Infinitesimal rigidity of collapsed gradient steady Ricci solitons in dimension three
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arXiv
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| Format: | Preprint |
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2014
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| _version_ | 1866911792847912960 |
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| author | Cao, Huai-Dong He, Chenxu |
| author_facet | Cao, Huai-Dong He, Chenxu |
| contents | The only known example of collapsed three-dimensional complete gradient steady Ricci solitons so far is the 3D cigar soliton $N^2\times \mathbb{R}$, the product of Hamilton's cigar soliton $N^2$ and the real line $\mathbb{R}$ with the product metric. R. Hamilton has conjectured that there should exist a family of collapsed positively curved three-dimensional complete gradient steady solitons, with $\mathsf{S}^1$-symmetry, connecting the 3D cigar soliton. In this paper, we make the first initial progress and prove that the infinitesimal deformation at the 3D cigar soliton is non-essential. In Appendix A, we show that the 3D cigar soliton is the unique complete nonflat gradient steady Ricci soliton in dimension three that admits two commuting Killing vector fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1412_2714 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Infinitesimal rigidity of collapsed gradient steady Ricci solitons in dimension three Cao, Huai-Dong He, Chenxu Differential Geometry 53C24, 53C10, 53C21, 53C25, 53C44 The only known example of collapsed three-dimensional complete gradient steady Ricci solitons so far is the 3D cigar soliton $N^2\times \mathbb{R}$, the product of Hamilton's cigar soliton $N^2$ and the real line $\mathbb{R}$ with the product metric. R. Hamilton has conjectured that there should exist a family of collapsed positively curved three-dimensional complete gradient steady solitons, with $\mathsf{S}^1$-symmetry, connecting the 3D cigar soliton. In this paper, we make the first initial progress and prove that the infinitesimal deformation at the 3D cigar soliton is non-essential. In Appendix A, we show that the 3D cigar soliton is the unique complete nonflat gradient steady Ricci soliton in dimension three that admits two commuting Killing vector fields. |
| title | Infinitesimal rigidity of collapsed gradient steady Ricci solitons in dimension three |
| topic | Differential Geometry 53C24, 53C10, 53C21, 53C25, 53C44 |
| url | https://arxiv.org/abs/1412.2714 |