Consider a $2$-torus $\mathbb{T}^{2}$ with a flat metric of area $A$ and let $U=\mathbb{T}^{2}_{x,y}\times\mathbb{R}_{z>0}$, where we have fixed a choice of an orthogonal frame $\{e_{1},e_{2}\}$ on $\mathbb{T}^{2}$ such that $g_{\mathbb{T}^{2}}=A(e_{1}^{2}+e_{2}^{2})$. Let $V(z)=\frac{2\pi bz}{A}$ for a positive integer $b>0$. Fixing a connection form $\theta$ such that $d\theta=\frac{2\pi b}{A}{\rm dvol}_{\mathbb{T}^{2}}$, the corresponding hyperkähler Gibbons-Hawking metric $g$ is given by

(2.19)

$\displaystyle g=\frac{2\pi bz}{A}(g_{\mathbb{T}^{2}}+dz^{2})+\frac{A}{2\pi bz}\theta^{2}.$

The Gibbons-Hawking space $(\mathfrak{M},g)$ has one complete end as $z\rightarrow\infty$ and one incomplete end as $z\rightarrow 0$. Moreover, for each $z_{0}>0$, the level set $\{z=z_{0}\}$ is a Heisenberg nilmanifold $\Nil^{3}_{b}(\epsilon,\tau)$ with a $z_{0}$-dependent left-invariant metric, where $\tau$ denotes the modulus of our flat $2$-torus in the upper half-plane and $A=\epsilon^{2}\tau_{2}$ as in Section 2.1. Making the substitution $z=(3/2)s^{2/3}$, and then scaling appropriately, the Gibbons-Hawking metric $g$ takes the form

(2.20)

$\displaystyle ds^{2}+s^{2/3}g_{\mathbb{T}^{2}}+s^{-2/3}\Big(\frac{A}{3\pi b}\theta\Big)^{2}.$

In this form, it is easy to see that the volume growth is $\sim s^{4/3}$ and that $|{\Rm}|\sim s^{-2}$ as $s\rightarrow\infty$. One can also show using the Chern-Gauss-Bonnet formula that the $L^{2}$ norm of $\Rm$ is finite.

Note that if we had instead taken $b=0$, the Gibbons-Hawking metric would be the product of $\mathbb{R}$ with a flat $3$-torus, i.e., the type of geometry known as ALH geometry in the literature.

View the flat torus $\mathbb{T}^{2}$ as an elliptic curve $E$ of modulus $\tau$ with respect to the complex structure $J$ defined by $Je_{1}=e_{2}$. Then there is exactly one $g$-parallel complex structure $J_{0}$ on the total space $\mathfrak{M}$ that makes the projection map to $E$ holomorphic. This choice of complex structure realizes $\mathfrak{M}$ as an open subset of the total space of a degree $b$ holomorphic line bundle $L$ over $E$ (more precisely, as a tubular neighborhood of the zero section of $L$ with the zero section removed). The Ricci-flat Kähler form with respect to $J_{0}$ is then given by (see Proposition 3.1)

(2.21)

$$\omega_{0}=\frac{2}{3}i\partial\bar{\partial}(-{\log|\xi|}_{h}^{2})^{3/2},$$

where $h$ is a hermitian metric on $L$ whose curvature form is a multiple of the flat Kähler form on $E$, and where the tautological section $\xi$ cuts out the zero section of $L$. This is an example of the Calabi construction, and we call the corresponding metric a Calabi model metric of degree $b$. We will discuss this construction (in all dimensions) in more detail in Section 3. In complex dimension $2$, the Gibbons-Hawking ansatz actually recovers the Calabi ansatz for all degree $b$ holomorphic line bundles over $E$. Indeed, any two such line bundles differ by a degree 0 line bundle and $\Pic^{0}(E)=H^{1}(E,\mathcal{O}_{E})/H^{1}(E,\mathbb{Z})$; on the other hand, the gauge equivalence classes of flat connections are parametrized by $H^{1}(E,\mathbb{R})/H^{1}(E,\mathbb{Z})$.

A very convenient property for our purposes is that if we do change the connection $1$-form $\theta$ by a flat connection, then the associated Gibbons-Hawking metric (2.19) actually only changes by a diffeomorphism (although this diffeomorphism necessarily breaks the $S^{1}$-bundle structure on $\mathfrak{M}$). To see this, fix any choice of $\theta$ such that $d\theta=\frac{2\pi b}{A}\dvol_{E}$. Note that we can always arrange by parallel transport in the $z$-direction that the $dz$-component of $\theta$ vanishes. Then we only need to consider the case that $\theta$ gets changed by the pullback (under the projection $\mathfrak{M}\stackrel{{\scriptstyle\pi}}{{\to}}U\to E$) of a parallel $1$-form $\eta$ on $E$. Write $\eta=v\,\lrcorner\,\dvol_{E}$, where $v$ is a parallel vector field on $E$. Let $\hat{v}$ be the $\theta$-horizontal lift of $v$ to $\mathfrak{M}$ and let $\hat{f}_{s}$ be the $1$-parameter group of diffeomorphisms generated by $\hat{v}$, which covers a $1$-parameter group of translations on $E$. Then

(2.22)

$\displaystyle\begin{split}\frac{d}{ds}(\hat{f}_{s}^{*}\theta)&=\hat{v}\,\lrcorner\,\hat{f}_{s}^{*}(d\theta)+d(\hat{v}\,\lrcorner\,\hat{f}_{s}^{*}\theta)=\frac{2\pi b}{A}(\hat{v}\,\lrcorner\,\pi^{*}({\rm dvol}_{E}))=\frac{2\pi b}{A}\pi^{*}\eta,\end{split}$

using the fact that $d\hat{f}_{s}|_{x}(\hat{v}|_{x})=\hat{v}|_{\hat{f}_{s}(x)}$ for all $x\in\mathfrak{M}$. Thus,

(2.23)

$$\hat{f}_{s}^{*}\theta=\theta+\frac{2\pi bs}{A}\pi^{*}\eta.$$

This shows that any two possible choices of $\theta$ with vanishing $dz$-component differ by the translation action of $E$ on itself, lifted to $\mathfrak{M}$, and then the corresponding hyperkähler metrics (2.19) are clearly isometric as well. We note that with the particular choice $\theta=\theta_{b}$ from (2.12), these diffeomorphisms can be written down explicitly as follows: for all $p,q\in\mathbb{R}$, the mapping

(2.24)

$\displaystyle\varphi(x,y,t,z)=(x-q,y+p,t+px,z)$

descends to a diffeomorphism of ${\rm Nil}^{3}_{b}(\epsilon,\tau)_{x,y,t}\times\mathbb{R}_{z}$ which satisfies

(2.25)

$\displaystyle\varphi^{*}\theta_{b}=\theta_{b}+\frac{2\pi b}{A}(pdx+qdy).$

The Calabi construction provides the model at infinity of the Tian-Yau metrics in our context. These metrics will also be studied in more detail in Section 3. Note that every Tian-Yau metric comes with its own preferred choice of a connection form $\theta$ determined by the Chern connection of the normal bundle of the compactifying divisor. The above gauge fixing construction then allows us to choose a new coordinate system at infinity that identifies this $\theta$ with the standard connection form $\theta_{b}$ (see the proof of Proposition 3.1 and also equation (6.6)).