Rigidity theorems for the area widths of Riemannian manifolds
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929474863366144 |
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| author | Ambrozio, Lucas Marques, Fernando C. Neves, André |
| author_facet | Ambrozio, Lucas Marques, Fernando C. Neves, André |
| contents | The volume spectrum of a compact Riemannian manifold is a sequence of critical values for the area functional, defined in analogy with the Laplace spectrum by Gromov. In this paper we prove that the canonical metric on the two-dimensional projective plane is determined modulo isometries by its volume spectrum. We also prove that the surface Zoll metrics on the three-dimensional sphere are characterized by the equality of the spherical area widths. These widths generalize to the surface case the Lusternik-Schnirelmann lengths of closed geodesics. We prove a new sharp area systolic inequality for metrics on the three-dimensional projective space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_14375 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rigidity theorems for the area widths of Riemannian manifolds Ambrozio, Lucas Marques, Fernando C. Neves, André Differential Geometry The volume spectrum of a compact Riemannian manifold is a sequence of critical values for the area functional, defined in analogy with the Laplace spectrum by Gromov. In this paper we prove that the canonical metric on the two-dimensional projective plane is determined modulo isometries by its volume spectrum. We also prove that the surface Zoll metrics on the three-dimensional sphere are characterized by the equality of the spherical area widths. These widths generalize to the surface case the Lusternik-Schnirelmann lengths of closed geodesics. We prove a new sharp area systolic inequality for metrics on the three-dimensional projective space. |
| title | Rigidity theorems for the area widths of Riemannian manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2408.14375 |