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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.18825 |
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| _version_ | 1866915408605347840 |
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| author | Shao, Guanhua Zou, Jiahua |
| author_facet | Shao, Guanhua Zou, Jiahua |
| contents | For each half-integer $J$ and large enough integer $m$ we construct by PDE gluing methods a self-shrinker $\breve{M}[J,m]$ with $2J+1$ ends and genus $2J(m-1)$. $\breve{M}[J,m]$ resembles the stacking of $2J+1$ levels of the plane $\mathbb{R}^2$ in $\mathbb{R}^3$ that have been connected by $2Jm$ catenoidal bridges with $m$ bridges connecting each pair of adjacent levels. It observes the symmetry of an $m$-gonal prism (when $J$ is a half integer) or an $m$-gonal antiprism (when $J$ is an integer). The construction is based on the Linearised Doubling (LD) methodology which was first introduced by Kapouleas in the construction of minimal surface doublings of $\mathbb{S}^2_{eq}$ in $\mathbb{S}^3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_18825 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Self-shrinkers with any number of ends in $\mathbb{R}^{3}$ by stacking $\mathbb{R}^{2}$ Shao, Guanhua Zou, Jiahua Differential Geometry 53A10 (Primary), 53E10 (Secondary) For each half-integer $J$ and large enough integer $m$ we construct by PDE gluing methods a self-shrinker $\breve{M}[J,m]$ with $2J+1$ ends and genus $2J(m-1)$. $\breve{M}[J,m]$ resembles the stacking of $2J+1$ levels of the plane $\mathbb{R}^2$ in $\mathbb{R}^3$ that have been connected by $2Jm$ catenoidal bridges with $m$ bridges connecting each pair of adjacent levels. It observes the symmetry of an $m$-gonal prism (when $J$ is a half integer) or an $m$-gonal antiprism (when $J$ is an integer). The construction is based on the Linearised Doubling (LD) methodology which was first introduced by Kapouleas in the construction of minimal surface doublings of $\mathbb{S}^2_{eq}$ in $\mathbb{S}^3$. |
| title | Self-shrinkers with any number of ends in $\mathbb{R}^{3}$ by stacking $\mathbb{R}^{2}$ |
| topic | Differential Geometry 53A10 (Primary), 53E10 (Secondary) |
| url | https://arxiv.org/abs/2507.18825 |