Extremality and rigidity for scalar curvature in dimension four
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866913386972839936 |
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| author | Bettiol, Renato G. Goodman, McFeely Jackson |
| author_facet | Bettiol, Renato G. Goodman, McFeely Jackson |
| contents | Following Gromov, a Riemannian manifold is called area-extremal if any modification that increases scalar curvature must decrease the area of some tangent 2-plane. We prove that large classes of compact 4-manifolds, with or without boundary, with nonnegative sectional curvature are area-extremal. We also show that all regions of positive sectional curvature on 4-manifolds are locally area-extremal. These results are obtained analyzing sections in the kernel of a twisted Dirac operator constructed from pairs of metrics, and using the Finsler--Thorpe trick for sectional curvature bounds in dimension 4. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_00543 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Extremality and rigidity for scalar curvature in dimension four Bettiol, Renato G. Goodman, McFeely Jackson Differential Geometry 53C21, 53C23, 53C24, 53C27 Following Gromov, a Riemannian manifold is called area-extremal if any modification that increases scalar curvature must decrease the area of some tangent 2-plane. We prove that large classes of compact 4-manifolds, with or without boundary, with nonnegative sectional curvature are area-extremal. We also show that all regions of positive sectional curvature on 4-manifolds are locally area-extremal. These results are obtained analyzing sections in the kernel of a twisted Dirac operator constructed from pairs of metrics, and using the Finsler--Thorpe trick for sectional curvature bounds in dimension 4. |
| title | Extremality and rigidity for scalar curvature in dimension four |
| topic | Differential Geometry 53C21, 53C23, 53C24, 53C27 |
| url | https://arxiv.org/abs/2205.00543 |