\times\mathbb{R}$ at infinity, which answers a long-standing question by Ilmanen. We further prove that $\mathbb{R}^{n+1}$ can be decomposed in many ways into a one-parameter family of closed sets $\coprod_{a\in \mathbb{R}} T_a$, and each closed set $T_a$ contains a complete translator asymptotic to $S\times\mathbb{R}$ at infinity. If the closed set $T_a$ fattens, namely it has nonempty interior, then there are at least two translators asymptotic to each other at an exponential rate, which can be viewed as a kind of nonuniqueness. We show that this fattening phenomenon is non-generic but indeed happens. |
| title | On mean curvature flow translators with prescribed ends |
| topic | Differential Geometry 53E10, 53A10 |
| url | https://arxiv.org/abs/2301.08224 |