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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.00595 |
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| _version_ | 1866910576074031104 |
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| author | Mazurowski, Liam Zhou, Xin |
| author_facet | Mazurowski, Liam Zhou, Xin |
| contents | Let $(M^{n+1},g)$ be a closed Riemannian manifold of dimension $3\le n+1\le 5$. We show that, if the metric $g$ is generic or if the metric $g$ has positive Ricci curvature, then $M$ contains infinitely many geometrically distinct constant mean curvature hypersurfaces, each enclosing half the volume of $M$. As an essential part of the proof, we develop an Almgren-Pitts type min-max theory for certain non-local functionals of the general form $$Ω\mapsto \operatorname{Area}(\partial Ω) - \int_Ωh + f(\operatorname{Vol}(Ω)).$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_00595 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Infinitely Many Half-Volume Constant Mean Curvature Hypersurfaces via Min-Max Theory Mazurowski, Liam Zhou, Xin Differential Geometry 53A10 Let $(M^{n+1},g)$ be a closed Riemannian manifold of dimension $3\le n+1\le 5$. We show that, if the metric $g$ is generic or if the metric $g$ has positive Ricci curvature, then $M$ contains infinitely many geometrically distinct constant mean curvature hypersurfaces, each enclosing half the volume of $M$. As an essential part of the proof, we develop an Almgren-Pitts type min-max theory for certain non-local functionals of the general form $$Ω\mapsto \operatorname{Area}(\partial Ω) - \int_Ωh + f(\operatorname{Vol}(Ω)).$$ |
| title | Infinitely Many Half-Volume Constant Mean Curvature Hypersurfaces via Min-Max Theory |
| topic | Differential Geometry 53A10 |
| url | https://arxiv.org/abs/2405.00595 |