Einstein metrics from the Calabi ansatz via Derdziński duality
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866909161490481152 |
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| author | Oliveira, Gonçalo Sena-Dias, Rosa |
| author_facet | Oliveira, Gonçalo Sena-Dias, Rosa |
| contents | Drawing on results of Derdziński's from the 80's, we classify conformally Kähler, $U(2)$-invariant, Einstein metrics on the total space of $\mathcal{O}(-m)$, for all $m \in \mathbb{N}$. This yields infinitely many $1$-parameter families of metrics exhibiting several different behaviours including asymptotically hyperbolic metrics (more specifically of Poincaré type), ALF metrics, and metrics which compactify to a Hirzebruch surface $\mathbb{H}_m$ with a cone singularity along the "divisor at infinity". This allows us to investigate transitions between behaviours yielding interesting results. For instance, we show that a Ricci--flat ALF metric known as the Taub-bolt metric can be obtained as the limit of a family of cone angle Einstein metrics on $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$ when the cone angle converges to zero.
We also construct Einstein metrics which are asymptotically hyperbolic and conformal to a scalar-flat Kähler metric. Such metrics cannot be obtained by applying Derdziński's theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_17328 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Einstein metrics from the Calabi ansatz via Derdziński duality Oliveira, Gonçalo Sena-Dias, Rosa Differential Geometry 53C25, 53C55 Drawing on results of Derdziński's from the 80's, we classify conformally Kähler, $U(2)$-invariant, Einstein metrics on the total space of $\mathcal{O}(-m)$, for all $m \in \mathbb{N}$. This yields infinitely many $1$-parameter families of metrics exhibiting several different behaviours including asymptotically hyperbolic metrics (more specifically of Poincaré type), ALF metrics, and metrics which compactify to a Hirzebruch surface $\mathbb{H}_m$ with a cone singularity along the "divisor at infinity". This allows us to investigate transitions between behaviours yielding interesting results. For instance, we show that a Ricci--flat ALF metric known as the Taub-bolt metric can be obtained as the limit of a family of cone angle Einstein metrics on $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$ when the cone angle converges to zero. We also construct Einstein metrics which are asymptotically hyperbolic and conformal to a scalar-flat Kähler metric. Such metrics cannot be obtained by applying Derdziński's theorem. |
| title | Einstein metrics from the Calabi ansatz via Derdziński duality |
| topic | Differential Geometry 53C25, 53C55 |
| url | https://arxiv.org/abs/2306.17328 |