A family of Kähler flying wing steady Ricci solitons
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911884417957888 |
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| author | Chan, Pak-Yeung Conlon, Ronan J. Lai, Yi |
| author_facet | Chan, Pak-Yeung Conlon, Ronan J. Lai, Yi |
| contents | In $1996$, H.-D. Cao constructed a $U(n)$-invariant steady gradient Kähler-Ricci soliton on $\mathbb{C}^{n}$ and asked whether every steady gradient Kähler-Ricci soliton of positive curvature on $\mathbb{C}^{n}$ is necessarily $U(n)$-invariant (and hence unique up to scaling). Recently, Apostolov-Cifarelli answered this question in the negative for $n=2$. Here, we construct a family of $U(1)\times U(n-1)$-invariant, but not $U(n)$-invariant, complete steady gradient Kähler-Ricci solitons with strictly positive curvature operator on real $(1,\,1)$-forms (in particular, with strictly positive sectional curvature) on $\mathbb{C}^{n}$ for $n\geq3$, thereby answering Cao's question in the negative for $n\geq3$. This family of steady Ricci solitons interpolates between Cao's $U(n)$-invariant steady Kähler-Ricci soliton and the product of the cigar soliton and Cao's $U(n-1)$-invariant steady Kähler-Ricci soliton. This provides the Kähler analog of the Riemannian flying wings construction of Lai. In the process of the proof, we also demonstrate that the almost diameter rigidity of $\mathbb{P}^{n}$ endowed with the Fubini-Study metric does not hold even if the curvature operator is bounded below by $2$ on real $(1,\,1)$-forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_04089 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A family of Kähler flying wing steady Ricci solitons Chan, Pak-Yeung Conlon, Ronan J. Lai, Yi Differential Geometry 53E20, 53E30 In $1996$, H.-D. Cao constructed a $U(n)$-invariant steady gradient Kähler-Ricci soliton on $\mathbb{C}^{n}$ and asked whether every steady gradient Kähler-Ricci soliton of positive curvature on $\mathbb{C}^{n}$ is necessarily $U(n)$-invariant (and hence unique up to scaling). Recently, Apostolov-Cifarelli answered this question in the negative for $n=2$. Here, we construct a family of $U(1)\times U(n-1)$-invariant, but not $U(n)$-invariant, complete steady gradient Kähler-Ricci solitons with strictly positive curvature operator on real $(1,\,1)$-forms (in particular, with strictly positive sectional curvature) on $\mathbb{C}^{n}$ for $n\geq3$, thereby answering Cao's question in the negative for $n\geq3$. This family of steady Ricci solitons interpolates between Cao's $U(n)$-invariant steady Kähler-Ricci soliton and the product of the cigar soliton and Cao's $U(n-1)$-invariant steady Kähler-Ricci soliton. This provides the Kähler analog of the Riemannian flying wings construction of Lai. In the process of the proof, we also demonstrate that the almost diameter rigidity of $\mathbb{P}^{n}$ endowed with the Fubini-Study metric does not hold even if the curvature operator is bounded below by $2$ on real $(1,\,1)$-forms. |
| title | A family of Kähler flying wing steady Ricci solitons |
| topic | Differential Geometry 53E20, 53E30 |
| url | https://arxiv.org/abs/2403.04089 |