Linear stability of the blowdown Ricci shrinker in 4D
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917097828777984 |
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| author | Naff, Keaton Ozuch, Tristan |
| author_facet | Naff, Keaton Ozuch, Tristan |
| contents | We prove that the four-dimensional blowdown shrinking Ricci soliton constructed by Feldman-Ilmanen-Knopf is strictly linearly stable in the sense of Cao-Hamilton-Ilmanen. This provides the first known example of a non-cylindrical linearly stable shrinking Ricci soliton. This offers new insights into the topological behavior of generic solutions to the Ricci flow in four dimensions: on top of reversing connected sums and handle surgeries, they should also undo complex blow-ups.
The proof starts from an explicit description of the metric and develops a tensor harmonic analysis, adapted to its weighted Lichnerowicz Laplacian and based on its $U(2)$-invariance. It further exploits the Kähler structure of the blowdown shrinking soliton and insights from four-dimensional selfduality. The main difficulty is that the weighted Lichnerowicz Laplacian of the soliton admits a $9$-dimensional set of eigentensors associated with nonnegative eigenvalues. We show that they correspond to the Ricci tensor and gauge transformations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_05470 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Linear stability of the blowdown Ricci shrinker in 4D Naff, Keaton Ozuch, Tristan Differential Geometry We prove that the four-dimensional blowdown shrinking Ricci soliton constructed by Feldman-Ilmanen-Knopf is strictly linearly stable in the sense of Cao-Hamilton-Ilmanen. This provides the first known example of a non-cylindrical linearly stable shrinking Ricci soliton. This offers new insights into the topological behavior of generic solutions to the Ricci flow in four dimensions: on top of reversing connected sums and handle surgeries, they should also undo complex blow-ups. The proof starts from an explicit description of the metric and develops a tensor harmonic analysis, adapted to its weighted Lichnerowicz Laplacian and based on its $U(2)$-invariance. It further exploits the Kähler structure of the blowdown shrinking soliton and insights from four-dimensional selfduality. The main difficulty is that the weighted Lichnerowicz Laplacian of the soliton admits a $9$-dimensional set of eigentensors associated with nonnegative eigenvalues. We show that they correspond to the Ricci tensor and gauge transformations. |
| title | Linear stability of the blowdown Ricci shrinker in 4D |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2509.05470 |