Eguchi--Hanson metrics arising from Kahler--Einstein edge metrics

Yuxiang Ji; Yanir A. Rubinstein; Kewei Zhang

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arXiv · 2111.00652

Eguchi--Hanson metrics arising from Kahler--Einstein edge metrics

Yuxiang Ji·Yanir A. Rubinstein·Kewei Zhang

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Abstract

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.

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BibTeX RIS

Yuxiang Ji, Yanir A. Rubinstein, Kewei Zhang. 2021-11-01. Eguchi--Hanson metrics arising from Kahler--Einstein edge metrics. https://arxiv.org/abs/2111.00652

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