Critical metrics for the quadratic curvature functional on complete four-dimensional manifolds
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912780508987392 |
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| author | Euh, Yunhee Park, JeongHyeong |
| author_facet | Euh, Yunhee Park, JeongHyeong |
| contents | We study critical metrics of the curvature functional $\A(g)=\int_M |R|^2\, \vol$, on complete four-dimensional Riemannian manifolds $(M,g)$ with finite energy, that is, $\A(g)<\infty$. Under the natural inequality condition on the curvature operator of the second kind associated with the trace-free Ricci tensor, we prove that $(M,g)$ is either Einstein or locally isometric to a Riemannian product of two-dimensional manifolds of constant Gaussian curvatures $c$ and $-c$ $(c\ne 0)$. This extends the compact classification of four-dimensional $\mathcal{A}$-critical metrics obtained in earlier work to the complete setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_18758 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Critical metrics for the quadratic curvature functional on complete four-dimensional manifolds Euh, Yunhee Park, JeongHyeong Differential Geometry 53C21, 53C24, 53C25 We study critical metrics of the curvature functional $\A(g)=\int_M |R|^2\, \vol$, on complete four-dimensional Riemannian manifolds $(M,g)$ with finite energy, that is, $\A(g)<\infty$. Under the natural inequality condition on the curvature operator of the second kind associated with the trace-free Ricci tensor, we prove that $(M,g)$ is either Einstein or locally isometric to a Riemannian product of two-dimensional manifolds of constant Gaussian curvatures $c$ and $-c$ $(c\ne 0)$. This extends the compact classification of four-dimensional $\mathcal{A}$-critical metrics obtained in earlier work to the complete setting. |
| title | Critical metrics for the quadratic curvature functional on complete four-dimensional manifolds |
| topic | Differential Geometry 53C21, 53C24, 53C25 |
| url | https://arxiv.org/abs/2512.18758 |