Before introducing weighted Hölder spaces and proving the main estimates, it is helpful to look more closely at the linearisation of (6.1). It involves the operator $D=d^{\ast}+2\,d^{+}\colon\thinspace\Omega^{1}(M_{\epsilon})\rightarrow\Omega^{0}(M_{\epsilon})\oplus\Omega^{+}(M_{\epsilon})$, where adjoints and projections are computed using the metric $g_{\epsilon}$.

We are interested in understanding the behaviour of the operator $D$ (in particular, the presence of small eigenvalues) for a sequence of collapsing metrics in the Gibbons–Hawking form (3.3a).

Consider then the metric

$$g^{\textup{gh}}_{\epsilon}=h_{\epsilon}\,g_{\mathbb{T}}+\epsilon^{2}h_{\epsilon}^{-1}\theta^{2}$$

of (4.6) over the circle bundle $\pi\colon\thinspace P\rightarrow\mathcal{U}_{\epsilon}\subset\mathbb{T}^{\ast}$. By Lemma 4.9 the open sets $\mathcal{U}_{\epsilon}$ form an exhaustion of $\mathbb{T}^{\ast}$ as $\epsilon\rightarrow 0$.

We first consider the geometry of $g^{\textup{gh}}_{\epsilon}$ and in particular calculate its Levi–Civita connection.

As before let $\theta_{1},\theta_{2},\theta_{3}$ denote closed $1$–forms on $\mathbb{T}$ such that $g_{\mathbb{T}}=\theta_{1}^{2}+\theta_{2}^{2}+\theta_{3}^{2}$. Let $\xi_{1},\xi_{2},\xi_{3}$ be the dual vector fields with respect to $g_{\mathbb{T}}$. We will not distinguish between a vector tangent to $\mathbb{T}$ and its horizontal lift to $P$ with respect to the connection $\theta$. In particular, $[\xi_{i},\xi_{j}]=-d\theta(\xi_{i},\xi_{j})\,\xi$ as vector fields on $P$. Finally, let $\xi$ be the vertical vector field normalised so that $\theta(\xi)=1$.

Since $\theta([\xi,\xi_{i}])=-d\theta(\xi,\xi_{i})=0$ and $\pi_{\ast}[\xi,\xi_{i}]=[\pi_{\ast}\xi,\xi_{i}]=0$, we have $[\xi,\xi_{i}]=0$. The Koszul formula

$$2\langle\nabla_{X}Y,Z\rangle=\langle[X,Y],Z\rangle-\langle[Y,Z],X\rangle+\langle[Z,X],Y\rangle+X\cdot\langle Y,Z\rangle+Y\cdot\langle X,Z\rangle-Z\cdot\langle X,Y\rangle$$

then allows to calculate the Levi–Civita connection $\nabla$ of the metric $g^{\textup{gh}}_{\epsilon}$:

$$\begin{gathered}\nabla_{\xi}\xi=-\tfrac{1}{4}\epsilon^{2}\nabla^{\mathbb{T}}h_{\epsilon}^{-2},\qquad\nabla_{\xi_{i}}\xi=-\tfrac{1}{2}h_{\epsilon}^{-1}(\xi_{i}\cdot h_{\epsilon})\,\xi+\tfrac{1}{2}\epsilon^{2}h_{\epsilon}^{-2}(\xi_{i}\lrcorner d\theta)^{\sharp_{\mathbb{T}}},\\ \nabla_{\xi}\xi_{i}=-\tfrac{1}{2}h_{\epsilon}^{-1}(\xi_{i}\cdot h_{\epsilon})\,\xi+\tfrac{1}{2}\epsilon^{2}h_{\epsilon}^{-2}(\xi_{i}\lrcorner d\theta)^{\sharp_{\mathbb{T}}},\\ \nabla_{\xi_{j}}\xi_{i}=-\tfrac{1}{2}d\theta(\xi_{i},\xi_{j})\,\xi+\tfrac{1}{2}h_{\epsilon}^{-1}\left((\xi_{j}\cdot h_{\epsilon})\,\xi_{i}-(\xi_{i}\cdot h_{\epsilon})\,\xi_{j}-\delta_{ij}\nabla^{\mathbb{T}}h_{\epsilon}\right).\end{gathered}$$

Here $\nabla^{\mathbb{T}}$ and ${}^{\sharp_{\mathbb{T}}}$ denote gradient and musical isomorphism with respect to the flat metric $g_{\mathbb{T}}$ and we used the fact that $h_{\epsilon}$ is $S^{1}$–invariant. Since $\nabla$ is a metric connection, we calculate the covariant derivatives of the $1$–forms $\theta,\theta_{i}$ by duality. Since we will need this later, we write out formulas for $\nabla_{\xi}\theta$ and $\nabla_{\xi}\theta_{i}$:

(6.2)

$$\nabla_{\xi}\theta=\tfrac{1}{2}h_{\epsilon}^{-1}dh_{\epsilon},\qquad\nabla_{\xi}\theta_{i}=-\tfrac{1}{2}\epsilon^{2}h_{\epsilon}^{-3}(\xi_{i}\cdot h_{\epsilon})\,\theta+\tfrac{1}{2}\epsilon^{2}h_{\epsilon}^{-2}(\xi_{i}\lrcorner d\theta).$$

Now, the cotangent bundle of $M^{\textup{gh}}_{\epsilon}$ is trivial as it is spanned by $\theta,\theta_{1},\theta_{2},\theta_{3}$. Thus we can write every $1$–form $a$ as

(6.3)

$$a=\epsilon\,a_{0}\,\theta+a_{1}\,\theta_{1}+a_{2}\,\theta_{2}+a_{3}\,\theta_{3}$$

for functions $a_{0},a_{1},a_{2},a_{3}$. Note that

$$|a|^{2}_{g^{\textup{gh}}_{\epsilon}}=h_{\epsilon}\,|a_{0}|^{2}+h_{\epsilon}^{-1}\left(|a_{1}|^{2}+|a_{2}|^{2}+|a_{3}|^{2}\right).$$

A direct computation using the fact that $\theta_{i}$ is closed for $i=1,2,3$ and $(h_{\epsilon},\epsilon\,d\theta)$ is a solution of the monopole equation (3.4) with respect to the flat metric $g_{\mathbb{T}}$ shows that

(6.4)

$$\begin{gathered}d^{\ast}a=-h_{\epsilon}^{-1}\left(\sum_{i=1}^{3}{\xi_{i}\cdot a_{i}}+\tfrac{1}{\epsilon}h_{\epsilon}^{2}\,\xi\cdot a_{0}\right),\\ 2\,d^{+}a=\sum_{i=1}^{3}{\left(\xi_{i}\cdot a_{0}+h_{\epsilon}^{-1}(\xi_{j}\cdot a_{k}-\xi_{k}\cdot a_{j})+h_{\epsilon}^{-1}(\xi_{i}\cdot h_{\epsilon})\,a_{0}-\tfrac{1}{\epsilon}(\xi\cdot a_{i})\right)\omega^{\textup{gh}}_{\epsilon,i}},\end{gathered}$$

where $\omega^{\textup{gh}}_{\epsilon,i}$, $i=1,2,3$, is the hyperkähler triple $\bm{\underline{\omega}}^{\textup{gh}}_{\epsilon}$ of (4.6).

By Fourier analysis along the circle fibres we define projections $\Pi_{0}$ and $\Pi_{\perp}$ onto $S^{1}$–invariant and oscillatory components of functions. Via the trivialisation (6.3) $\Pi_{0}$ and $\Pi_{\perp}$ extend to $1$–forms. Since $h_{\epsilon}$ is $S^{1}$–invariant we see from (6.4) that the operator $D$ respects this decomposition.

For any fixed $\tau\in(0,1)$ restrict attention to the region in $M^{\textup{gh}}_{\epsilon}$ where $\rho_{j},\rho_{i}\geq c\,\epsilon^{\frac{1-\tau}{2}}$ for all $i=1,\dots,n$ and $j=1,\dots,8$. Then

$$\|h_{\epsilon}-1\|_{C^{0}}\leq C\epsilon^{\frac{1+\tau}{2}},\qquad\|\nabla^{\mathbb{T}}h_{\epsilon}\|_{C^{0}}\leq C\epsilon^{\tau}$$

by Lemma 4.10. We conclude that the operator $D$ of (6.4) acting on $S^{1}$–invariant $1$–forms approaches the Dirac operator

(6.5)

$$D_{0}\colon\thinspace\Omega^{0}(\mathbb{T})\oplus\Omega^{1}(\mathbb{T})\rightarrow\Omega^{0}(\mathbb{T})\oplus\Omega^{1}(\mathbb{T}),\qquad(f,\gamma)\mapsto(d^{\ast}\gamma,df+\ast d\gamma)$$

of the flat torus $\mathbb{T}$.

Moreover, using the expressions (6.2) for $\nabla_{\xi}\theta$ and $\nabla_{\xi}\theta_{i}$ we find

$$\epsilon^{2}h_{\epsilon}^{-1}|\nabla a|^{2}_{g^{\textup{gh}}_{\epsilon}}\geq|\nabla_{\xi}a|^{2}_{g^{\textup{gh}}_{\epsilon}}\geq h_{\epsilon}|\xi\cdot a_{0}|^{2}+h_{\epsilon}^{-1}\sum_{i=1}^{3}{|\xi\cdot a_{i}|^{2}}-Ch_{\epsilon}^{-4}|\nabla^{\mathbb{T}}h_{\epsilon}|^{2}_{g_{\mathbb{T}}}\left(h_{\epsilon}|a_{0}|^{2}+h_{\epsilon}^{-1}\sum_{i=1}^{3}{|a_{i}|^{2}}\right).$$

Thus in the region where $\rho_{j},\rho_{i}\geq c\,\epsilon^{\frac{1-\tau}{2}}$ for some $\tau>0$ we have

(6.6)

$$\|\Pi_{\perp}a\|_{C^{0,\alpha}(S^{1})}\leq C\epsilon\,\|\nabla a\|_{C^{0,\alpha}(S^{1})}$$

on each fibre for all $\epsilon$ sufficiently small.

These two observations — the convergence of the operator $D$ to the Dirac operator $D_{0}$ of the flat $3$–torus as $\epsilon\rightarrow 0$ and the strong control of the oscillatory part of $1$–forms — will be crucial in the rest of the section. We will exploit the same remarks when considering blow-downs of ALF gravitational instantons. Let $(M,g_{M})$ be a complete ALF space. Given a sequence $R_{i}\rightarrow\infty$ consider the blow down $R_{i}^{-2}g_{M}$. Since by Definition 3.6 $R_{i}^{-2}g_{M}$ is asymptotic (up to a double cover in the dihedral case) to the Gibbons–Hawking metric

$$\left(1+R_{i}^{-1}\frac{k}{2\rho}\right)\,g_{\mathbb{R}^{3}}+R_{i}^{-2}\left(1+R_{i}^{-1}\frac{k}{2\rho}\right)^{-1}\theta^{2},$$

as $i\rightarrow\infty$ the behaviour of the operator $D$ with respect to the metric $R_{i}^{-2}g_{M}$ is the same as the one observed for the metric $g^{\textup{gh}}_{\epsilon}$ as $\epsilon\rightarrow 0$ with the flat $\mathbb{R}^{3}$ in place of the flat $3$–torus $\mathbb{T}$. Namely, on the region $\rho\geq c\,R_{i}^{-\frac{1-\tau}{2}}$ (6.6) holds with $\epsilon=R_{i}^{-1}$ and the operator $D$ converges to the Dirac operator $D_{0}$ of flat space $\mathbb{R}^{3}$.

In order to control the growth of differential forms close to the punctures on $\mathbb{T}^{\ast}$ and on the end of an ALF space we will now introduce weighted Hölder spaces.