Rigidity of spin fill-ins with non-negative scalar curvature
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910045384474624 |
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| author | Cecchini, Simone Hirsch, Sven Zeidler, Rudolf |
| author_facet | Cecchini, Simone Hirsch, Sven Zeidler, Rudolf |
| contents | We establish new mean curvature rigidity theorems for spin fill-ins with non-negative scalar curvature using two different spinorial techniques. Our results address two questions by Miao and Gromov, respectively. The first technique is based on extending boundary spinors satisfying a generalized eigenvalue equation via the Fredholm alternative for an APS boundary value problem, while the second is a comparison result in the spirit of Llarull and Lott using index theory. We also show that the latter implies a new Witten-type integral inequality for the mass of an asymptotically Schwarzschild manifold, which holds even when the scalar curvature is not assumed to be non-negative. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_17533 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rigidity of spin fill-ins with non-negative scalar curvature Cecchini, Simone Hirsch, Sven Zeidler, Rudolf Differential Geometry General Relativity and Quantum Cosmology 53C27, 53C24 (Primary), 53C21, 53C23, 58J20 (Secondary) We establish new mean curvature rigidity theorems for spin fill-ins with non-negative scalar curvature using two different spinorial techniques. Our results address two questions by Miao and Gromov, respectively. The first technique is based on extending boundary spinors satisfying a generalized eigenvalue equation via the Fredholm alternative for an APS boundary value problem, while the second is a comparison result in the spirit of Llarull and Lott using index theory. We also show that the latter implies a new Witten-type integral inequality for the mass of an asymptotically Schwarzschild manifold, which holds even when the scalar curvature is not assumed to be non-negative. |
| title | Rigidity of spin fill-ins with non-negative scalar curvature |
| topic | Differential Geometry General Relativity and Quantum Cosmology 53C27, 53C24 (Primary), 53C21, 53C23, 58J20 (Secondary) |
| url | https://arxiv.org/abs/2404.17533 |