The Gibbons–Hawking ansatz describes $4$–dimensional hyperkähler metrics with an isometric $S^{1}$–action that also preserves the whole hyperkähler structure. Such an $S^{1}$ action is therefore called triholomorphic.

Let $U$ be an open set of $\mathbb{R}^{3}$ and $\pi\colon\thinspace P\rightarrow U$ be a principal $U(1)$–bundle. Suppose that there exists a positive harmonic function $h$ on $U$ such that $\ast dh$ is the curvature $d\theta$ of a connection $\theta$ on $P$. Then

(3.3a)		$$g^{\textup{gh}}=h\,\pi^{\ast}g_{\mathbb{R}^{3}}+h^{-1}\theta^{2}$$
is a hyperkähler metric. Indeed, we can exhibit an explicit hyperkähler triple $\bm{\underline{\omega}}^{\textup{gh}}$ that induces the metric $g^{\textup{gh}}$. Fix coordinates $(x_{1},x_{2},x_{3})$ on $U\subset\mathbb{R}^{3}$ and define
(3.3b)		$$\omega^{\textup{gh}}_{i}=dx_{i}\wedge\theta+h\,dx_{j}\wedge dx_{k}.$$

Here and in the rest of the paper we use the convention that for every $i=1,2,3$ the indices $j,k$ are chosen so that $\epsilon_{ijk}=1$. One can check explicitly that $\bm{\underline{\omega}}^{gh}$ defines an $SU(2)$–structure and it induces the Riemannian metric $g^{gh}$. Moreover, the requirement that $\bm{\underline{\omega}}^{gh}$ is also closed is equivalent to the abelian monopole equation

(3.4)

$$\ast dh=d\theta$$

The fibre-wise circle action on $P$ preserves $\bm{\underline{\omega}}^{\textup{gh}}$ and $\pi$ is nothing but a hyperkähler moment map for this action. Conversely, every $4$–dimensional hyperkähler metric with a triholomorphic circle action is described by (3.3).

The basic example of the Gibbons–Hawking construction is given in terms of so-called Dirac monopoles on $\mathbb{R}^{3}$. Fix a set of distinct points $p_{1},\dots,p_{n}$ in $\mathbb{R}^{3}$ and consider the harmonic function

$$h=\lambda+\sum_{j=1}^{n}{\frac{k_{j}}{2|x-p_{j}|}},$$

where $\lambda>0$ and $k_{1},\dots,k_{n}$ are constants. Since $\mathbb{R}^{3}\setminus\{p_{1},\dots,p_{n}\}$ has non-trivial second homology, we must require $k_{j}\in\mathbb{Z}$ for all $j$ in order to be able to solve (3.4). If these integrality constraints are satisfied then $\ast dh$ defines the curvature $d\theta$ of a connection $\theta$ (unique up to gauge transformations) on a principal $U(1)$–bundle $P$ over $\mathbb{R}^{3}\setminus\{p_{1},\dots,p_{n}\}$ which restricts to the principal $U(1)$–bundle associated with the line bundle $\mathcal{O}(k_{j})\rightarrow S^{2}$ on a small punctured neighbourhood of $p_{j}$. The pair $(h,\theta)$ is a solution of (3.4) which we call a Dirac monopole with singularities at $p_{1},\dots,p_{n}$.

The Gibbons–Hawking ansatz (3.3) associates a hyperkähler metric $g^{\textup{gh}}$ to every Dirac monopole on the open set where $h>0$. When $k_{j}>0$ then $g^{\textup{gh}}$ is certainly defined on the restriction of $P$ to a small punctured neighbourhood of $p_{j}$. By a change of variables one can check that $g^{gh}$ can be extended to a smooth (orbifold) metric modelled on $\mathbb{C}^{2}/\mathbb{Z}_{k_{j}}$ by adding a single point. In particular $g^{\textup{gh}}$ is a complete metric whenever $\lambda\geq 0$ and $k_{j}=1$ for all $j=1,\dots,n$. One can check that $g^{\textup{gh}}$ is an ALE metric when $\lambda=0$ and an ALF metric of cyclic type when $\lambda>0$. Note also that when $\lambda>0$ we can always rescale the metric so that $\lambda=1$.

We are now going to use the Gibbons–Hawking ansatz to define a refined asymptotic model for ALF gravitational instantons.

Let $H^{k}$ be the total space of the principal $U(1)$–bundle associated with the line bundle $\mathcal{O}(k)$ over $S^{2}$ radially extended to $\mathbb{R}^{3}\setminus B_{R}$ for any $R>0$. $\theta_{k}$ will denote the $SO(3)$–invariant connection on $H^{k}$. The Gibbons–Hawking ansatz (3.3) yields a hyperkähler metric

(3.5)

$$g_{k}=\left(1+\frac{k}{2\rho}\right)(d\rho^{2}+\rho^{2}g_{\mathbb{S}^{2}})+\left(1+\frac{k}{2\rho}\right)^{-1}\theta_{k}^{2}$$

on $H^{k}$ for all $k\in\mathbb{Z}$. Here $\rho$ is a radial function on $\mathbb{R}^{3}$. We denote by $\bm{\underline{\omega}}_{k}$ the associated hyperkähler triple defined by (3.3b). Note that we could replace the harmonic function $1+\frac{k}{2\rho}$ with $\lambda+\frac{k}{2\rho}$ for any $\lambda>0$ but we can always reduce to the case $\lambda=1$ by scaling.

Finally, on $H^{2k}$ we consider the $\mathbb{Z}_{2}$–action which is defined as the simultaneous standard involutions on the base $\mathbb{R}^{3}$ and the fibre. Here the involution on the fibre $S^{1}=\mathbb{R}/2\pi\mathbb{Z}$ is the one induced by the standard involution on the universal cover $\mathbb{R}$. Throughout the paper we refer to this as the standard involution of $S^{1}$.

By [11, Theorem 1.1] every ALF gravitational instanton is either of type $A_{k}$ for some $k\geq-1$ or $D_{m}$ for some $m\geq 0$ (the constraints $k\geq-1$ and $m\geq 0$ follow from [33, Theorem 0.1] and [8, Corollary 3.2], respectively).