The First Eigenvalue of Embedded Minimal Hypersurfaces in the Unit Sphere I: Yau's Conjecture
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866911098383368192 |
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| author | Zeng, Lingzhong |
| author_facet | Zeng, Lingzhong |
| contents | In this paper, by meticulously constructing a minimizing sequence within a suitable Sobolev space and leveraging the variational principle, we establish that the first non-zero eigenvalue of the Laplace-Beltrami operator on an embedded minimal hypersurface in the unit sphere equals the dimension of the hypersurface. This result furnishes an affirmative resolution to a renowned conjecture posed by Yau, which had remained unresolved for an extended period. As some important applications, several rigidity theorems are established via eigenvalue characterization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_06123 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The First Eigenvalue of Embedded Minimal Hypersurfaces in the Unit Sphere I: Yau's Conjecture Zeng, Lingzhong Differential Geometry In this paper, by meticulously constructing a minimizing sequence within a suitable Sobolev space and leveraging the variational principle, we establish that the first non-zero eigenvalue of the Laplace-Beltrami operator on an embedded minimal hypersurface in the unit sphere equals the dimension of the hypersurface. This result furnishes an affirmative resolution to a renowned conjecture posed by Yau, which had remained unresolved for an extended period. As some important applications, several rigidity theorems are established via eigenvalue characterization. |
| title | The First Eigenvalue of Embedded Minimal Hypersurfaces in the Unit Sphere I: Yau's Conjecture |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2508.06123 |