Cohomogeneity one 4-dimensional gradient Ricci solitons
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908275020136448 |
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| author | Donovan, Patrick |
| author_facet | Donovan, Patrick |
| contents | Simply-connected four-dimensional gradient Ricci solitons that are invariant under a compact cohomogeneity one group action have been studied extensively. However, the special case where the group is $SU(2)$ (the smallest possible example) has received comparatively little attention. The purpose of this article is to give a comprehensive study of simply-connected $SU(2)$-invariant expanding and shrinking cohomogeneity one gradient Ricci solitons. The first result is the construction of new 3-parameter families of complete $SU(2)$-invariant asymptotically conical expanding gradient Ricci solitons. New shrinking Kähler $U(2)$-invariant gradient Ricci solitons in dimension 4 with orbifold singularities are also constructed, leading to a classification of such metrics when the base space of the orbifold is a simply-connected smooth manifold. Finally, we highlight numerical evidence that all the compact cohomogeneity one shrinking gradient Ricci solitons are known. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_15033 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cohomogeneity one 4-dimensional gradient Ricci solitons Donovan, Patrick Differential Geometry 53E20 Simply-connected four-dimensional gradient Ricci solitons that are invariant under a compact cohomogeneity one group action have been studied extensively. However, the special case where the group is $SU(2)$ (the smallest possible example) has received comparatively little attention. The purpose of this article is to give a comprehensive study of simply-connected $SU(2)$-invariant expanding and shrinking cohomogeneity one gradient Ricci solitons. The first result is the construction of new 3-parameter families of complete $SU(2)$-invariant asymptotically conical expanding gradient Ricci solitons. New shrinking Kähler $U(2)$-invariant gradient Ricci solitons in dimension 4 with orbifold singularities are also constructed, leading to a classification of such metrics when the base space of the orbifold is a simply-connected smooth manifold. Finally, we highlight numerical evidence that all the compact cohomogeneity one shrinking gradient Ricci solitons are known. |
| title | Cohomogeneity one 4-dimensional gradient Ricci solitons |
| topic | Differential Geometry 53E20 |
| url | https://arxiv.org/abs/2503.15033 |