Quotients of gravitational instantons (Q658359)

The author obtains a classification result concerning the Ricci-flat anti-self-dual asymptotically locally Euclidean 4-manifolds. These manifolds are either hyper-Kähler or they are cyclic quotients of a Gibbons-Hawking space. Theorem A. Every Ricci-flat anti-self-dual ALE (asymptotically locally Euclidean) 4-manifold \(X\) which is not simply connected and nonflat is a finite isometric quotient of a Gibbons-Hawking space \(\widetilde X\), and is actually Kähler. Moreover, if the monopole set \(F\subset \mathbb{R}^3\) of \(\widetilde X\) is normalized to have Euclidean centre of mass at the origin, then the isometric quotients of \(\widetilde X\) are in one-to-one correspondence with the cyclic subgroup of \(SO(3)\) which preserves \(F\) and acts freely on it. At the end, there are two corollaries in which \(\int _X|Rm|^2d\mu\) is calculated and another result is also obtained concerning the structure of the above manifolds in the case \(b_2(X)=0\).