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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.16635 |
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| _version_ | 1866918345723346944 |
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| author | Gaia, Filippo Li, Xuanyu |
| author_facet | Gaia, Filippo Li, Xuanyu |
| contents | We establish the existence of a non-trivial, branched immersion of a closed Riemann surface $Σ$ with constant mean curvature (CMC) $H$ into any closed, orientable 3-manifold $\mathcal{M}$, for almost every prescribed value of $H$. The genus of the surface $Σ$ is bounded from above by the Heegaard genus $h$ of $\mathcal{M}$.
Starting from a family of sweep-outs of $\mathcal{M}$ by surfaces of genus $h$, we apply a min-max construction for a family $\{E_{H,σ}\}_σ$ of perturbations of the energy involving the second fundamental form of the immersions to produce almost-critical points $u_k$ of $E_{H,σ}$. We then show, following ideas developed by Pigati and Rivière, that the maps $u_k$ converge to a "CMC-parametrized varifold". This limiting object is then shown to be a smooth, branched immersion with the prescribed mean curvature $H$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_16635 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Existence of constant mean curvature surfaces with controlled topology in 3-manifolds Gaia, Filippo Li, Xuanyu Differential Geometry 53A10, 58E12 We establish the existence of a non-trivial, branched immersion of a closed Riemann surface $Σ$ with constant mean curvature (CMC) $H$ into any closed, orientable 3-manifold $\mathcal{M}$, for almost every prescribed value of $H$. The genus of the surface $Σ$ is bounded from above by the Heegaard genus $h$ of $\mathcal{M}$. Starting from a family of sweep-outs of $\mathcal{M}$ by surfaces of genus $h$, we apply a min-max construction for a family $\{E_{H,σ}\}_σ$ of perturbations of the energy involving the second fundamental form of the immersions to produce almost-critical points $u_k$ of $E_{H,σ}$. We then show, following ideas developed by Pigati and Rivière, that the maps $u_k$ converge to a "CMC-parametrized varifold". This limiting object is then shown to be a smooth, branched immersion with the prescribed mean curvature $H$. |
| title | Existence of constant mean curvature surfaces with controlled topology in 3-manifolds |
| topic | Differential Geometry 53A10, 58E12 |
| url | https://arxiv.org/abs/2602.16635 |