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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.21997 |
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| _version_ | 1866914114416148480 |
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| author | Kim, Dain Ozuch, Tristan |
| author_facet | Kim, Dain Ozuch, Tristan |
| contents | We prove that on ALF $n$-manifolds with $n\ge 4$ the Ricci flow preserves the ALF structure, and develop a weighted Fredholm framework adapted to ALF manifolds. Motivated by Perelman's $λ$-functional, we define a renormalized functional $λ_{\mathrm{ALF}}$ whose gradient flow is the Ricci flow. It is built from a relative mass with respect to a reference Ricci-flat metric at infinity. This yields a natural notion of variational and linear stability for Ricci-flat ALF $4$-metrics and lets us show that the conformally Kähler, non-hyperkähler examples are dynamically unstable along Ricci flow. We finally relate the sign of $λ_{\mathrm{ALF}}$ to positive relative mass statements for ALF metrics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_21997 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ricci Flow on ALF manifolds Kim, Dain Ozuch, Tristan Differential Geometry We prove that on ALF $n$-manifolds with $n\ge 4$ the Ricci flow preserves the ALF structure, and develop a weighted Fredholm framework adapted to ALF manifolds. Motivated by Perelman's $λ$-functional, we define a renormalized functional $λ_{\mathrm{ALF}}$ whose gradient flow is the Ricci flow. It is built from a relative mass with respect to a reference Ricci-flat metric at infinity. This yields a natural notion of variational and linear stability for Ricci-flat ALF $4$-metrics and lets us show that the conformally Kähler, non-hyperkähler examples are dynamically unstable along Ricci flow. We finally relate the sign of $λ_{\mathrm{ALF}}$ to positive relative mass statements for ALF metrics. |
| title | Ricci Flow on ALF manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2510.21997 |