Eguchi--Hanson metrics arising from Kahler--Einstein edge metrics
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866914692524408832 |
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| author | Ji, Yuxiang Rubinstein, Yanir A. Zhang, Kewei |
| author_facet | Ji, Yuxiang Rubinstein, Yanir A. Zhang, Kewei |
| contents | Calabi--Hirzebruch manifolds are higher-dimensional generalizations of both the football and Hirzebruch surfaces. We construct a family of Kahler--Einstein edge metrics singular along two disjoint divisors on the Calabi--Hirzebruch manifolds and study their Gromov--Hausdorff limits when either cone angle tends to its extreme value. As a very special case, we show that the celebrated Eguchi--Hanson metric arises in this way naturally as a Gromov--Hausdorff limit. We also completely describe all other (possibly rescaled) Gromov--Hausdorff limits which exhibit a wide range of behaviors, resolving in this setting a conjecture of Cheltsov--Rubinstein. This gives a new interpretation of both the Eguchi--Hanson space and Calabi's Ricci flat spaces as limits of compact singular Einstein spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_00652 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Eguchi--Hanson metrics arising from Kahler--Einstein edge metrics Ji, Yuxiang Rubinstein, Yanir A. Zhang, Kewei Differential Geometry Calabi--Hirzebruch manifolds are higher-dimensional generalizations of both the football and Hirzebruch surfaces. We construct a family of Kahler--Einstein edge metrics singular along two disjoint divisors on the Calabi--Hirzebruch manifolds and study their Gromov--Hausdorff limits when either cone angle tends to its extreme value. As a very special case, we show that the celebrated Eguchi--Hanson metric arises in this way naturally as a Gromov--Hausdorff limit. We also completely describe all other (possibly rescaled) Gromov--Hausdorff limits which exhibit a wide range of behaviors, resolving in this setting a conjecture of Cheltsov--Rubinstein. This gives a new interpretation of both the Eguchi--Hanson space and Calabi's Ricci flat spaces as limits of compact singular Einstein spaces. |
| title | Eguchi--Hanson metrics arising from Kahler--Einstein edge metrics |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2111.00652 |