Classification results for conformally Kähler gravitational instantons
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866917804768231424 |
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| author | Li, Mingyang |
| author_facet | Li, Mingyang |
| contents | We investigate the asymptotic geometry of Hermitian non-Kähler Ricci-flat metrics with finite $\int|Rm|^2$ at infinity. Specifically, we prove:
1. Any such metric is asymptotic to an ALE, ALF-A, AF, skewed special Kasner, ALH* model at infinity.
2. Any Hermitian non-Kähler gravitational instanton with non-Euclidean volume growth is one of the following: the Kerr family, the Chen-Teo family, the Taub-bolt space, the reversed Taub-NUT space. This particularly confirms a conjecture by Aksteiner-Andersson. It includes the well-known Kerr family from general relativity.
3. All Hermitian non-Kähler gravitational instantons can be compactified to log del Pezzo surfaces. This explains a curious relation to compact Hermitian non-Kähler Einstein 4-manifolds.
For a 4-dimensional Ricci-flat metric, being Hermitian non-Kähler is equivalent to being non-trivially conformally Kähler. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_13197 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Classification results for conformally Kähler gravitational instantons Li, Mingyang Differential Geometry Mathematical Physics We investigate the asymptotic geometry of Hermitian non-Kähler Ricci-flat metrics with finite $\int|Rm|^2$ at infinity. Specifically, we prove: 1. Any such metric is asymptotic to an ALE, ALF-A, AF, skewed special Kasner, ALH* model at infinity. 2. Any Hermitian non-Kähler gravitational instanton with non-Euclidean volume growth is one of the following: the Kerr family, the Chen-Teo family, the Taub-bolt space, the reversed Taub-NUT space. This particularly confirms a conjecture by Aksteiner-Andersson. It includes the well-known Kerr family from general relativity. 3. All Hermitian non-Kähler gravitational instantons can be compactified to log del Pezzo surfaces. This explains a curious relation to compact Hermitian non-Kähler Einstein 4-manifolds. For a 4-dimensional Ricci-flat metric, being Hermitian non-Kähler is equivalent to being non-trivially conformally Kähler. |
| title | Classification results for conformally Kähler gravitational instantons |
| topic | Differential Geometry Mathematical Physics |
| url | https://arxiv.org/abs/2310.13197 |