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On 4-dimensional Ricci-flat ALE manifolds

Name: On 4-dimensional Ricci-flat ALE manifolds
Creator: Li, Mingyang
Keywords: Mathematics - Differential Geometry, Differential Geometry (math.DG), FOS: Mathematics

descriptionPublicationkeyboard_double_arrow_right Article , Preprint 01 Jan 2023Embargo end date: 01 Jan 2023Publisher:arXivFunded by:FCT | D4, NSF | Singularity Formation in ...

Authors: Li, Mingyang;

doi: 10.48550/arxiv.2304.01609

arXiv: http://arxiv.org/abs/2304.01609

On 4-dimensional Ricci-flat ALE manifolds

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Abstract

In this paper, we prove: 1. There is a one-to-one correspondence between: Hermitian non-Kähler ALE gravitational instantons $(M,h)$, and Bach-flat Kähler orbifolds $(\widehat{M},\widehat{g})$ of complex dimension 2 with exactly one orbifold point $q$, such that the scalar curvature $s_{\widehat{g}}$ satisfies $s_{\widehat{g}}(q)=0$ while being positive elsewhere. 2. There is no Hermitian non-Kähler ALE gravitational instanton $(M,h)$ with structure group contained in $SU(2)$, except for the Eguchi-Hanson metric with reversed orientation. A 4-dimensional Ricci-flat metric being Hermitian non-Kähler is equivalent to being non-trivially conformally Kähler.

Several computation mistakes corrected

Related Organizations

University of California, Berkeley
United States

Keywords

Mathematics - Differential Geometry, Differential Geometry (math.DG), FOS: Mathematics

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Funded by

FCT| D4, NSF| Singularity Formation in Kahler Geometry and Yang-Mills Instantons