Mountain Pass Critical Points of the Volume Constrained Area Functional
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909872645210112 |
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| author | Chambers, Gregory R. Marx-Kuo, Jared |
| author_facet | Chambers, Gregory R. Marx-Kuo, Jared |
| contents | We construct mountain pass critical points of the perimeter functional on sets of fixed volume. For a generic metric, this gives rise to a smooth almost embedded hypersurface with non-zero constant mean curvature. Our work utilizes recent techniques of Mazurwoski--Zhou \cite{mazurowski2024infinitely}, and a new result on the connectedness of Cacciopoli sets: any two smooth Cacciopoli sets can be connected by an $\mathbf{F}$-continuous map. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_09098 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mountain Pass Critical Points of the Volume Constrained Area Functional Chambers, Gregory R. Marx-Kuo, Jared Analysis of PDEs Mathematical Physics Differential Geometry We construct mountain pass critical points of the perimeter functional on sets of fixed volume. For a generic metric, this gives rise to a smooth almost embedded hypersurface with non-zero constant mean curvature. Our work utilizes recent techniques of Mazurwoski--Zhou \cite{mazurowski2024infinitely}, and a new result on the connectedness of Cacciopoli sets: any two smooth Cacciopoli sets can be connected by an $\mathbf{F}$-continuous map. |
| title | Mountain Pass Critical Points of the Volume Constrained Area Functional |
| topic | Analysis of PDEs Mathematical Physics Differential Geometry |
| url | https://arxiv.org/abs/2509.09098 |