3.2.2. ALF spaces of dihedral type [02H0]

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3.2.2. ALF spaces of dihedral type

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.

Remark.

The fact that the double cover of the Atiyah–Hitchin manifold admits a $3$ –parameter family of $D_{1}$ ALF deformations can also be shown using methods similar to the ones developed in this paper. Indeed, it is known [24, §5.4] that the rotationally invariant $D_{1}$ ALF metric admits a unique $L^{2}$ –integrable (in fact, exponentially decaying) anti-self-dual harmonic form $\eta$ . This form yields a $3$ –dimensional space of infinitesimal hyperkähler deformations and an extension of the analysis needed for the proof of Theorem 6.15 could be used to integrate these infinitesimal deformations to genuine $D_{1}$ ALF metrics. In fact Dancer [14] has constructed a $3$ –parameter family of hyperkähler deformations of the rotationally invariant $D_{1}$ ALF metric using Nahm’s equations and hyperkähler quotient techniques: there exists a hyperkähler $8$ –manifold $\mathcal{N}$ constructed as a moduli space of solutions to Nahm’s equations which admits a triholomorphic $U(1)$ –action. Denote by $\mu\colon\thinspace\mathcal{N}\rightarrow\mathbb{R}^{3}$ the corresponding hyperkähler moment map. Dancer identifies the rotationally symmetric $D_{1}$ ALF metric with the hyperkähler quotient $\mu^{-1}(0)/U(1)$ . By varying the level set of the moment map he then obtains a $3$ –parameter family of hyperkähler deformations of the Atiyah–Hitchin metric. By a general formula for the infinitesimal deformation of the symplectic form of a symplectic quotient corresponding to varying the level set of the moment map [17], the infinitesimal deformations of the Atiyah–Hitchin metric corresponding to Dancer’s metrics coincide with those determined by the $L^{2}$ harmonic form $\eta$ , which is interpreted in this context as the curvature of the natural hyperholomorphic connection on the $U(1)$ –bundle $\mu^{-1}(0)\rightarrow\mu^{-1}(0)/U(1)$ induced by the Levi–Civita connection of $\mathcal{N}$ .

$m=2$ :

$D_{2}$ 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.

Remark.

Biquard–Minerbe [8, Theorem 2.4] use singular perturbation methods to solve a complex Monge–Ampère equation on the minimal resolution of $(\mathbb{R}^{3}\times\mathbb{S}^{1})/\mathbb{Z}_{2}$ . Using the more general approach adopted in this paper to glue hyperkähler structures one could extend their construction to recover a $6$ –dimensional family of $D_{2}$ ALF metrics.

$m\geq 3$ :

$D_{m}$ 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 $\mathbb{C}^{2}/\mathcal{D}_{m}$ .

Remark 3.7.

The gluing construction presented in this paper could be extended to the non-compact setting to yield yet another construction of dihedral ALF metrics. Indeed, one considers a Gibbons–Hawking metric obtained from the harmonic function

h=\lambda-\frac{2}{|x|}+\sum_{i=1}^{m}{\frac{1}{2|x-x_{i}|}+\frac{1}{2|x+x_{i}|}}

for $m$ distinct points $x_{1},\dots,x_{m}\in\mathbb{R}^{3}\setminus\{0\}$ . Observe that for $\lambda>0$ sufficiently large $h>0$ outside an arbitrarily small neighbourhood of the origin. Since the configuration of punctures is invariant under the standard involution of $\mathbb{R}^{3}$ , this (incomplete) metric descends to a hyperkähler metric on a $\mathbb{Z}_{2}$ quotient. For $\lambda$ sufficiently large one can then complete this metric by gluing in a copy of the $D_{0}$ ALF space close to the origin. This approximate solution could then be deformed to an exact hyperkähler metric in a way similar to the proof of Theorem 6.15.

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

$m$	$\pi_{1}(M)$	$b_{2}(M)$	$\chi(M)$	$\text{dim}(\mathcal{M})$
$0$	$\mathbb{Z}_{2}$	$0$	$1$	$0$
$m>0$	$1$	$m$	$m+1$	$3m$

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