We saw that gravitational instantons of type $A_{k}$ can be constructed from Dirac monopoles on $\mathbb{R}^{3}$ with $k+1$ singularities via the Gibbons–Hawking ansatz. These are usually called multi-Taub–NUT metrics. The case $k=0$ is the Taub–NUT metric on $\mathbb{R}^{4}$ and $k=-1$ is $\mathbb{R}^{3}\times\mathbb{S}^{1}$ with its flat metric. Minerbe [35, Theorem 0.2] has shown that every ALF space of cyclic type must be isometric to a multi-Taub–NUT metric.

From their explicit description one can easily compute basic information about cyclic ALF spaces: the fundamental group $\pi_{1}(M)$, the second Betti number $b_{2}(M)$, the Euler characteristic and the dimension of the moduli space $\mathcal{M}$ of $A_{k}$ metrics:

Here we assume that the asymptotic length of the circle fibre is normalised to be $1$ so that $\text{dim}(\mathcal{M})$ does not include rescalings.

ALF metrics of dihedral type are not globally given by the Gibbons–Hawking construction and in most cases are not explicit. A number of different constructions have appeared over the past 30 years, but only recently Chen–Chen [11, Theorem 1.2] have shown that all these constructions yield equivalent families of ALF metrics. We distinguish the cases $m=0,1,2$ and $m\geq 3$.

• $m=0$:

  The $D_{0}$ ALF manifold is the moduli space of centred charge $2$ monopoles on $\mathbb{R}^{3}$ with its natural $L^{2}$–metric, known as the Atiyah–Hitchin manifold. The metric admits a cohomogeneity one isometric action of $SU(2)$ and is explicitly given in terms of elliptic integrals [5, Chapter 11]. The Atiyah–Hitchin manifold is diffeomorphic to the complement of a Veronese $\mathbb{R}\mathbb{P}^{2}$ in $\mathbb{S}^{4}$ and therefore it retracts to $\mathbb{R}\mathbb{P}^{2}$. The Atiyah–Hitchin metric does not admit deformations as a $D_{0}$ ALF metric except for scaling.

• $m=1$:

  The double cover of the Atiyah–Hitchin manifold is a $D_{1}$ ALF space. As a smooth manifold it is diffeomorphic to the complement of $\mathbb{R}\mathbb{P}^{2}$ in $\mathbb{C}\mathbb{P}^{2}$, or equivalently to the total space of $\mathcal{O}(-4)$ over $S^{2}$. Exploiting the rotational invariance of the metric it can be shown [30, Proposition 5.5] that the $2$–sphere in the interior is a strictly stable minimal sphere which is not holomorphic with respect to any complex structure compatible with the metric, a fact that will play a crucial role in the proof of Theorem 7.1. This rotationally invariant $D_{1}$ ALF metric admits a $3$–dimensional family of $D_{1}$ ALF deformations, sometimes referred to as the Dancer metrics.

• $m=2$:

  $D_{2}$ ALF metrics were constructed by Hitchin [23, §7] using twistor methods and by Biquard–Minerbe [8, Theorem 2.4] using a non-compact version of the Kummer construction: one considers the quotient of $\mathbb{R}^{3}\times\mathbb{S}^{1}$ by an involution and resolves the two singularities gluing in copies of the Eguchi–Hanson metric.

• $m\geq 3$:

  $D_{m}$ ALF metrics (for all $m\geq 1$) appeared in the work of Cherkis–Kapustin [13] on moduli spaces of singular monopoles on $\mathbb{R}^{3}$ and were rigorously constructed by Cherkis–Hitchin [12] using twistor methods and the generalised Legendre transform. In the case $m\geq 3$ a more transparent construction due to Biquard–Minerbe [8, Theorem 2.5] yields $D_{m}$ ALF metrics by desingularising the quotient of the Taub–NUT metric by the binary dihedral group $\mathcal{D}_{m}$ of order $4(m-2)$ using ALE dihedral spaces. Using complex Monge–Ampère methods Auvray [6, 7] has then constructed $3m$–dimensional families of $D_{m}$ ALF metrics on the smooth $4$–manifold underlying the minimal resolution of $\mathbb{C}^{2}/\mathcal{D}_{m}$.

We summarise some of the properties of a $D_{m}$ ALF gravitational instanton $M$ in the following table:

As in the cyclic case, $\mathcal{M}$ is the moduli space of $D_{m}$ ALF metrics modulo scaling.