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Gravitational instantons with faster than quadratic curvature decay. I

Gao Chen Institute of Geometry and Physics, University of Science and Technology of China, Hefei, China Xiuxiong Chen Institute of Geometry and Physics, University of Science and Technology of China, Hefei, China; and Department of Mathematics, Stony Brook University, Stony Brook, New York, U.S.A.

TBD mathscidoc:2203.43010

Acta Mathematica, 227, (2), 263-307, 2021.12
In this paper, we study gravitational instantons (i.e., complete hyperkähler 4‑manifolds with faster than quadratic curvature decay). We prove three main theorems: (1) Any gravitational instanton must have one of the following known ends: ALE, ALF, ALG, and ALH. (2) In the ALG and ALH non-splitting cases, it must be biholomorphic to a compact complex elliptic surface minus a divisor. Thus, we confirm a long-standing question of Yau in the ALG and ALH cases. (3) In the ALF‑D_k case, it must have an O(4)‑multiplet.
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@inproceedings{gao2021gravitational,
  title={Gravitational instantons with faster than quadratic curvature decay. I},
  author={Gao Chen, and Xiuxiong Chen},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220310114513134302927},
  booktitle={Acta Mathematica},
  volume={227},
  number={2},
  pages={263-307},
  year={2021},
}
Gao Chen, and Xiuxiong Chen. Gravitational instantons with faster than quadratic curvature decay. I. 2021. Vol. 227. In Acta Mathematica. pp.263-307. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220310114513134302927.
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