Minimal surfaces with low genus in lens spaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929401155813376 |
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| author | Li, Xingzhe Wang, Tongrui Yao, Xuan |
| author_facet | Li, Xingzhe Wang, Tongrui Yao, Xuan |
| contents | Given a Riemannian $\mathbb{RP}^3$ with a bumpy metric or a metric of positive Ricci curvature, we show that there either exist four distinct minimal real projective planes, or exist one minimal real projective plane together with two distinct minimal $2$-spheres. Our proof is based on a variant multiplicity one theorem for the Simon-Smith min-max theory under certain equivariant settings. In particular, we show under the positive Ricci assumption that $\mathbb{RP}^3$ contains at least four distinct minimal real projective planes and four distinct minimal tori. Additionally, the number of minimal tori can be improved to five for a generic positive Ricci metric on $\mathbb{RP}^3$ by the degree method. Moreover, using the same strategy, we show that in the lens space $L(4m,2m\pm 1)$, $m\geq 1$, with a bumpy metric or a metric of positive Ricci curvature, there either exist $N(m)$ numbers of distinct minimal Klein bottles, or exist one minimal Klein bottle and three distinct minimal $2$-spheres, where $N(1)=4$, $N(m)=2$ for $m\geq 2$, and the first case happens under the positive Ricci assumption. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_12584 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minimal surfaces with low genus in lens spaces Li, Xingzhe Wang, Tongrui Yao, Xuan Differential Geometry Geometric Topology 53A10, 53C42 Given a Riemannian $\mathbb{RP}^3$ with a bumpy metric or a metric of positive Ricci curvature, we show that there either exist four distinct minimal real projective planes, or exist one minimal real projective plane together with two distinct minimal $2$-spheres. Our proof is based on a variant multiplicity one theorem for the Simon-Smith min-max theory under certain equivariant settings. In particular, we show under the positive Ricci assumption that $\mathbb{RP}^3$ contains at least four distinct minimal real projective planes and four distinct minimal tori. Additionally, the number of minimal tori can be improved to five for a generic positive Ricci metric on $\mathbb{RP}^3$ by the degree method. Moreover, using the same strategy, we show that in the lens space $L(4m,2m\pm 1)$, $m\geq 1$, with a bumpy metric or a metric of positive Ricci curvature, there either exist $N(m)$ numbers of distinct minimal Klein bottles, or exist one minimal Klein bottle and three distinct minimal $2$-spheres, where $N(1)=4$, $N(m)=2$ for $m\geq 2$, and the first case happens under the positive Ricci assumption. |
| title | Minimal surfaces with low genus in lens spaces |
| topic | Differential Geometry Geometric Topology 53A10, 53C42 |
| url | https://arxiv.org/abs/2406.12584 |