New Rigidity Results for Critical Metrics of Some Quadratic Curvature Functionals
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916143708504064 |
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| author | Bernardini, Marco |
| author_facet | Bernardini, Marco |
| contents | We prove a new rigidity result for metrics defined on closed smooth $ n $-manifolds that are critical for the quadratic functional $ \mathfrak{F}_{t} $, which depends on the Ricci curvature $ Ric $ and the scalar curvature $ R $, and that satisfy a pinching condition of the form $ Sec > εR $, where $ ε$ is a function of $ t $ and $ n $, while $ Sec $ denotes the sectional curvature. In particular, we show that Bach-flat metrics with constant scalar curvature satisfying $ Sec > \frac{1}{48} R $ are Einstein and, by a known result, are isometric to $ \mathbb{S}^{4} $, $ \mathbb{RP}^{4} $ or $ \mathbb{CP}^{2} $. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_00388 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | New Rigidity Results for Critical Metrics of Some Quadratic Curvature Functionals Bernardini, Marco Differential Geometry 53C24 (Primary), 53C25 (Secondary) We prove a new rigidity result for metrics defined on closed smooth $ n $-manifolds that are critical for the quadratic functional $ \mathfrak{F}_{t} $, which depends on the Ricci curvature $ Ric $ and the scalar curvature $ R $, and that satisfy a pinching condition of the form $ Sec > εR $, where $ ε$ is a function of $ t $ and $ n $, while $ Sec $ denotes the sectional curvature. In particular, we show that Bach-flat metrics with constant scalar curvature satisfying $ Sec > \frac{1}{48} R $ are Einstein and, by a known result, are isometric to $ \mathbb{S}^{4} $, $ \mathbb{RP}^{4} $ or $ \mathbb{CP}^{2} $. |
| title | New Rigidity Results for Critical Metrics of Some Quadratic Curvature Functionals |
| topic | Differential Geometry 53C24 (Primary), 53C25 (Secondary) |
| url | https://arxiv.org/abs/2403.00388 |