Acta Mathematica

Volume 227 (2021)

Number 2

Gravitational instantons with faster than quadratic curvature decay. I

Pages: 263 – 307

DOI: https://dx.doi.org/10.4310/ACTA.2021.v227.n2.a2

Authors

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.)

Abstract

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.

Received 15 November 2018

Published 10 January 2022