We are now going to define an approximately hyperkähler triple $\bm{\underline{\omega}}_{\epsilon}$ on the $4$–manifold $M_{\epsilon}$.

For each $j=1,\dots,8$ denote by $\bm{\underline{\omega}}_{q_{j},\epsilon}$ the hyperkähler triple obtained from the Gibbons–Hawking ansatz (3.3) using the harmonic function

By abuse of notation we think of $\bm{\underline{\omega}}_{q_{j},\epsilon}$ as defined both on the circle bundle $H^{2m_{j}-4}\rightarrow\mathbb{R}^{3}\setminus B_{8\epsilon}(0)$ as well as on its quotient by the involution that acts as the simultaneous standard involution on $\mathbb{R}^{3}$ and the circle fibres.

Similarly, for each $i=1,\dots,n$ let $\bm{\underline{\omega}}_{p_{i},\epsilon}$ be the hyperkähler triple obtained from the Gibbons–Hawking ansatz using the harmonic function

Here $\lambda_{j},\lambda_{i}$ and $\ell_{i}$ are the constants and linear functions appearing in Lemma 4.7. We will assume that $\epsilon$ is small enough to guarantee that $\tfrac{1}{2}<\lambda_{i},\lambda_{j}<\tfrac{3}{2}$.

For all $j=1,\dots,8$ let $(M_{j},\bm{\underline{\omega}}_{M_{j}})$ be a complete $D_{m_{j}}$ ALF space. By Definition 3.6 there exists a compact set $K\subset M_{j}$, $R_{0}>0$ and a diffeomorphism $M_{j}\setminus K\simeq H^{2m_{j}-4}/\mathbb{Z}_{2}$ such that

for $\rho>\epsilon R_{0}$ with $\bm{\underline{\eta}}_{q_{j},\epsilon}=O(\epsilon^{3}\rho^{-3})$ and similar estimates on the derivatives.

For each $i=1,\dots,n$ let $(N_{i},\bm{\underline{\omega}}_{N_{i}})$ be a complete $A_{k_{i}-1}$ ALF space. By the classification of ALF spaces of cyclic type [35] the hyperkähler structure on $N_{i}$ is explicitly given via the Gibbons–Hawking ansatz starting from a harmonic function on $\mathbb{R}^{3}$ with $k_{i}$ singularities. We can add to this function the smooth harmonic function $\epsilon^{2}\ell_{i}$. Over a ball in $\mathbb{R}^{3}$ of radius much smaller than $\epsilon^{-2}$ we can regard the resulting hyperkähler structure as a small perturbation of the ALF $A_{k_{i}-1}$ hyperkähler structure $\bm{\underline{\omega}}_{N_{i}}$. By abuse of notation we denote this perturbed hyperkähler structure with the same symbol $\bm{\underline{\omega}}_{N_{i}}$. The advantage of this modification is that now $\epsilon^{2}\bm{\underline{\omega}}_{N_{i}}$ approaches $\bm{\underline{\omega}}_{p_{i},\epsilon}$ with a smaller error: by Definition 3.6 there exists a compact set $K\subset N_{i}$, $R_{0}>0$ and a diffeomorphism $N_{i}\setminus K\simeq H^{k_{i}}|_{\mathbb{R}^{3}\setminus B_{R_{0}}}$ such that

with (by scaling) $\bm{\underline{\eta}}_{p_{i},\epsilon}=O(\epsilon^{3}\rho^{-3})$ and similar estimates on the derivatives.

We will assume that $\epsilon$ is chosen so small as to make sure that $\epsilon R_{0}\ll\rho_{0}$, where $\rho_{0}>0$ was fixed in Lemma 4.7. By choosing $R_{0}$ larger if necessary we can assume that the harmonic functions $h_{q_{j}}$ and $h_{p_{i}}$ in (5.2) and (5.3) are as close to constant functions as we please for $\epsilon R_{0}\leq\rho\leq\rho_{0}$. Then the hyperkähler triples $\bm{\underline{\omega}}_{q_{j},\epsilon}$ and $\bm{\underline{\omega}}_{p_{i},\epsilon}$ define metrics $g_{q_{j},\epsilon}$ and $g_{p_{i},\epsilon}$ which are uniformly equivalent to $g_{\mathbb{T}}+\epsilon^{2}\theta^{2}$ in the regions $\epsilon R_{0}\leq\rho_{j},\rho_{i}\leq 2\rho_{0}$. In the rest of the section all norms and covariant derivatives will be computed with respect to this metric.

A crucial observation is that we can take $\bm{\underline{\eta}}_{p_{i},\epsilon}$ and $\bm{\underline{\eta}}_{q_{j},\epsilon}$ to be exact.

By a similar radial integration, Lemma 4.7.(i) implies that in the regions $\epsilon R_{0}\leq\rho_{j}\leq 2\rho_{0}$ and $\epsilon R_{0}\leq\rho_{i}\leq 2\rho_{0}$, respectively, we can write

for $k=0,1,2,3$.

Now, let $\chi_{q_{j}}$ and $\chi_{p_{i}}$ be cut-off functions with the following properties:

We finally define a triple of closed $2$–forms on $M_{\epsilon}$ by