1.2.1 Gibbons–Hawking versus LeBrun ansätze

Gibbons–Hawking ansatz. As recalled in the Introduction, the Taub-NUT metric on $\mathbb{R}^{4}$ is often described via the Gibbons–Hawking ansatz as follows: given $m\in (0,\infty )$ , set

$$\begin{eqnarray}\mathbf{f}_{m}=V(dy_{1}^{2}+dy_{2}^{2}+dy_{3}^{2})+V^{-1}\unicode[STIX]{x1D702}^{2},\end{eqnarray}$$

where $(y_{1},y_{2},y_{3})$ is a circle fibration of $\mathbb{R}^{4}\backslash \{0\}$ over $\mathbb{R}^{3}\backslash \{0\}$ , $V$ is the function $(1+4mR)/2R$ (harmonic in the $y_{j}$ coordinates) with $R^{2}=y_{1}^{2}+y_{2}^{2}+y_{3}^{2}$ , and where $\unicode[STIX]{x1D702}$ is a connection 1-form for this fibration such that $d\unicode[STIX]{x1D702}=\ast _{\mathbb{R}^{3}}dV$ . Thus defined, the metric $\mathbf{f}_{m}$ confers length $\unicode[STIX]{x1D70B}\sqrt{2/m}$ to the fibres at infinity, and is hermitian for the almost-complex structures

$$\begin{eqnarray}J_{a}:\left\{\begin{array}{@{}l@{}}Vdy_{a}\longmapsto \unicode[STIX]{x1D702},\quad \\ dy_{b}\longmapsto dy_{c},\quad \end{array}\right.\end{eqnarray}$$

with $(a,b,c)\in \{(1,2,3),(2,3,1),(3,1,2)\}$ . These are in fact complex structures, verifying the quaternionic relations $J_{a}J_{b}J_{c}=-1$ , for which $\mathbf{f}_{m}$ is Kähler, thanks to the harmonicity of $V$ : $\mathbf{f}_{m}$ is thus hyperkähler. One checks moreover that this way, the metric $\mathbf{f}_{m}$ and the complex structures extend as such through $0\in \mathbb{R}^{4}$ .

We now switch point of view to a description better adapted to our construction.

LeBrun’s potential. As depicted in [Reference LeBrunLeB91] and reviewed in detail in Appendix A, one can give a somehow more concrete support of the description of $\mathbf{f}_{m}$ , through which the complex structure $J_{1}$ mentioned above is the standard $I_{1}$ on $\mathbb{C}^{2}$ , and $\operatorname{vol}^{\mathbf{f}_{m}}=\unicode[STIX]{x1D6FA}_{\mathbf{e}}$ , the standard Euclidean volume form. One starts with the following implicit formulas:

(5) $$\begin{eqnarray}\begin{array}{@{}c@{}}|z_{1}|=e^{m(u^{2}-v^{2})}u,\\ |z_{2}|=e^{m(v^{2}-u^{2})}v,\end{array}\end{eqnarray}$$

defining functions $u,v:\mathbb{C}^{2}\rightarrow \mathbb{R}$ , invariant under the circle action $e^{i\unicode[STIX]{x1D703}}\cdot (z_{1},z_{2})=(e^{i\unicode[STIX]{x1D703}}z_{1},e^{-i\unicode[STIX]{x1D703}}z_{2})$ which makes $\mathbb{S}^{1}$ as a subgroup of $\operatorname{SU}(2)$ ; notice the role of $m$ in these formulas, which enlightens our choice of taking it as the parameter of the upcoming construction. One then sets $y_{1}=\frac{1}{2}(u^{2}-v^{2})$ , $y_{2}+iy_{3}=-iz_{1}z_{2}$ , $R=\frac{1}{2}(u^{2}+v^{2})=(y_{1}^{2}+y_{2}^{2}+y_{3}^{2})^{1/2}$ ; these are $\mathbb{S}^{1}$ -invariant functions, making $(y_{1},y_{2},y_{3})$ as a principal- $\mathbb{S}^{1}$ fibration $\mathbb{C}^{2}\rightarrow \mathbb{R}^{3}$ away from the origins. One finally defines

(6) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{m}:={\textstyle \frac{1}{4}}(u^{2}+v^{2}+m(u^{4}+v^{4}))={\textstyle \frac{1}{2}}(R+m(R^{2}+y_{1}^{2})).\end{eqnarray}$$

One can then check (see Appendix A) that $dd_{I_{1}}^{c}\unicode[STIX]{x1D711}_{m}$ is positive in the sense of $I_{1}$ -hermitian 2-forms, and that $(dd_{I_{1}}^{c}\unicode[STIX]{x1D711}_{m})^{2}=2\unicode[STIX]{x1D6FA}_{\mathbf{e}}$ . If one sets moreover $V=(1+4mR)/2R$ , and $\unicode[STIX]{x1D702}=I_{1}Vdy_{1}$ , noticing by passing that $\unicode[STIX]{x1D702}$ is then a connection 1-form for the fibration with $d\unicode[STIX]{x1D702}=\ast _{\mathbb{R}^{3}}\,dV$ , one has $\mathbf{f}_{m}:=V(dy_{1}^{2}+dy_{2}^{2}+dy_{3}^{2})+V^{-1}\unicode[STIX]{x1D702}^{2}=(dd_{I_{1}}^{c}\unicode[STIX]{x1D711}_{m})(\cdot ,I_{1}\cdot )$ . This metric is well defined at $0\in \mathbb{C}^{2}$ , as $(dd_{I_{1}}^{c}\unicode[STIX]{x1D711}_{m})(\cdot ,I_{1}\cdot )=\mathbf{e}$ at that point.

The metric $\mathbf{f}_{m}$ is therefore Kähler for $I_{1}$ with volume form $\operatorname{vol}^{\mathbf{f}_{m}}=\unicode[STIX]{x1D6FA}_{\mathbf{e}}$ on the whole $\mathbb{C}^{2}$ ; by the standard properties of Kähler metrics, it is thus Ricci-flat. One recovers a complete hyperkähler data after checking that the defining equations

(7) $$\begin{eqnarray}\mathbf{f}_{m}(J_{j}\cdot ,\cdot )=\unicode[STIX]{x1D714}_{j}^{\mathbf{e}}\quad \text{where}~\unicode[STIX]{x1D714}_{j}^{\mathbf{e}}=\mathbf{e}(I_{j}\cdot ,\cdot ),\quad j=2,3,\end{eqnarray}$$

with $I_{2}$ , $I_{3}$ the other two standard complex structures on $\mathbb{R}^{4}\cong \mathbb{H}$ , give rise to integrable complex structures, verifying respectively $J_{j}:Vdy_{j}\mapsto \unicode[STIX]{x1D702}$ , $dy_{k}\mapsto dy_{\ell }$ for $(j,k,\ell )=(2,3,1),(3,1,2)$ , as well as the quaternionic relations together with $I_{1}$ .

Let us now take a look at the length of the $\mathbb{S}^{1}$ -fibres at infinity. Consider the vector field $\unicode[STIX]{x1D709}:=i(z_{1}(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}z_{1})-\overline{z_{1}}(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\overline{z_{1}})-z_{2}(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}z_{2})+\overline{z_{2}}(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\overline{z_{2}}))$ giving the infinitesimal action of $\mathbb{S}^{1}$ . One has $dy_{1}(-I_{1}\unicode[STIX]{x1D709})=V^{-1}$ , thus $\unicode[STIX]{x1D702}(\unicode[STIX]{x1D709})=1$ , and $dy_{j}(\unicode[STIX]{x1D709})=0$ , $j=1,2,3$ ; since $R$ is $\mathbb{S}^{1}$ -invariant, the length of the fibres is just $2\unicode[STIX]{x1D70B}V^{-1/2}$ , which tends to $\unicode[STIX]{x1D70B}\sqrt{2/m}$ .

From now on, we see the mass parameter $m$ as fixed, and we drop the indices $m$ when there is no risk of confusion.

The Taub-NUT metric and the action of the binary dihedral group on  $\mathbb{C}^{2}$ . For $k\geqslant 2$ , which we fix until the end of this part, the action of the binary dihedral group ${\mathcal{D}}_{k}$ of order $4k$ seen as a subgroup of $\operatorname{SU}(2)=\operatorname{Sp}(1)$ is generated by the matrices $\unicode[STIX]{x1D701}_{k}:=\big(\!\begin{smallmatrix}e^{i\unicode[STIX]{x1D70B}/k} & 0\\ 0 & e^{-i\unicode[STIX]{x1D70B}/k}\end{smallmatrix}\!\big)$ and $\unicode[STIX]{x1D70F}:=\big(\!\begin{smallmatrix}0 & -1\\ 1 & 0\end{smallmatrix}\!\big)$ . One has $\unicode[STIX]{x1D701}_{k}^{\ast }y_{j}=y_{j}$ , $j=1,2,3$ , and thus $\unicode[STIX]{x1D701}_{k}^{\ast }R=R$ , and $\unicode[STIX]{x1D701}_{k}^{\ast }\unicode[STIX]{x1D702}=\unicode[STIX]{x1D702}$ , whereas $\unicode[STIX]{x1D70F}^{\ast }y_{j}=-y_{j}$ , $j=1,2,3$ , thus $\unicode[STIX]{x1D70F}^{\ast }R=R$ , and $\unicode[STIX]{x1D70F}^{\ast }\unicode[STIX]{x1D702}=-\unicode[STIX]{x1D702}$ . The Taub-NUT metric $\mathbf{f}$ is therefore ${\mathcal{D}}_{k}$ -invariant, and descends smoothly to $(\mathbb{R}^{4}\backslash \{0\})/{\mathcal{D}}_{k}$ : this is the metric we are going to glue at infinity of ${\mathcal{D}}_{k}$ -ALE instantons in the next section. Before though, we need a few more analytical tools for the Taub-NUT metric as we describe it here.

1.2.2 Orthonormal frames, covariant derivatives and curvature

In addition to the above relations between the vector field $\unicode[STIX]{x1D709}$ , and the 1-forms $\unicode[STIX]{x1D702}$ and $dy_{j}$ , $j=1,2,3$ , one has that the data

(8) $$\begin{eqnarray}(e_{0},e_{1},e_{2},e_{3}):=(V^{1/2}\unicode[STIX]{x1D709},-I_{1}V^{1/2}\unicode[STIX]{x1D709},V^{-1/2}\unicode[STIX]{x1D701},V^{-1/2}I_{1}\unicode[STIX]{x1D701}),\end{eqnarray}$$

is the dual frame of the orthonormal frame of 1-forms

(9) $$\begin{eqnarray}(e_{0}^{\ast },e_{1}^{\ast },e_{2}^{\ast },e_{3}^{\ast }):=(V^{-1/2}\unicode[STIX]{x1D702},V^{1/2}dy_{1},V^{1/2}dy_{2},V^{1/2}dy_{3})\end{eqnarray}$$

on $\mathbb{C}^{2}\backslash \{0\}$ , provided that the vector field $\unicode[STIX]{x1D701}$ is defined by

(10) $$\begin{eqnarray}\unicode[STIX]{x1D701}:=\frac{1}{2iR}\biggl(e^{4my_{1}}\biggl(z_{2}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\overline{z_{1}}}-\overline{z_{2}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z_{1}}\biggr)+e^{-4my_{1}}\biggl(z_{1}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\overline{z_{2}}}-\overline{z_{1}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z_{2}}\biggr)\biggr),\end{eqnarray}$$

see Appendix A; we keep the notation $(e_{j})_{j=0,\ldots ,3}$ and $(e_{j}^{\ast })_{j=0,\ldots ,3}$ throughout this part. An explicit computation made in Appendix A then gives the estimates

$$\begin{eqnarray}|(\unicode[STIX]{x1D6FB}^{\mathbf{f}})^{\ell }e_{j}|_{\mathbf{f}}=O(R^{-1-\ell })\quad \text{near infinity for all }\ell \geqslant 1\text{ and }j=0,\ldots ,3.\end{eqnarray}$$

Consequently, for all $\ell \geqslant 0$ , $|(\unicode[STIX]{x1D6FB}^{\mathbf{f}})^{\ell }\operatorname{Rm}^{\mathbf{f}}\!|_{\mathbf{f}}=O(R^{-3-\ell })$ , this justifies the terminology ‘Asymptotically Locally Flat’ for $\mathbf{f}$ ; this estimate, done using the Gibbons–Hawking ansatz, can also be found e.g. in [Reference MinerbeMin07, § 1.0.3].

We close this section with two further useful estimates, giving an idea of the geometric gap between $\mathbf{e}$ and $\mathbf{f}$ : first, at the level of distance functions, rearranging (5) gives $R\leqslant 2r^{2}$ , which is sharp is general; second, there exists $C=C(m)>0$ such that outside the unit ball of $\mathbb{C}^{2}$ , $C^{-1}r^{-2}\mathbf{e}\leqslant \mathbf{f}\leqslant Cr^{2}\mathbf{e}$ , which, again, is sharp in general. Details are given in § A.2, in Appendix A.

1.3.1 Approximation of the ALE Kähler form as a complex hessian

Asymptotics of the Kähler form and the complex structure. In view of steps (iii) and (iv) of the program developed in § 1.1, since we are performing our gluing on some specific ALE instantons, we fix for the rest of this part

(11) $$\begin{eqnarray}\unicode[STIX]{x1D709}\in \mathfrak{h}-D,\quad \text{such that:}~|\unicode[STIX]{x1D709}_{2}|^{2}-|\unicode[STIX]{x1D709}_{3}|^{2}=\langle \unicode[STIX]{x1D709}_{2},\unicode[STIX]{x1D709}_{3}\rangle =0,\end{eqnarray}$$

and consider the associated ALE instanton $(Y,g_{Y},I_{1}^{Y},I_{2}^{Y},I_{3}^{Y})$ . Lemma 1.1 gives an ALE diffeomorphism $\unicode[STIX]{x1D6F7}_{Y}:Y\backslash K\rightarrow (\mathbb{R}^{4}\backslash B)/{\mathcal{D}}_{k}$ , where $K$ is some compact subset of $Y$ and $B$ a ball in $\mathbb{R}^{4}$ centred at the origin; recall that by ‘ALE diffeomorphism’ we mean that for all $\ell \geqslant 0$ ,

$$\begin{eqnarray}|(\unicode[STIX]{x1D6FB}^{\mathbf{e}})^{\ell }({\unicode[STIX]{x1D6F7}_{Y}}_{\ast }g_{Y}-\mathbf{e})|_{\mathbf{e}}=O(r^{-4-\ell }),\end{eqnarray}$$

and likewise on the complex structures. Before using the more specific properties of $\unicode[STIX]{x1D6F7}_{Y}$ at the level of complex structures, let us mention the following: we want to proceed to a gluing of Kähler metrics, and the convenient way of doing so is to glue the Kähler forms, via their potentials. We already have a candidate for the potential of an ALF metric at infinity at hand: as evoked, this would be ${\unicode[STIX]{x1D6F7}_{Y}}_{\ast }\unicode[STIX]{x1D711}^{\flat }$ (see point (iv) in § 1.1). Conversely, we need to kill the ALE metric near infinity, and for this we want a sharp enough potential, in a sense that we make clear below, see Proposition 1.12. We thus need for this a sharp knowledge of the Kähler form $\unicode[STIX]{x1D714}_{1}^{Y}:=g_{Y}(I_{1}^{Y}\cdot ,\cdot )$ , and since we are about to compute $I_{1}^{Y}$ -complex hessians as well, we also need a precise description of the complex structure $I_{1}^{Y}$ . These are given by the following, from which Lemma 1.1 actually follows as we shall see at the end of this section, with the same $\unicode[STIX]{x1D6F7}_{Y}$ .

Lemma 1.6. One can choose the ALE diffeomorphism $\unicode[STIX]{x1D6F7}_{Y}$ such that

(12) $$\begin{eqnarray}{\unicode[STIX]{x1D6F7}_{Y}}_{\ast }\unicode[STIX]{x1D714}_{1}^{Y}=\unicode[STIX]{x1D714}_{1}^{\mathbf{e}}-c(|\unicode[STIX]{x1D709}_{1}|^{2}\unicode[STIX]{x1D703}_{1}+2\langle \unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{2}\rangle \unicode[STIX]{x1D703}_{2}+2\langle \unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{3}\rangle \unicode[STIX]{x1D703}_{3})+O(r^{-8})\end{eqnarray}$$

where $c>0$ is some universal constant, $\unicode[STIX]{x1D703}_{j}=\frac{1}{4}dd_{I_{j}}^{c}(r^{-2})$ , $j=1,2,3$ , on the one hand, and if $\unicode[STIX]{x1D704}_{1}^{Y}$ denotes ${\unicode[STIX]{x1D6F7}_{Y}}_{\ast }I_{1}^{Y}-I_{1}$ , then it is given by:

$$\begin{eqnarray}\mathbf{e}(\unicode[STIX]{x1D704}_{1}^{Y}\cdot ,\cdot )=-c(|\unicode[STIX]{x1D709}_{2}|^{2}+|\unicode[STIX]{x1D709}_{3}|^{2})\frac{rdr\cdot \unicode[STIX]{x1D6FC}_{1}}{r^{6}}+O(r^{-8})\end{eqnarray}$$

where $c$ is the same constant as above and $\unicode[STIX]{x1D6FC}_{1}=I_{1}r\,dr$ , on the other hand.

We can moreover assume that ${\unicode[STIX]{x1D6F7}_{Y}}_{\ast }\unicode[STIX]{x1D6FA}_{Y}=\unicode[STIX]{x1D6FA}_{\mathbf{e}}$ , where $\unicode[STIX]{x1D6FA}_{Y}=\operatorname{vol}^{g_{Y}}$ .

In this statement the error terms $O(r^{-8})$ are understood in the ‘Euclidean way’, namely for any $\ell \geqslant 0$ , the $\ell$ th $\unicode[STIX]{x1D6FB}^{\mathbf{e}}$ -derivatives of these tensors are $O(r^{-8-\ell })$ . This lemma requires further notions on Kronheimer’s construction, and is more precisely a direct application of Theorem 2.1 of Part 2 to $Y=X_{\unicode[STIX]{x1D709}}$ with $\unicode[STIX]{x1D709}$ verifying (11). Notice however the error term order $-8$ , whereas one would expect $-6$ , if one thinks for instance about the Eguchi–Hanson metric [Reference JoyceJoy00, Example 7.2.2]; this estimate is crucial in proving Lemma 1.1, and is specific to (groups containing) dihedral binary groups. Besides, the assertion on the volume forms is only needed in the next paragraph.

Approximating  $\unicode[STIX]{x1D714}_{1}^{Y}$ as an $I_{1}^{Y}$ -complex hessian. We shall see for now how Lemma 1.6 allows us to approximate the Kähler form $\unicode[STIX]{x1D714}_{1}^{Y}$ as an $I_{1}^{Y}$ -complex hessian, with respect to the Taub-NUT metric pushed-forward to $Y$ .

Proposition 1.7. Take $\unicode[STIX]{x1D6F7}_{Y}$ as in Lemmas 1.1 and 1.6, and denote by $\tilde{\mathbf{f}}$ a smooth extension of ${\unicode[STIX]{x1D6F7}_{Y}}^{\ast }\mathbf{f}$ on $Y$ . Then there exists a function $\unicode[STIX]{x1D6F9}$ on $Y$ such that near infinity,

(13) $$\begin{eqnarray}|(\unicode[STIX]{x1D6FB}^{\tilde{\mathbf{f}}})^{\ell }(\unicode[STIX]{x1D714}_{1}^{Y}-dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D6F9})|_{\tilde{\boldsymbol{ f}}}=O(R^{-2}),\quad \ell =0,1,2.\end{eqnarray}$$

More precisely, $\unicode[STIX]{x1D6F9}$ can be decomposed as a sum ${\unicode[STIX]{x1D6F7}_{Y}}^{\ast }\unicode[STIX]{x1D6F9}_{\operatorname{euc}}+{\unicode[STIX]{x1D6F7}_{Y}}^{\ast }\unicode[STIX]{x1D6F9}_{\operatorname{mxd}}$ , where on the one hand, $\unicode[STIX]{x1D6F9}_{\operatorname{euc}}=O(r^{2})$ , $|d\unicode[STIX]{x1D6F9}_{\operatorname{euc}}|_{\mathbf{e}}=O(r)$ , and

(14) $$\begin{eqnarray}|(\unicode[STIX]{x1D6FB}^{\mathbf{e}})^{\ell }(\unicode[STIX]{x1D714}_{1}^{\mathbf{e}}-c|\unicode[STIX]{x1D709}_{1}|^{2}\unicode[STIX]{x1D703}_{1}-dd_{{\unicode[STIX]{x1D6F7}_{Y}}_{\ast }I_{1}^{Y}}^{c}\unicode[STIX]{x1D6F9}_{\operatorname{euc}})|_{\mathbf{e}}=O(r^{-8-\ell })\quad \text{for all }\ell \geqslant 0,\end{eqnarray}$$

and on the other hand, $\unicode[STIX]{x1D6F9}_{\operatorname{mxd}}=O(R^{-1})$ , $|d\unicode[STIX]{x1D6F9}_{\operatorname{mxd}}|_{\mathbf{f}}=O(R^{-1})$ , and

(15) $$\begin{eqnarray}|(\unicode[STIX]{x1D6FB}^{\mathbf{f}})^{\ell }(-2c(\langle \unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{2}\rangle \unicode[STIX]{x1D703}_{2}+\langle \unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{3}\rangle \unicode[STIX]{x1D703}_{3})-dd_{{\unicode[STIX]{x1D6F7}_{Y}}_{\ast }I_{1}^{Y}}^{c}\unicode[STIX]{x1D6F9}_{\operatorname{mxd}})\!|_{\mathbf{f}}=O(R^{-2}),\quad \ell =0,1,2.\end{eqnarray}$$

This lemma takes into account estimates such as $R=O(r^{2})$ and $C^{-1}r^{-2}\mathbf{e}\leqslant \mathbf{f}\leqslant Cr^{2}\mathbf{e}$ of Proposition A.9 at level $\ell =0$ , and follows for positive $\ell$ from explicit computations using the material given in Appendix A, where the proof of Lemma 1.8 is thus postponed.

We hence come to the statements on $\unicode[STIX]{x1D6F9}_{\operatorname{euc}}$ and $\unicode[STIX]{x1D6F9}_{\operatorname{mxd}}$ . We consider before starting a large constant $K$ such that the image of $\unicode[STIX]{x1D6F7}_{Y}$ is contained in both $\{r\geqslant K\}\subset \mathbb{R}^{4}/{\mathcal{D}}_{k}$ and $\{R\geqslant K\}\subset \mathbb{R}^{4}/{\mathcal{D}}_{k}$ , and define a cut-off function $\unicode[STIX]{x1D712}:\mathbb{R}\rightarrow [0,1]$ such that

$$\begin{eqnarray}\unicode[STIX]{x1D712}(t)=\left\{\begin{array}{@{}ll@{}}0\quad & \text{if}~t\leqslant K-1,\\ 1\quad & \text{if}~t\geqslant K,\end{array}\right.\end{eqnarray}$$

which will be useful when defining functions to be pulled-back to $Y$ via $\unicode[STIX]{x1D6F7}_{Y}$ .

The Euclidean component $\unicode[STIX]{x1D6F9}_{\operatorname{euc}}$ . In an asymptotically Euclidean setting, a natural first candidate for the potential of a Kähler form is $1/4r^{2}$ . Now remember we are working with $I_{1}^{Y}$ , or more exactly with ${\unicode[STIX]{x1D6F7}_{Y}}_{\ast }I_{1}^{Y}$ , but we forget about the push-forward here for simplicity of notation; following Lemma 1.6, a straightforward computation gives, near infinity in $\mathbb{R}^{4}$ :

$$\begin{eqnarray}\displaystyle dd_{I_{1}^{Y}}^{c}({\textstyle \frac{1}{4}}r^{2}) & = & \displaystyle {\textstyle \frac{1}{2}}d[(I_{1}+\unicode[STIX]{x1D704}_{1}^{Y})rdr]={\textstyle \frac{1}{2}}d[\unicode[STIX]{x1D6FC}_{1}+c(|\unicode[STIX]{x1D709}_{2}|^{2}+|\unicode[STIX]{x1D709}_{3}|^{2})r^{-4}\unicode[STIX]{x1D6FC}_{1}+O(r^{-7})]\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D714}_{1}^{\mathbf{e}}-c(|\unicode[STIX]{x1D709}_{2}|^{2}+|\unicode[STIX]{x1D709}_{3}|^{2})\unicode[STIX]{x1D703}_{1}+O(r^{-8}),\nonumber\end{eqnarray}$$

where the $O$ are understood in the Euclidean way. On the other hand observe that $I_{1}d(r^{-2})=-2r^{-4}\unicode[STIX]{x1D6FC}_{1}$ , and thus

$$\begin{eqnarray}dd_{I_{1}^{Y}}^{c}(r^{-2})=d[(I_{1}+\unicode[STIX]{x1D704}_{1}^{Y})d(r^{-2})]=d[-2r^{-4}\unicode[STIX]{x1D6FC}_{1}+O(r^{-7})]=4\unicode[STIX]{x1D703}_{1}+O(r^{-8}).\end{eqnarray}$$

Now define

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}_{\operatorname{euc}}={\textstyle \frac{1}{4}}\unicode[STIX]{x1D712}(r)(r^{2}+c(|\unicode[STIX]{x1D709}_{2}|^{2}+|\unicode[STIX]{x1D709}_{3}|^{2}-|\unicode[STIX]{x1D709}_{1}|^{2})r^{-2});\end{eqnarray}$$

on $\mathbb{R}^{4}/{\mathcal{D}}_{k}$ (it is ${\mathcal{D}}_{k}$ -invariant); it has support in the image of $\unicode[STIX]{x1D6F7}_{Y}$ , has the growth stated in the lemma as well as its differential, and by the previous two estimates we get that $\unicode[STIX]{x1D714}_{1}^{\mathbf{e}}-c|\unicode[STIX]{x1D709}_{1}|^{2}\unicode[STIX]{x1D703}_{1}-dd_{{\unicode[STIX]{x1D6F7}_{Y}}_{\ast }I_{1}^{Y}}^{c}\unicode[STIX]{x1D6F9}_{\operatorname{euc}}=O(r^{-8})$ for $\mathbf{e}$ with according decay on the derivatives, as wanted.

The mixed component $\unicode[STIX]{x1D6F9}_{\operatorname{mxd}}$ . The main reason why we could construct $\unicode[STIX]{x1D6F9}_{\operatorname{euc}}$ such as to reach estimates (14) is essentially that $\unicode[STIX]{x1D703}_{1}$ can be realised as an $I_{1}$ -complex hessian, at least away from 0. Now realising $\unicode[STIX]{x1D703}_{2}$ and $\unicode[STIX]{x1D703}_{3}$ as $I_{1}$ -complex hessians as well does not seem possible: see [Reference JoyceJoy00, p. 202] on that matter. Nonetheless, $\unicode[STIX]{x1D703}_{2}$ and $\unicode[STIX]{x1D703}_{3}$ may not be so problematic when looked at via $\mathbf{f}$ . We can indeed approximate them precisely enough with respect to this metric by the $I_{1}$ or $I_{1}^{Y}$ -complex hessians of some well-chosen ${\mathcal{D}}_{k}$ -invariant functions, provided that we partially leave the Euclidean world and use also functions coming from Taub-NUT geometry, e.g. $y_{1}$ and $R$ (hence the previous dichotomy ‘Euclidean/mixed’).

Lemma 1.10. Consider the complex valued function

$$\begin{eqnarray}\unicode[STIX]{x1D713}_{c}:=-2\frac{(y_{2}+iy_{3})\sinh (4my_{1})}{r^{2}R}\end{eqnarray}$$

on $\mathbb{R}^{4}\backslash \{0\}$ . Then near infinity:

1. (i) $|(\unicode[STIX]{x1D6FB}^{\mathbf{f}})^{\ell }\unicode[STIX]{x1D713}_{c}|_{\mathbf{f}}=O(R^{-1})$ for $\ell =0,\ldots ,4$ ;

2. (ii) $|(\unicode[STIX]{x1D6FB}^{\mathbf{f}})^{\ell }(dd_{I_{1}}^{c}\unicode[STIX]{x1D713}_{c}-(\unicode[STIX]{x1D703}_{2}+i\unicode[STIX]{x1D703}_{3}))|_{\mathbf{f}}=O(R^{-2})$ for $\ell =0,1,2$ , and these estimates hold with $I_{1}$ replaced by $I_{1}^{Y}$ as well.

The proof of this crucial lemma is essentially computational, which is why we postpone it to § 1.5; let us just say at this stage that it will appear clearly along this proof that the main point is to identify some function $\unicode[STIX]{x1D713}$ such that $\unicode[STIX]{x2202}\unicode[STIX]{x1D713}/\unicode[STIX]{x2202}y_{1}$ is proportional to $1/r^{4}$ , up to higher order terms, and this turns out to happen precisely for $\unicode[STIX]{x1D713}=\sinh (4my_{1})/r^{2}R$ , see (30) below.

For now set $\unicode[STIX]{x1D713}_{2}=\mathfrak{R}\mathfrak{e}(\unicode[STIX]{x1D713}_{c})$ and $\unicode[STIX]{x1D713}_{3}=\mathfrak{I}\mathfrak{m}(\unicode[STIX]{x1D713}_{c})$ , and define

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}_{\operatorname{mxd}}:=-2c\unicode[STIX]{x1D712}(R)(\langle \unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{2}\rangle \unicode[STIX]{x1D713}_{2}+\langle \unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{3}\rangle \unicode[STIX]{x1D713}_{3}).\end{eqnarray}$$

In view of Lemma 1.10, such a function, defined on the image of $\unicode[STIX]{x1D6F7}_{Y}$ , verifies the growth assertions of Proposition 1.7, as well as the estimates (15): Proposition 1.7 is proved.◻

We are now in position to perform the gluing advertised in point (iv) of the program of § 1.1. This is done in the next section to which the reader may jump directly, since we conclude the current section by the proof of Lemma 1.1, assuming Lemma 1.6 (and more precisely the assertion on $I_{1}^{Y}$ in that statement).

Proof of Lemma 1.1 following Lemma 1.6.

We fix $\unicode[STIX]{x1D6F7}_{Y}$ as in Lemma 1.6; we work on $\mathbb{R}^{4}$ , and to simplify notation we forget about the push-forwards by $\unicode[STIX]{x1D6F7}_{Y}$ .

We are thus looking for a diffeomorphism $\beth$ of $\mathbb{R}^{4}$ such that $|I_{1}^{Y}-\beth ^{\ast }I_{1}|_{\mathbf{e}}=O(r^{-8})$ , with according decay on Euclidean derivatives, until the end of this proof we forget about ALF geometry and stick to the Euclidean setting; we will thus content ourselves with using $O$ in this Euclidean meaning. An explicit formula is given for $\beth$ in the statement of Lemma 1.1, which is $\beth :(z_{1},z_{2})\mapsto (1+(a/r^{4}))(z_{1},z_{2})$ with $(z_{1},z_{2})$ the standard complex coordinates on $(\mathbb{C}^{2},I_{1})$ ; up to determining the value of the constant $a$ , we could thus simply check that such a $\beth$ meets our requirement, in light of the asymptotics for $I_{1}^{Y}$ stated in Lemma 1.6.

We prefer nonetheless the following more constructive approach. In terms of Kodaira–Spencer theory, if we set $\beth =\operatorname{id}_{\mathbb{C}^{2}}+\unicode[STIX]{x1D700}$ , seeing thus $\unicode[STIX]{x1D700}=(\unicode[STIX]{x1D700}_{1}(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}z_{1}),\unicode[STIX]{x1D700}_{2}(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}z_{2}))$ as the direction of a deformation of $\operatorname{id}_{\mathbb{C}^{2}}$ , the condition $I_{1}^{Y}-\beth ^{\ast }I_{1}=O(r^{-8})$ becomes, neglecting the $O(r^{-8})$ error term, $\overline{\unicode[STIX]{x2202}}_{I_{1}}\unicode[STIX]{x1D700}=\unicode[STIX]{x1D704}_{1}^{Y}$ . Now $\unicode[STIX]{x1D704}_{1}^{Y}$ is an $\mathbf{e}$ -harmonic $I_{1}\text{-}(0,1)$ form with values in $T^{(1,0)}\mathbb{C}^{2}$ and with $r^{-4}$ -decay; a multiple of the $(1,0)$ -gradient of the Green function $1/r^{2}$ is thus the best-placed candidate for the role of $\unicode[STIX]{x1D700}$ . This actually works, with the choice

$$\begin{eqnarray}\unicode[STIX]{x1D700}=\frac{c}{8}(|\unicode[STIX]{x1D709}_{2}|^{2}+|\unicode[STIX]{x1D709}_{3}|^{2})\biggl[\text{grad}^{\mathbf{e}}\biggl(\frac{1}{r^{2}}\biggr)\biggr]^{1,0},\end{eqnarray}$$

that is

$$\begin{eqnarray}\unicode[STIX]{x1D700}_{1}=-\frac{c(|\unicode[STIX]{x1D709}_{2}|^{2}+|\unicode[STIX]{x1D709}_{3}|^{2})z_{1}}{4r^{4}}\quad \text{and}\quad \unicode[STIX]{x1D700}_{2}=-\frac{c(|\unicode[STIX]{x1D709}_{2}|^{2}+|\unicode[STIX]{x1D709}_{3}|^{2})z_{2}}{4r^{4}},\end{eqnarray}$$

with $c$ the constant given by Lemma 1.6.

The last point to be dealt with is $\beth =\operatorname{id}_{\mathbb{C}^{2}}+(\unicode[STIX]{x1D700}_{1},\unicode[STIX]{x1D700}_{2})$ being a diffeomorphism between infinities of $\mathbb{C}^{2}$ ; we leave it to the reader as an easy exercise.

The estimate $\beth ^{\ast }\unicode[STIX]{x1D6FA}_{\mathbf{e}}-\unicode[STIX]{x1D6FA}_{\mathbf{e}}=O(r^{-8})$ amounts to seeing that $\mathfrak{R}\mathfrak{e}(\unicode[STIX]{x2202}\unicode[STIX]{x1D700}_{1}/\unicode[STIX]{x2202}z_{1}+\unicode[STIX]{x2202}\unicode[STIX]{x1D700}_{2}/\unicode[STIX]{x2202}z_{2})=O(r^{-8})$ : extend $\operatorname{id}(z_{1}+\unicode[STIX]{x1D700}_{1})\wedge d(\overline{z_{1}}+\overline{\unicode[STIX]{x1D700}_{1}})\wedge \operatorname{id}(z_{2}+\unicode[STIX]{x1D700}_{2})\wedge d(\overline{z_{2}}+\overline{\unicode[STIX]{x1D700}_{2}})$ , and look at the linear terms in $\unicode[STIX]{x1D700}_{1}$ , $\unicode[STIX]{x1D700}_{2}$ . Since after multiplication by $a:=-c(|\unicode[STIX]{x1D709}_{2}|^{2}+|\unicode[STIX]{x1D709}_{3}|^{2})/4$ the error would again be $O(r^{-8})$ , we can do this computation with $z_{1}/r^{4}$ and $z_{2}/r^{4}$ playing the respective roles of $\unicode[STIX]{x1D700}_{1}$ and $\unicode[STIX]{x1D700}_{2}$ . Now $(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}z_{j})(z_{j}/r^{4})=1/r^{4}-2|z_{j}|^{2}/r^{6}$ , $j=1,2$ . Since these are real, we only need to compute the sum $(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}z_{1})(z_{1}/r^{4})+(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}z_{2})(z_{2}/r^{4})$ , which is $2/r^{4}-2|z_{1}|^{2}/r^{6}-2|z_{2}|^{2}/r^{6}=0$ .

The ${\mathcal{D}}_{k}$ -invariance of $\beth$ thus constituted is clear.◻

1.3.2 The gluing

We keep the notation of the previous section: $(Y,g_{Y},(I_{j}^{Y})_{j=1,2,3})$ is a ${\mathcal{D}}_{k}$ -ALE instanton with parameter $\unicode[STIX]{x1D709}$ verifying (11), $\unicode[STIX]{x1D6F7}_{Y}$ an asymptotic isometry between infinities of $Y$ and $\mathbb{R}^{4}/{\mathcal{D}}_{k}$ fixed by Lemma 1.6, and $\beth$ is given by Lemma 1.1 which we may also see as as diffeomorphism of $(\mathbb{R}^{4}\backslash \{0\})/{\mathcal{D}}_{k}$ .

As alluded to above, the form we want to glue $\unicode[STIX]{x1D714}_{1}^{Y}=g_{Y}(I_{1}^{Y}\cdot ,\cdot )$ with at infinity is $dI_{1}^{Y}d\unicode[STIX]{x1D711}^{\flat }$ , where $\unicode[STIX]{x1D711}^{\flat }=\beth ^{\ast }\unicode[STIX]{x1D711}$ , with $\unicode[STIX]{x1D711}=\unicode[STIX]{x1D711}_{m}$ LeBrun’s $I_{1}$ -potential for $\mathbf{f}$ given by (6). We set likewise $\mathbf{f}^{\flat }=\beth ^{\ast }\mathbf{f}$ , both on $\mathbb{R}^{4}$ and its quotient. Recall that $\unicode[STIX]{x1D6F9}=\unicode[STIX]{x1D6F9}_{\operatorname{euc}}+\unicode[STIX]{x1D6F9}_{\operatorname{mxd}}$ is defined in Proposition 1.7 as an approximate $I_{1}^{Y}$ -complex potential of $\unicode[STIX]{x1D714}_{1}^{Y}$ . The next proposition explains how to glue $dI_{1}^{Y}d\unicode[STIX]{x1D711}^{\flat }$ to $\unicode[STIX]{x1D714}_{1}^{Y}$ , so as to obtain an ALF metric on $Y$ at the end; as we need it below, let us mention here that we see $\unicode[STIX]{x1D6F9}_{\text{mxd}}$ on $Y$ via $\unicode[STIX]{x1D6F7}_{Y}$ , and we more precisely consider a smooth extension of $\unicode[STIX]{x1D6F7}_{Y}^{\ast }\unicode[STIX]{x1D6F9}_{\text{mxd}}$ , which is assumed to be bounded in absolute value by $\frac{1}{2}$ .

Proposition 1.12. Take $K\geqslant 0$ so that the identification $\unicode[STIX]{x1D6F7}_{Y}$ between infinities of $\mathbb{R}^{4}/{\mathcal{D}}_{k}$ and $Y$ , as well as the diffeomorphism $\beth$ , are defined on $\unicode[STIX]{x1D711}\geqslant K$ . Consider $r_{0},R_{0}\gg 1$ , $\unicode[STIX]{x1D6FD}\in (0,1]$ and set

(16) $$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{m}^{\flat }=\unicode[STIX]{x1D6F7}_{\text{a lf}}^{+}-\unicode[STIX]{x1D6F7}_{\text{a le}}-\unicode[STIX]{x1D6F7}_{\text{alf}}^{-},\end{eqnarray}$$

where

(17) $$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}\unicode[STIX]{x1D6F7}_{\text{alf}}^{+}=\unicode[STIX]{x1D6F7}_{\text{alf}}^{+}(\,\cdot \,;\unicode[STIX]{x1D6FD},K)=\unicode[STIX]{x1D705}\circ (\unicode[STIX]{x1D711}^{\flat }+\unicode[STIX]{x1D6FD}^{-1}(\unicode[STIX]{x1D711}^{\flat })^{1-\unicode[STIX]{x1D6FD}}-\unicode[STIX]{x1D6F9}_{\operatorname{mxd}}-(K+\unicode[STIX]{x1D6FD}^{-1}K^{1-\unicode[STIX]{x1D6FD}})),\quad \\ \unicode[STIX]{x1D6F7}_{\text{ale}}=\unicode[STIX]{x1D6F7}_{\text{ale}}(\,\cdot \,;r_{0})=\unicode[STIX]{x1D712}(r-r_{0})\unicode[STIX]{x1D6F9}_{\operatorname{euc}},\quad \\ \unicode[STIX]{x1D6F7}_{\text{alf}}^{-}=\unicode[STIX]{x1D6F7}_{\text{alf}}^{-}(\,\cdot \,;\unicode[STIX]{x1D6FD},R_{0})=\unicode[STIX]{x1D712}\biggl({\displaystyle \frac{R-R_{0}}{R_{0}}}\biggr)\unicode[STIX]{x1D6FD}^{-1}(\unicode[STIX]{x1D711}^{\flat })^{1-\unicode[STIX]{x1D6FD}},\quad \end{array}\right.\end{eqnarray}$$

with $\unicode[STIX]{x1D705}:\mathbb{R}\rightarrow \mathbb{R}$ a convex function which is constant on $(-\infty ,\frac{1}{2}]$ and equal to $\operatorname{id}_{\mathbb{R}}$ on $[1,\infty )$ and $\unicode[STIX]{x1D712}$ the cut-off function $d\unicode[STIX]{x1D705}/dt$ . Then if the parameters $K$ , $r_{0}$ and $R_{0}$ (respectively $\unicode[STIX]{x1D6FD}$ ) are chosen large enough (respectively small enough), the symmetric 2-tensor $g_{m}$ associated via $I_{1}^{Y}$ to the $I_{1}^{Y}$ -(1, 1)-form

$$\begin{eqnarray}\unicode[STIX]{x1D714}_{m}:=\unicode[STIX]{x1D714}_{1}^{Y}+dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D6F7}_{m}^{\flat }\end{eqnarray}$$

is well defined on the whole $Y,$ is a Kähler metric for $I_{1}^{Y}$ , is ALF in the sense that $|(\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }})^{\ell }(g_{m}-\mathbf{f}^{\flat })|_{\mathbf{f}^{\flat }}=O(R^{-2})$ for $\ell =0,1,2$ , and its volume form $\unicode[STIX]{x1D6FA}_{m}$ verifies

(18) $$\begin{eqnarray}|(\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }})^{\ell }(\unicode[STIX]{x1D6FA}_{m}-\unicode[STIX]{x1D6FA}_{Y})|_{\mathbf{f}^{\flat }}=O(R^{-2})\end{eqnarray}$$

for $\ell =0,1,2$ , where $\unicode[STIX]{x1D6FA}_{Y}$ is the volume form of the ALE metric $g_{Y}$ .

The proof of this lemma is postponed to § 1.5.2, as we focus on that of Proposition 1.12.

Step 1.1. For now, we work first with the parameter $\unicode[STIX]{x1D6FD}$ in (17) equal to 1, and consider the closed $I_{1}^{Y}$ -hermitian form $dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D6F7}_{\text{alf}}^{+}=dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D705}\circ (\unicode[STIX]{x1D711}^{\flat }-\unicode[STIX]{x1D6F9}_{\operatorname{mxd}}-K)$ on $Y$ . Even though $K$ is not fixed yet, this form is equal to $dd_{I_{1}^{Y}}^{c}(\unicode[STIX]{x1D711}^{\flat }-\unicode[STIX]{x1D6F9}_{\operatorname{mxd}})$ on $\{\unicode[STIX]{x1D711}^{\flat }-\unicode[STIX]{x1D6F9}_{\operatorname{mxd}}\geqslant K+1\}$ seen on $Y$ via $\unicode[STIX]{x1D6F7}_{Y}$ ; this is possible for $K$ large enough since $\unicode[STIX]{x1D711}^{\flat }-\unicode[STIX]{x1D6F9}_{\operatorname{mxd}}$ is proper on $\mathbb{R}^{4}$ as $\unicode[STIX]{x1D711}^{\flat }\geqslant \beth ^{\ast }R\sim R$ (by Lemma 1.14) and $\unicode[STIX]{x1D6F9}_{\operatorname{mxd}}=O(R^{-1})$ . Moreover $\unicode[STIX]{x1D705}$ is convex, and thus $dd_{I_{1}^{Y}}^{c}[\unicode[STIX]{x1D705}\circ (\unicode[STIX]{x1D711}^{\flat }-\unicode[STIX]{x1D6F9}_{\operatorname{mxd}}-K)]$ is non-negative wherever $dd_{I_{1}^{Y}}^{c}(\unicode[STIX]{x1D711}^{\flat }-\unicode[STIX]{x1D6F9}_{\operatorname{mxd}})$ is, which we claim is the case near infinity. Since indeed $|dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D6F9}_{\operatorname{mxd}}|_{\mathbf{f}}=O(R^{-1})$ , our claim will be checked if we prove the estimate

(19) $$\begin{eqnarray}|dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D711}^{\flat }-{\textstyle \frac{1}{2}}[\mathbf{f}^{\flat }(I_{1}^{Y}\cdot ,\cdot )-\mathbf{f}^{\flat }(\cdot ,I_{1}^{Y}\cdot )]|_{\boldsymbol{ f}^{\flat }}=O(R^{-2}),\end{eqnarray}$$

as $\unicode[STIX]{x1D71B}_{\mathbf{f}^{\flat }}:=\frac{1}{2}[\mathbf{f}^{\flat }(I_{1}^{Y}\cdot ,\cdot )-\mathbf{f}^{\flat }(\cdot ,I_{1}^{Y}\cdot )]$ is nothing but the $I_{1}^{Y}$ -hermitian form associated to the $I_{1}^{Y}$ -hermitian metric $\frac{1}{2}[\mathbf{f}^{\flat }+\mathbf{f}^{\flat }(I_{1}^{Y}\cdot ,I_{1}^{Y}\cdot )]$ ; notice $\unicode[STIX]{x1D71B}_{\mathbf{f}^{\flat }}$ is not closed in general. Pushing-forward by $\beth$ , proving estimate (19) amounts to seeing that

$$\begin{eqnarray}|dd_{\beth _{\ast }I_{1}^{Y}}^{c}\unicode[STIX]{x1D711}-{\textstyle \frac{1}{2}}[\mathbf{f}(\beth _{\ast }I_{1}^{Y}\cdot ,\cdot )-\mathbf{f}(\cdot ,\beth _{\ast }I_{1}^{Y}\cdot )]|_{\boldsymbol{ f}^{\flat }}=O(R^{-2}).\end{eqnarray}$$

Now $dd_{\beth _{\ast }I_{1}^{Y}}^{c}\unicode[STIX]{x1D711}=d\beth _{\ast }I_{1}^{Y}d\unicode[STIX]{x1D711}=\unicode[STIX]{x1D714}_{\mathbf{f}}+d\jmath d\unicode[STIX]{x1D711}$ , where $\unicode[STIX]{x1D714}_{\mathbf{f}}=\mathbf{f}(I_{1}\cdot ,\cdot )$ and $\jmath =\beth _{\ast }I_{1}^{Y}-I_{1}$ . Let us estimate $d\jmath d\unicode[STIX]{x1D711}$ ; by Lemma 1.1 and by Lemma 1.8, for all $\ell =0,1$ , $|(\unicode[STIX]{x1D6FB}^{\mathbf{f}})^{\ell }\jmath |_{\mathbf{f}}=O(R^{-3})$ , whereas $|(\unicode[STIX]{x1D6FB}^{\mathbf{f}})^{\ell }\unicode[STIX]{x1D711}|_{\mathbf{f}}=O(R^{2-\ell })$ , $\ell =1,2$ ; therefore $|d\jmath d\unicode[STIX]{x1D711}|_{\mathbf{f}}=O(R^{-2})$ . On the other hand, still from $\beth _{\ast }I_{1}^{Y}=I_{1}+\jmath$ , $\mathbf{f}(\beth _{\ast }I_{1}^{Y}\cdot ,\cdot )-\mathbf{f}(\cdot ,\beth _{\ast }I_{1}^{Y}\cdot )=2\unicode[STIX]{x1D714}_{\mathbf{f}}+\mathbf{f}(\jmath \cdot ,\cdot )-\mathbf{f}(\cdot ,\jmath \cdot )$ . The error term $\mathbf{f}(\jmath \cdot ,\cdot )-\mathbf{f}(\cdot ,\jmath \cdot )$ is controlled by $|\jmath |_{\mathbf{f}}$ , which is $O(R^{-3})$ . We have thus proved estimate (19). Thanks to the general formal formula

(20) $$\begin{eqnarray}\unicode[STIX]{x1D6FB}^{g+h}T=\unicode[STIX]{x1D6FB}^{g}T+(g+h)^{-1}\ast \unicode[STIX]{x1D6FB}^{g}h\ast T,\end{eqnarray}$$

(see e.g. [Reference Gursky and ViaclovskyGV16], formula (3.39)) for any metrics $g$ and $g+h$ ( $h$ is thus seen here as a perturbation) and any tensor $T$ , with Lemma 1.14 we can take $g=\mathbf{f}$ , $g+h=\mathbf{f}^{\flat }$ and $T$ the tensor in play, and prove with the same techniques an estimate similar to (19) up to order 2, that is

$$\begin{eqnarray}|(\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }})^{\ell }(dd_{\beth _{\ast }I_{1}^{Y}}^{c}\unicode[STIX]{x1D711}-\unicode[STIX]{x1D71B}_{\boldsymbol{ f}^{\flat }})|_{\mathbf{f}^{\flat }}=O(R^{-2}),\end{eqnarray}$$

for $\ell =1,2$ ; this also uses $|(\unicode[STIX]{x1D6FB}^{\mathbf{f}})^{\ell }\unicode[STIX]{x1D711}|_{\mathbf{f}}=O(R^{-2-\ell })$ for $\ell =0,\ldots ,4$ , and $|(\unicode[STIX]{x1D6FB}^{\mathbf{f}})^{\ell }\jmath |_{\mathbf{f}}=O(R^{-3})$ for $\ell =0,\ldots ,4$ , which follows from Lemma 1.8. If therefore $K$ is chosen large enough, and taking moreover the contribution of $\unicode[STIX]{x1D6F9}_{\operatorname{mxd}}$ into account, $\unicode[STIX]{x1D714}_{1}^{Y}+dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D705}\circ (\unicode[STIX]{x1D711}^{\flat }-\unicode[STIX]{x1D6F9}_{\operatorname{mxd}}-K)$ is well defined and is an $I_{1}^{Y}$ -Kähler form, and is equal to $(\unicode[STIX]{x1D714}_{1}^{Y}-dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D6F9}_{\operatorname{mxd}})+\unicode[STIX]{x1D71B}_{\mathbf{f}^{\flat }}$ up to a $O(R^{-2})$ error at orders 0, 1 and 2 for $\mathbf{f}^{\flat }$ .

Step 1.2. We can now deal with the $\unicode[STIX]{x1D6FD}<1$ case (our intention is to make $\unicode[STIX]{x1D6FD}$ small); as

$$\begin{eqnarray}dd_{I_{1}^{Y}}^{c}(\unicode[STIX]{x1D6FD}^{-1}(\unicode[STIX]{x1D711}^{\flat })^{1-\unicode[STIX]{x1D6FD}})=(1-\unicode[STIX]{x1D6FD})(\unicode[STIX]{x1D711}^{\flat })^{-\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6FD}^{-1}dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D711}^{\flat }-(\unicode[STIX]{x1D711}^{\flat })^{-1}d\unicode[STIX]{x1D711}^{\flat }\wedge d_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D711}^{\flat }),\end{eqnarray}$$

as $\unicode[STIX]{x1D711}^{\flat }$ grows at least like $m(R^{\flat })^{2}$ , and as $d\unicode[STIX]{x1D711}^{\flat }$ and $d_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D711}^{\flat }$ are $O(R^{\flat })$ for $\mathbf{f}^{\flat }$ according to what precedes, we have $|(\unicode[STIX]{x1D711}^{\flat })^{-1}d\unicode[STIX]{x1D711}^{\flat }\wedge d_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D711}^{\flat }|_{\mathbf{f}^{\flat }}\leqslant A$ near infinity for some $A>0$ (independent of $K,\unicode[STIX]{x1D6FD},R_{0},r_{0}$ ), whereas $dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D711}^{\flat }\geqslant \frac{1}{2}\unicode[STIX]{x1D71B}_{\mathbf{f}^{\flat }}$ near infinity. This way, assuming that $\unicode[STIX]{x1D6FD}\in (0,A^{-1})$ ,

$$\begin{eqnarray}dd_{I_{1}^{Y}}^{c}(\unicode[STIX]{x1D6FD}^{-1}(\unicode[STIX]{x1D711}^{\flat })^{1-\unicode[STIX]{x1D6FD}})\geqslant (1-\unicode[STIX]{x1D6FD})(\unicode[STIX]{x1D711}^{\flat })^{-\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6FD}^{-1}-A)dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D711}^{\flat }\geqslant {\textstyle \frac{1}{2}}(1-\unicode[STIX]{x1D6FD})(\unicode[STIX]{x1D711}^{\flat })^{-\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6FD}^{-1}-A)\unicode[STIX]{x1D71B}_{\boldsymbol{ f}^{\flat }}\end{eqnarray}$$

near infinity, and, more precisely, as soon as $|(\unicode[STIX]{x1D711}^{\flat })^{-1}d\unicode[STIX]{x1D711}^{\flat }\wedge d_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D711}^{\flat }|_{\mathbf{f}^{\flat }}\leqslant A$ and $dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D711}^{\flat }\geqslant \frac{1}{2}\unicode[STIX]{x1D71B}_{\mathbf{f}^{\flat }}$ , hence on $\{\unicode[STIX]{x1D711}^{\flat }\geqslant K\}$ , say; we fix such a  $K$ once and for all. Therefore, as $|\unicode[STIX]{x1D6F9}_{\text{mxd}}|\leqslant \frac{1}{2}$ on $Y$ :

• ∙ $dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D6F7}_{\text{alf}}^{+}$ is trivial where $(\unicode[STIX]{x1D711}^{\flat }+\unicode[STIX]{x1D6FD}^{-1}(\unicode[STIX]{x1D711}^{\flat })^{1-\unicode[STIX]{x1D6FD}}-\unicode[STIX]{x1D6F9}_{\text{mxd}}-K-\unicode[STIX]{x1D6FD}^{-1}K^{1-\unicode[STIX]{x1D6FD}})<\frac{1}{2}$ ;

• ∙ on $\{(\unicode[STIX]{x1D711}^{\flat }+\unicode[STIX]{x1D6FD}^{-1}(\unicode[STIX]{x1D711}^{\flat })^{1-\unicode[STIX]{x1D6FD}}-\unicode[STIX]{x1D6F9}_{\text{mxd}}-K-\unicode[STIX]{x1D6FD}^{-1}K^{1-\unicode[STIX]{x1D6FD}})\geqslant \frac{1}{2}\}\subset \{\unicode[STIX]{x1D711}^{\flat }\geqslant K\}$ (this inclusion being independent of  $\unicode[STIX]{x1D6FD}$ ), $dd_{I_{1}^{Y}}^{c}[\unicode[STIX]{x1D711}^{\flat }+\unicode[STIX]{x1D6FD}^{-1}(\unicode[STIX]{x1D711}^{\flat })^{1-\unicode[STIX]{x1D6FD}}-\unicode[STIX]{x1D6F9}_{\text{mxd}}]\geqslant \frac{1}{2}\unicode[STIX]{x1D71B}_{\mathbf{f}^{\flat }}+(\unicode[STIX]{x1D6FD}^{-1}-A)(\unicode[STIX]{x1D711}^{\flat })^{-1}\unicode[STIX]{x1D71B}_{\mathbf{f}^{\flat }}>0$ , and thus $dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D6F7}_{\text{alf}}^{+}\geqslant 0$ . More precisely, on $\{\unicode[STIX]{x1D711}^{\flat }\geqslant K+2\}$ , where $(\unicode[STIX]{x1D711}^{\flat }+\unicode[STIX]{x1D6FD}^{-1}(\unicode[STIX]{x1D711}^{\flat })^{1-\unicode[STIX]{x1D6FD}}-\unicode[STIX]{x1D6F9}_{\text{mxd}}-K-\unicode[STIX]{x1D6FD}^{-1}K^{1-\unicode[STIX]{x1D6FD}})\geqslant 1$ , $dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D6F7}_{\text{alf}}^{+}=dd_{I_{1}^{Y}}^{c}[\unicode[STIX]{x1D711}^{\flat }+\unicode[STIX]{x1D6FD}^{-1}(\unicode[STIX]{x1D711}^{\flat })^{1-\unicode[STIX]{x1D6FD}}-\unicode[STIX]{x1D6F9}_{\text{mxd}}]$ . Thus $\unicode[STIX]{x1D714}_{1}^{Y}+dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D6F7}_{\text{alf}}^{+}\geqslant \unicode[STIX]{x1D714}_{1}^{Y}$ on $Y$ , refined as $\unicode[STIX]{x1D714}_{1}^{Y}+dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D6F7}_{\text{alf}}^{+}\geqslant \unicode[STIX]{x1D714}_{1}^{Y}+\frac{1}{2}(1+(1-\unicode[STIX]{x1D6FD})(\unicode[STIX]{x1D6FD}^{-1}-A)(\unicode[STIX]{x1D711}^{\flat })^{-\unicode[STIX]{x1D6FD}})\unicode[STIX]{x1D71B}_{\boldsymbol{ f}^{\flat }}$ on $\{\unicode[STIX]{x1D711}^{\flat }\geqslant K+2\}$ ;

• ∙ $\unicode[STIX]{x1D714}_{1}^{Y}+dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D6F7}_{\text{alf}}^{+}$ is furthermore asymptotic to

  (21) $$\begin{eqnarray}(\unicode[STIX]{x1D714}_{1}^{Y}-dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D6F9}_{\text{mxd}})+\unicode[STIX]{x1D71B}_{\mathbf{f}^{\flat }}+\unicode[STIX]{x1D6FD}^{-1}dd_{I_{1}^{Y}}^{c}(\unicode[STIX]{x1D711}^{\flat })^{1-\unicode[STIX]{x1D6FD}},\end{eqnarray}$$
  with an error of size $O(R^{-2})$ , as well as its first and second $\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }}$ -derivatives.

Notice that at this stage, we have not used $\unicode[STIX]{x1D6F9}_{\operatorname{mxd}}$ , but instead only checked it did not impede us to reach a desired positivity assertion on $\unicode[STIX]{x1D714}_{1}^{Y}+dd^{c}\unicode[STIX]{x1D6F7}_{\text{alf}}^{+}$ .

Step 2. We now deal with the summand $-\unicode[STIX]{x1D6F7}_{\text{ale}}$ of $\unicode[STIX]{x1D6F7}_{m}^{\flat }$ , which is meant to kill the ALE part of the Kähler form $\unicode[STIX]{x1D714}_{1}^{Y}+dd^{c}\unicode[STIX]{x1D6F7}_{\text{alf}}^{+}$ , or equivalently of the $I_{1}^{Y}$ -hermitian form $(\unicode[STIX]{x1D714}_{1}^{Y}-dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D6F9}_{\operatorname{mxd}})+\unicode[STIX]{x1D71B}_{\mathbf{f}^{\flat }}$ (we see the remaining terms in (21) as ALF perturbations, corrected below with the help of $\unicode[STIX]{x1D6F7}_{\text{alf}}^{-}$ ). There are again two issues here: the positivity of the resulting $I_{1}^{Y}$ -(1, 1) form $\unicode[STIX]{x1D714}_{1}^{Y}+dd_{I_{1}^{Y}}^{c}(\unicode[STIX]{x1D6F7}_{\text{alf}}^{+}-\unicode[STIX]{x1D6F7}_{\text{ale}})$ on $Y$ , and its asymptotics. The asymptotics are independent of $r_{0}$ , $R_{0}$ and, to a certain extent, $\unicode[STIX]{x1D6FD}$ , since we are only looking at what happens near infinity. Indeed, for any value of these parameters, and provided that $r_{0}$ (respectively $R_{0}$ ) is chosen much larger than $K$ (respectively $r_{0}$ ), we have $\{R\geqslant R_{0}\}\subset \{r\geqslant r_{0}+1\}\subset \{\unicode[STIX]{x1D711}^{\flat }\geqslant K+2\}$ and on that region, by definition of $\unicode[STIX]{x1D6F7}_{\text{alf}}^{+}$ and $\unicode[STIX]{x1D6F7}_{\text{ale}}$ ,

$$\begin{eqnarray}\unicode[STIX]{x1D714}_{1}^{Y}+dd_{I_{1}^{Y}}^{c}(\unicode[STIX]{x1D6F7}_{\text{a lf}}^{+}-\unicode[STIX]{x1D6F7}_{\text{a le}})=(\unicode[STIX]{x1D714}_{1}^{Y}-dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D6F9})+dd_{I_{1}^{Y}}^{c}(\unicode[STIX]{x1D711}^{\flat }+\unicode[STIX]{x1D6FD}^{-1}(\unicode[STIX]{x1D711}^{\flat })^{1-\unicode[STIX]{x1D6FD}}),\end{eqnarray}$$

with that $\unicode[STIX]{x1D6F9}=\unicode[STIX]{x1D6F9}_{\operatorname{mxd}}+\unicode[STIX]{x1D6F9}_{\operatorname{euc}}$ of Proposition 1.7. The term $(\unicode[STIX]{x1D714}_{1}^{Y}-dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D6F9})$ in the right-hand side is thus $O(R^{-2})$ for $\mathbf{f}$ by this proposition, which is exactly where $\unicode[STIX]{x1D6F9}_{\operatorname{mxd}}$ , as well as $\unicode[STIX]{x1D6F9}_{\operatorname{euc}}$ as we defined it, are used; again, these asymptotics hold for $\mathbf{f}^{\flat }$ by Lemma 1.14. We have already dealt with the asymptotics of $dd_{I_{1}^{Y}}^{c}(\unicode[STIX]{x1D711}^{\flat }+\unicode[STIX]{x1D6FD}^{-1}(\unicode[STIX]{x1D711}^{\flat })^{1-\unicode[STIX]{x1D6FD}})$ in the previous step. As in (21), we hence have that $\unicode[STIX]{x1D714}_{1}^{Y}+dd_{I_{1}^{Y}}^{c}(\unicode[STIX]{x1D6F7}_{\text{alf}}^{+}-\unicode[STIX]{x1D6F7}_{\text{ale}})$ is asymptotic to $\unicode[STIX]{x1D71B}_{\mathbf{f}^{\flat }}+\unicode[STIX]{x1D6FD}^{-1}dd_{I_{1}^{Y}}^{c}(\unicode[STIX]{x1D711}^{\flat })^{1-\unicode[STIX]{x1D6FD}}$ , with an error of size $O(R^{-2})$ for $\mathbf{f}^{\flat }$ up to its second $\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }}$ -derivatives.

We are therefore left with the positivity assertion, which has to be proved carefully since we essentially have to subtract a metric to another one; this is where we fix $r_{0}$ and $\unicode[STIX]{x1D6FD}$ . This boils down to the following:

• ∙ take $r_{0}$ so that on $r\geqslant r_{0}$ , $dd_{I_{1}^{Y}}^{c}(\unicode[STIX]{x1D711}^{\flat }+\unicode[STIX]{x1D6FD}^{-1}(\unicode[STIX]{x1D711}^{\flat })^{1-\unicode[STIX]{x1D6FD}}-\unicode[STIX]{x1D6F9}_{\operatorname{mxd}})\geqslant \frac{1}{2}[1+(1-\unicode[STIX]{x1D6FD})(\unicode[STIX]{x1D6FD}^{-1}-A)(\unicode[STIX]{x1D711}^{\flat })^{-\unicode[STIX]{x1D6FD}}]\unicode[STIX]{x1D71B}_{\boldsymbol{ f}^{\flat }}$ (as underlined above, this can be done independently of  $\unicode[STIX]{x1D6FD}$ );

• ∙ consider $\unicode[STIX]{x1D714}_{1}^{Y}-dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D6F7}_{\text{ale}}$ , rewritten as $[1-\unicode[STIX]{x1D712}(r-r_{0})]\unicode[STIX]{x1D714}_{1}^{Y}+\unicode[STIX]{x1D712}(r-r_{0})(\unicode[STIX]{x1D714}_{1}^{Y}-dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D6F9}_{\text{euc}})+{\mathcal{R}}_{r_{0}}$ , where ${\mathcal{R}}_{r_{0}}=\unicode[STIX]{x1D712}^{\prime }(r-r_{0})(\unicode[STIX]{x1D6F9}_{\text{euc}}dd_{I_{1}^{Y}}^{c}r+dr\wedge d_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D6F9}_{\text{euc}}+d\unicode[STIX]{x1D6F9}_{\text{euc}}\wedge d_{I_{1}^{Y}}^{c}r)+\unicode[STIX]{x1D712}^{\prime \prime }(r-r_{0})\unicode[STIX]{x1D6F9}_{\text{euc}}dr\wedge d_{I_{1}^{Y}}^{c}r$ has support in $\{r_{0}\leqslant r\leqslant r_{0}+1\}$ ;

• ∙ as $|\unicode[STIX]{x1D714}_{1}^{Y}-dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D6F9}_{\text{euc}}|_{\mathbf{e}}=O(r^{-4})$ , we get $|\unicode[STIX]{x1D714}_{1}^{Y}-dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D6F9}_{\text{euc}}|_{\mathbf{f}^{\flat }}=O(R^{-1})$ by Lemma 1.8; we can thus fix  $r_{0}$  once and for all so that $|\unicode[STIX]{x1D712}(r-r_{0})(\unicode[STIX]{x1D714}_{1}^{Y}-dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D6F9}_{\text{euc}})|_{\mathbf{f}^{\flat }}\leqslant \frac{1}{6}\unicode[STIX]{x1D71B}_{\mathbf{f}^{\flat }}$ on $Y$ ;

• ∙ we now fix  $\unicode[STIX]{x1D6FD}>0$ small enough so that $(1-\unicode[STIX]{x1D6FD})(\unicode[STIX]{x1D6FD}^{-1}-A)(\unicode[STIX]{x1D711}^{\flat })^{-\unicode[STIX]{x1D6FD}}\geqslant 6\sup _{r_{0}\leqslant r\leqslant r_{0}+1}|{\mathcal{R}}_{r_{0}}|_{\mathbf{f}^{\flat }}$ ; this way, $\unicode[STIX]{x1D714}_{1}^{Y}+dd_{I_{1}^{Y}}^{c}(\unicode[STIX]{x1D6F7}_{\text{alf}}^{+}-\unicode[STIX]{x1D6F7}_{\text{ale}})$ , which equals $\unicode[STIX]{x1D714}_{1}^{Y}+dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D6F7}_{\text{alf}}^{+}>0$ on $Y\backslash \{r\geqslant r_{0}\}$ , is bounded below by $(\frac{1}{2}-\frac{1}{6}-\frac{1}{6})\unicode[STIX]{x1D71B}_{\mathbf{f}^{\flat }}>0$ on $\{r_{0}\leqslant r\leqslant r_{0}+1\}$ , and by $(\frac{1}{2}-\frac{1}{6})\unicode[STIX]{x1D71B}_{\mathbf{f}^{\flat }}>0$ on $\{r\geqslant r_{0}+1\}$ .

Step 3. We conclude by analysing the term $\unicode[STIX]{x1D6F7}_{\text{alf}}^{-}$ of $\unicode[STIX]{x1D6F7}_{m}^{\flat }$ , and by fixing $R_{0}$ . First, by the known asymptotics on $\unicode[STIX]{x1D714}_{1}^{Y}+dd_{I_{1}^{Y}}^{c}(\unicode[STIX]{x1D6F7}_{\text{alf}}^{+}-\unicode[STIX]{x1D6F7}_{\text{ale}})$ before the above list, and the very shape of $\unicode[STIX]{x1D6F7}_{\text{alf}}^{-}$ , we get that $\unicode[STIX]{x1D714}_{m}=\unicode[STIX]{x1D714}_{1}^{Y}+dd_{I_{1}^{Y}}^{c}(\unicode[STIX]{x1D6F7}_{\text{alf}}^{+}-\unicode[STIX]{x1D6F7}_{\text{ale}})-dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D6F7}_{\text{alf}}^{-}$ is asymptotic to $\unicode[STIX]{x1D71B}_{\mathbf{f}^{\flat }}$ (the error being of size $O(R^{-2})$ for $\mathbf{f}^{\flat }$ up to two $\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }}$ -derivatives). Second, recall that $R_{0}$ is supposed large enough so that $\{R\geqslant R_{0}\}\subset \{r\geqslant r_{0}+1\}$ ; running the first three points in the above list, and taking supports into account, one can also say that $\unicode[STIX]{x1D714}_{1}^{Y}+dd_{I_{1}^{Y}}^{c}(\unicode[STIX]{x1D6F7}_{\text{ALF}}^{+}-\unicode[STIX]{x1D6F7}_{\text{ALE}})$ , which equals $dd^{c}\unicode[STIX]{x1D711}^{\flat }+dd_{I_{1}^{Y}}^{c}(\unicode[STIX]{x1D6FD}^{-1}(\unicode[STIX]{x1D711}^{\flat })^{1-\unicode[STIX]{x1D6FD}})+(\unicode[STIX]{x1D714}_{1}^{Y}-dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D6F9})$ on $\{R\geqslant R_{0}\}$ , is thus ${\geqslant}\frac{1}{2}\unicode[STIX]{x1D71B}_{\mathbf{f}^{\flat }}+dd_{I_{1}^{Y}}^{c}(\unicode[STIX]{x1D6FD}^{-1}(\unicode[STIX]{x1D711}^{\flat })^{1-\unicode[STIX]{x1D6FD}})-\frac{1}{6}\unicode[STIX]{x1D71B}_{\mathbf{f}^{\flat }}$ , i.e. ${\geqslant}\frac{1}{3}\unicode[STIX]{x1D71B}_{\mathbf{f}^{\flat }}+dd_{I_{1}^{Y}}^{c}(\unicode[STIX]{x1D6FD}^{-1}(\unicode[STIX]{x1D711}^{\flat })^{1-\unicode[STIX]{x1D6FD}})$ on this region, where we recall that $dd_{I_{1}^{Y}}^{c}(\unicode[STIX]{x1D6FD}^{-1}(\unicode[STIX]{x1D711}^{\flat })^{1-\unicode[STIX]{x1D6FD}})\geqslant 0$ . Now, $dd_{I_{1}^{Y}}^{c}\unicode[STIX]{x1D6F7}_{\text{alf}}^{-}$ (which has support in $\{R\geqslant R_{0}+1\}$ ) can be rewritten as $\unicode[STIX]{x1D712}((R-R_{0})/R_{0})dd_{I_{1}^{Y}}^{c}(\unicode[STIX]{x1D6FD}^{-1}(\unicode[STIX]{x1D711}^{\flat })^{1-\unicode[STIX]{x1D6FD}})+{\mathcal{R}}_{R_{0}}^{-}$ , where ${\mathcal{R}}_{R_{0}}^{-}=(1/\unicode[STIX]{x1D6FD}R_{0})\unicode[STIX]{x1D712}^{\prime }((R-R_{0})/R_{0}){\mathcal{R}}_{1}^{-}+(1/\unicode[STIX]{x1D6FD}R_{0}^{2})\unicode[STIX]{x1D712}^{\prime \prime }((R-R_{0})/R_{0})(\unicode[STIX]{x1D711}^{\flat })^{1-\unicode[STIX]{x1D6FD}}{\mathcal{R}}_{2}^{-}$ has support in $\{R_{0}\leqslant R\leqslant 2R_{0}\}$ ; here ${\mathcal{R}}_{1}^{-}$ stands for $(\unicode[STIX]{x1D711}^{\flat })^{1-\unicode[STIX]{x1D6FD}}dd_{I_{1}^{Y}}^{c}R+dR\wedge d_{I_{1}^{Y}}^{c}(\unicode[STIX]{x1D711}^{\flat })^{1-\unicode[STIX]{x1D6FD}}+d(\unicode[STIX]{x1D711}^{\flat })^{1-\unicode[STIX]{x1D6FD}}\wedge d_{I_{1}^{Y}}^{c}R$ and ${\mathcal{R}}_{2}^{-}$ for $dR\wedge d_{I_{1}^{Y}}^{c}R$ . As ${\mathcal{R}}_{1}^{-}=O(R^{1-2\unicode[STIX]{x1D6FD}})$ , and ${\mathcal{R}}_{2}^{}=O(R^{2-2\unicode[STIX]{x1D6FD}})$ , independently of  $R_{0}$ , we can fix $R_{0}$  large enough so that $\sup _{Y}|{\mathcal{R}}_{R_{0}}^{-}|_{\mathbf{f}^{\flat }}=\sup _{R_{0}\leqslant R\leqslant 2R_{0}}|{\mathcal{R}}_{R_{0}}^{-}|_{\mathbf{f}^{\flat }}\leqslant \frac{1}{6}$ .

We sum all this up as: on $\{R\geqslant R_{0}\}$ , $\unicode[STIX]{x1D714}_{m}=\unicode[STIX]{x1D714}_{1}^{Y}+dd_{I_{1}^{Y}}^{c}(\unicode[STIX]{x1D6F7}_{\text{alf}}^{+}-\unicode[STIX]{x1D6F7}_{\text{ale}})-\unicode[STIX]{x1D712}((R-R_{0})/R_{0})dd_{I_{1}^{Y}}^{c}(\unicode[STIX]{x1D6FD}^{-1}(\unicode[STIX]{x1D711}^{\flat })^{1-\unicode[STIX]{x1D6FD}})-{\mathcal{R}}_{R_{0}}^{-}\geqslant \frac{1}{3}\unicode[STIX]{x1D71B}_{\mathbf{f}^{\flat }}+[1-\unicode[STIX]{x1D712}((R-R_{0})/R_{0})]dd_{I_{1}^{Y}}^{c}(\unicode[STIX]{x1D6FD}^{-1}(\unicode[STIX]{x1D711}^{\flat })^{1-\unicode[STIX]{x1D6FD}})-\frac{1}{6}\unicode[STIX]{x1D71B}_{\mathbf{f}^{\flat }}\geqslant \frac{1}{6}\unicode[STIX]{x1D71B}_{\mathbf{f}^{\flat }}>0$ ; moreover, on $Y\backslash \{R\geqslant R_{0}\}$ , $\unicode[STIX]{x1D714}_{m}$ equals $\unicode[STIX]{x1D714}_{1}^{Y}+dd_{I_{1}^{Y}}^{c}(\unicode[STIX]{x1D6F7}_{\text{alf}}^{+}-\unicode[STIX]{x1D6F7}_{\text{ale}})$ , positive on the whole $Y$ . In conclusion, $\unicode[STIX]{x1D714}_{m}>0$ on $Y$ .

The last part of the statements concerns volume forms, and is a direct consequence of the estimates on the metrics, after observing that (on $\mathbb{R}^{4}$ , say; recall that ${\unicode[STIX]{x1D6F7}_{Y}}_{\ast }\unicode[STIX]{x1D6FA}_{Y}=\unicode[STIX]{x1D6FA}_{\mathbf{e}}$ ): $\operatorname{vol}^{\mathbf{f}^{\flat }}-\unicode[STIX]{x1D6FA}_{Y}=\beth ^{\ast }\operatorname{vol}^{\mathbf{f}}-\unicode[STIX]{x1D6FA}_{\mathbf{e}}=\beth ^{\ast }\unicode[STIX]{x1D6FA}_{\mathbf{e}}-\unicode[STIX]{x1D6FA}_{\mathbf{e}}$ , which can be written as $\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FA}_{\mathbf{e}}$ with $|(\unicode[STIX]{x1D6FB}^{\mathbf{e}})^{\ell }\unicode[STIX]{x1D700}|_{\mathbf{e}}=O(r^{-8})$ , $\ell \geqslant 0$ , by Lemma 1.1. This converts into $|(\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }})^{\ell }\unicode[STIX]{x1D700}|_{\mathbf{f}^{\flat }}=O(R^{-4})$ , $\ell \geqslant 0$ , which is better than wanted.◻

1.4.1 A Calabi–Yau type theorem

We want to correct our $I_{1}^{Y}$ -Kähler metric $g_{m}$ from Proposition 1.12 into a Ricci-flat Kähler metric. For this it is sufficient to correct it into an $I_{1}^{Y}$ -Kähler metric with volume form $\unicode[STIX]{x1D6FA}_{Y}$ , since this is the volume of the $I_{1}^{Y}$ -Kähler metric $g_{Y}$ , and, as is well known, once the complex structure is fixed, the Ricci tensor of a Kähler metric depends only on its volume form. As suggested by the program ending to Theorem 1.3, at the level of $I_{1}^{Y}$ -Kähler forms, we want to stay in the same class; in other words, we are looking for the $I_{1}^{Y}$ -complex hessian of some function to be the desired correction.

The tool we are willing to use to determine this function is the ALF Calabi–Yau type theorem of the Introduction, which we state precisely now (we call the manifold in play $Y$ for more genericity).

Theorem 1.16. Let $\unicode[STIX]{x1D6FD}\in (0,1)$ and let $(Y,g_{Y},J_{Y},\unicode[STIX]{x1D714}_{Y})$ an ALF Kähler 4-manifold of dihedral type of order $\unicode[STIX]{x1D6FD}$ . Let $f\in C_{\unicode[STIX]{x1D6FD}+2}^{\infty }(Y,g_{Y})$ . Then there exists $\unicode[STIX]{x1D711}\in C_{\unicode[STIX]{x1D6FD}}^{\infty }(Y,g_{Y})$ such that $\unicode[STIX]{x1D714}_{Y}+dd_{J_{Y}}^{c}\unicode[STIX]{x1D711}$ is Kähler, and

(22) $$\begin{eqnarray}(\unicode[STIX]{x1D714}_{Y}+dd_{J_{Y}}^{c}\unicode[STIX]{x1D711})^{2}=e^{f}\unicode[STIX]{x1D714}_{Y}^{2}.\end{eqnarray}$$

The weighted spaces of this statement follow a classical definition, and our statement simply means that for all $\ell \geqslant 0$ , $|(\unicode[STIX]{x1D6FB}^{g_{Y}})^{\ell }f|_{Y}=O(R^{-2-\unicode[STIX]{x1D6FD}-\ell })$ and $|(\unicode[STIX]{x1D6FB}^{g_{Y}})^{\ell }\unicode[STIX]{x1D711}|_{Y}=O(R^{-\unicode[STIX]{x1D6FD}-\ell })$ ; the proof of Theorem 1.16, rather classical yet a bit involved, is postponed to Part 3 below.

Let us now make the following remark: since we want to construct a metric with volume form $\unicode[STIX]{x1D6FA}_{Y}$ , this is tempting to take $f=\log (\unicode[STIX]{x1D6FA}_{Y}/\text{vol}_{g_{m}})$ to apply Theorem 1.16. But so far we only control such an $f$ up to two derivatives (see Proposition 1.12, estimates (18)); also, even in the $C^{0}$ sense, we only have $f=O(R^{-2})$ instead of $O(R^{-2-\unicode[STIX]{x1D6FD}})$ .

The other issue is that $Y$ being a ‘ALF Kähler manifold of dihedral type of order $\unicode[STIX]{x1D6FD}$ ’ means that outside a compact subset, $Y$ is diffeomorphic to the complement of a ball in $\mathbb{R}^{4}/{\mathcal{D}}_{k}$ , and that one can choose the diffeomorphism $\unicode[STIX]{x1D6F7}_{Y}$ between infinities of $Y$ and $\mathbb{R}^{4}/{\mathcal{D}}_{k}$ such that for all $\ell \geqslant 0$ , $|(\unicode[STIX]{x1D6FB}^{g_{Y}})^{\ell }({\unicode[STIX]{x1D6F7}_{Y}}_{\ast }g_{Y}-\mathbf{f})|_{g_{Y}}=O(\unicode[STIX]{x1D70C}^{-\unicode[STIX]{x1D6FD}-\ell })$ , and $|(\unicode[STIX]{x1D6FB}^{g_{Y}})^{\ell }({\unicode[STIX]{x1D6F7}_{Y}}_{\ast }J^{Y}-I_{1})|_{g_{Y}}=O(\unicode[STIX]{x1D70C}^{-\unicode[STIX]{x1D6FD}-\ell }).$ Here again, a reading of Proposition 1.12 indicates that the asymptotics at our disposal do not allow us to take immediately $\unicode[STIX]{x1D6F7}_{Y}=\unicode[STIX]{x1D6F7}_{Y}$ or $\unicode[STIX]{x1D6F7}_{Y}\circ \beth$ .

We remedy to these technical problems as follows. First we correct $g_{m}$ into an $I_{1}^{Y}$ -Kähler metric with volume form $\unicode[STIX]{x1D6FA}_{Y},$ which is nothing but a Ricci-flat $I_{1}^{Y}$ -Kähler metric, outside a compact subset of $Y$ , which gives us an $f$ with compact support; then we put this corrected metric into so-called Bianchi gauge with respect to ${\unicode[STIX]{x1D6F7}_{Y}}^{\ast }\mathbf{f}^{\flat }$ , which corresponds to correct $\unicode[STIX]{x1D6F7}_{Y}$ itself so as to fit into the definition of an ALF Kähler manifold of dihedral type up to the desired order.

1.4.2 Ricci-flatness outside a compact subset

To correct $g_{m}$ into an $I_{1}^{Y}$ -Kähler metric with volume form $\unicode[STIX]{x1D6FA}_{Y}$ outside a compact subset of $Y$ , we use the inverse function theorem on Monge–Ampère operators, between relevant Hölder spaces. Namely, we extend $\mathbf{f}^{\flat }$ on $Y$ as a smooth metric and define on $Y$ the following weighted Hölder spaces:

(23) $$\begin{eqnarray}C_{\unicode[STIX]{x1D6FF}}^{\ell ,\unicode[STIX]{x1D6FC}}(Y,\mathbf{f}^{\flat }):=\{f\in C_{\text{loc}}^{\ell ,\unicode[STIX]{x1D6FC}}\mid \Vert f\Vert _{C_{\unicode[STIX]{x1D6FF}}^{\ell ,\unicode[STIX]{x1D6FC}}(\mathbf{f}^{\flat })}<\infty \},\end{eqnarray}$$

for $\ell \in \mathbb{N}$ , $\unicode[STIX]{x1D6FC}\in (0,1]$ , $\unicode[STIX]{x1D6FF}\in \mathbb{R}$ , and where

$$\begin{eqnarray}\Vert f\Vert _{C_{\unicode[STIX]{x1D6FF}}^{\ell ,\unicode[STIX]{x1D6FC}}(\mathbf{f}^{\flat })}:=\Vert R^{\unicode[STIX]{x1D6FF}}f\Vert _{C^{0}}+\cdots +\Vert R^{\unicode[STIX]{x1D6FF}+\ell }(\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }})^{\ell }f\Vert _{C^{0}}+\sup _{x\in Y}[R^{\ell }(\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }})^{\ell }f]_{\unicode[STIX]{x1D6FF}}^{\unicode[STIX]{x1D6FC}},\end{eqnarray}$$

with

(24) $$\begin{eqnarray}[u]_{\unicode[STIX]{x1D6FF}}^{\unicode[STIX]{x1D6FC}}=\sup _{\substack{ (x,y)\in Y, \\ d_{\mathbf{f}^{\flat }}(x,y)<\operatorname{inj}_{\mathbf{f}^{\flat }}}}\bigg|\max (R(x)^{\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FF}},R(y)^{\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FF}})\frac{u(x)-u(y)}{d_{\mathbf{f}^{\flat }}(x,y)^{\unicode[STIX]{x1D6FC}}}\bigg|_{\mathbf{f}^{\flat }}\end{eqnarray}$$

for $u$ a $C_{\text{loc}}^{\ell ,\unicode[STIX]{x1D6FC}}$ tensor ( $u(x)-u(y)$ interpreted via parallel transport), with $R$ a smooth positive extension of ${\unicode[STIX]{x1D6F7}_{Y}}^{\ast }R$ on $Y$ , and $C^{0}$ -norms of the tensors computed with $\mathbf{f}^{\flat }$ .

We then state the following, indicating the type of functions which can help correcting $\unicode[STIX]{x1D714}_{m}$ in the sense raised above.

In the statement, the intention is to take the orders of regularity and decay as small as possible, or, more precisely, to keep $\unicode[STIX]{x1D6FC}$ in the range $(0,1-\unicode[STIX]{x1D6FF})$ , to make the rest of the argument (a forthcoming use of the implicit function theorem, followed by a gauge process) work, so as to minimise the efforts involved in the construction of the reference metric $g_{m}$ of Proposition 1.12.

1.4.3 Bianchi gauge for $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}}$

Motivation. We are now willing to deduce regularity statements on $g_{\unicode[STIX]{x1D713}}$ , using its Ricci-flatness near infinity. However this cannot be done immediately. The reason is that the Ricci-flatness condition is invariant under diffeomorphisms, and consequently the linearisation of the Ricci tensor seen as an operator on metrics is not (strongly) elliptic, which is problematic when looking for regularity.

One can however bypass this difficulty by fixing a gauge, which infinitesimally corresponds to looking at metrics with good diffeomorphisms. We introduce the diffeomorphisms we shall work with in next paragraph; then the gauge is fixed, and regularity is deduced from this process (Propositions 1.21 and 1.23). Notice that the Ricci-flatness of $g_{\unicode[STIX]{x1D713}}$ is an indispensable prerequisite in this procedure, since the gauge alone is not enough in general to obtain the regularity statement we are seeking here.

ALF diffeomorphisms of  $\mathbb{C}^{2}$ . The class of diffeomorphisms we work with to perform our gauge enters into the following definition; we define the dual frames $(e_{0}^{\flat },\ldots ,e_{3}^{\flat })$ and $((e_{0}^{\ast })^{\flat },\ldots ,(e_{3}^{\ast })^{\flat })$ as the pull-backs by $\beth$ of the frames $(e_{i})$ and $(e_{i}^{\ast })$ defined in § 1.2.2 by (8), (9). We fix $R_{0}\geqslant 1$ so that $\beth$ induces a diffeomorphism between $\{R^{\flat }\geqslant R_{0}\}$ and $\{R\geqslant R_{0}\}$ .

The Hölder spaces are those defined for (some smooth extension of) $\mathbf{f}^{\flat }$ on $\mathbb{C}^{2}$ , in the same way as those of defining equation (23). Notice that we authorise the distance between a point and its image by a $\unicode[STIX]{x1D719}\in \mathbf{Diff}_{\unicode[STIX]{x1D708},R_{0}}^{\ell ,\unicode[STIX]{x1D6FC}}$ to go to $\infty$ when $\unicode[STIX]{x1D708}<0$ ; observe nonetheless that the allowed rate of blow-up is sub-linear, and thus the diffeomorphisms are proper.

Diffeomorphisms as Riemannian exponential maps. We now parametrise our diffeomorphisms via vector fields.

Apart from the 0 indices indicating vanishing of the vector fields along $\{R^{\flat }=R_{0}\}$ , the weighted spaces of vector fields are defined analogously to that of the previous paragraph, or equivalently $Z\in C_{\unicode[STIX]{x1D708},0}^{\ell ,\unicode[STIX]{x1D6FC}}(\{R\geqslant R_{0}\},\mathbf{f}^{\flat })$ if and only if $Z\in C_{\text{loc}}^{\ell ,\unicode[STIX]{x1D6FC}}(\{R^{\flat }\geqslant R_{0}\})$ , $Z|_{R^{\flat }=R_{0}}\equiv 0$ and $\unicode[STIX]{x1D712}(R^{\flat }-R_{0})(e_{i}^{\flat })^{\ast }(Z)\in C_{\unicode[STIX]{x1D708}}^{\ell ,\unicode[STIX]{x1D6FC}}(\mathbb{C}^{2},\mathbf{f}^{\flat })$ , $i=0,\ldots ,3$ (with $\unicode[STIX]{x1D712}$ a cut-off function as in Proposition 1.12).

Proof. The regularity assertions are rather standard; we moreover use the fact that the injectivity radius of $\mathbf{f}^{\flat }$ is bounded from below (that of $\mathbf{f}$ is), to get that any diffeomorphism $C^{0}$ -close to identity can be written as a $\unicode[STIX]{x1D719}_{Z}$ for some (small) continuous $Z$ , which boils down to joining any two points $x$ and $y$ with $d_{\mathbf{f}^{\flat }}(x,y)\leqslant \frac{1}{2}\operatorname{inj}_{\mathbf{f}^{\flat }}$ by a unique minimising geodesic $t\mapsto \exp _{x}(tZ_{x})$ . We shall nonetheless pay particular attention to the fact that in the general case, we authorise vector fields blowing up at infinity, when verifying the injectivity of $\unicode[STIX]{x1D719}_{Z}$ for a given $Z$ close to 0 in $C_{\unicode[STIX]{x1D708}}^{\ell ,\unicode[STIX]{x1D6FC}}$ ; in the same vein, one should keep in mind that even on compact manifolds, a $\unicode[STIX]{x1D719}_{W}$ might not be injective, unless $|\unicode[STIX]{x1D6FB}W|$ is small. In this respect, the key here is the decay of the derivatives of $Z$ at infinity, combined with the decay of $\operatorname{Rm}^{\mathbf{f}^{\flat }}$ . Suppose $(\ell ,\unicode[STIX]{x1D6FC})=(1,0)$ to fix ideas; for simplicity, we work on the whole $\mathbb{C}^{2}$ , where we extend $\mathbf{f}^{\flat }$ smoothly. For the injectivity of $\unicode[STIX]{x1D719}_{Z}$ with fixed $Z\in C_{\unicode[STIX]{x1D708}}^{1,0}$ (defined on $\mathbb{C}^{2}$ ) and $\Vert Z\Vert _{C_{\unicode[STIX]{x1D708}}^{1,0}}\leqslant 1$ say, we claim that there exists a constant $C$ independent of $Z$ such that for any triple $(x,y,z)$ such that $\unicode[STIX]{x1D719}_{Z}(x)=\unicode[STIX]{x1D719}_{Z}(y)=:z$ ,

$$\begin{eqnarray}d_{\mathbf{f}^{\flat }}(x,y)\leqslant C(1+R(z))^{-4-3\unicode[STIX]{x1D708}}\Vert Z\Vert _{C_{\unicode[STIX]{x1D708}}^{1,0}}d_{\mathbf{f}^{\flat }}(x,y),\end{eqnarray}$$

from which the injectivity of $\unicode[STIX]{x1D719}_{Z}$ follows at once provided $\Vert Z\Vert _{C_{\unicode[STIX]{x1D708}}^{1,0}}$ is small enough. We reach this claim thanks to the estimate $|\operatorname{Rm}^{\mathbf{f}^{\flat }}|=O(R^{-3})$ , as follows. For $x,y$ as in the claim, call respectively $\unicode[STIX]{x1D6FE}_{x}$ and $\unicode[STIX]{x1D6FE}_{y}$ the geodesics $t\mapsto \exp _{x}^{\mathbf{f}^{\flat }}(tZ(x))$ and $t\mapsto \exp _{y}^{\mathbf{f}^{\flat }}(tZ(y))$ , and denote by $p_{\unicode[STIX]{x1D6FE}_{x}}$ , $p_{\unicode[STIX]{x1D6FE}_{y}}$ the attached parallel transports. Using [Reference Buser and KarcherBK81, Proposition 6.6], control first $d_{\mathbf{f}^{\flat }}(x,y)$ by $|p_{\unicode[STIX]{x1D6FE}_{x}}(1)(Z(x))-p_{\unicode[STIX]{x1D6FE}_{y}}(1)(Z(y))|_{\mathbf{f}^{\flat }}(1+R(z))^{-3-2\unicode[STIX]{x1D708}}$ . Then control $|p_{\unicode[STIX]{x1D6FE}_{x}}(1)(Z(x))-p_{\unicode[STIX]{x1D6FE}_{y}}(1)(Z(y))|_{\mathbf{f}^{\flat }}$ by $d_{\mathbf{f}^{\flat }}(x,y)(1+R(z))^{-1-\unicode[STIX]{x1D708}}\Vert Z\Vert _{C_{\unicode[STIX]{x1D708}}^{1,0}}$ ; for this interpolate between $\unicode[STIX]{x1D6FE}_{x}$ and $\unicode[STIX]{x1D6FE}_{y}$ by $\unicode[STIX]{x1D6FE}_{s}(t):=\exp _{\unicode[STIX]{x1D6FC}(s)}^{\mathbf{f}^{\flat }}[tZ(\unicode[STIX]{x1D6FC}(s))]$ , where $\unicode[STIX]{x1D6FC}$ is a minimising geodesic for $\mathbf{f}^{\flat }$ joining $x$ and $y$ . This is where one uses the estimates on the derivatives of $Z$ .◻

With similar techniques, one establishes the following lemma.

The gauge. Denote by $B^{h}=\unicode[STIX]{x1D6FF}^{h}+\frac{1}{2}d\operatorname{tr}^{h}$ the Bianchi operator associated to any smooth metric $h$ on (an open subset of) $\mathbb{R}^{4}$ . The gauge process now states the following.

Proposition 1.21. Let $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FF})$ fixed in Proposition 1.17 and $\unicode[STIX]{x1D6FF}$ assumed ${>}\frac{1}{2}$ . If $R_{0}$ is large enough, there exists a smooth diffeomorphism $\unicode[STIX]{x1D719}\in \mathbf{Diff}_{\unicode[STIX]{x1D6FF}-1,R_{0}}^{1,\unicode[STIX]{x1D6FC}}$ , commuting with the action of ${\mathcal{D}}_{k}$ hence descending to $\mathbb{R}^{4}/{\mathcal{D}}_{k}$ near infinity, such that

$$\begin{eqnarray}B^{\unicode[STIX]{x1D719}^{\ast }\boldsymbol{ f}^{\flat }}((\unicode[STIX]{x1D6F7}_{Y})_{\ast }g_{\unicode[STIX]{x1D713}})=0\end{eqnarray}$$

near infinity on $\mathbb{C}^{2}$ , where $g_{\unicode[STIX]{x1D713}}$ stands for the $I_{1}^{Y}$ -Kähler metric associated to the Kähler form $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}}$ of Proposition 1.17. As a consequence, $\mathbf{f}^{\flat }-(\unicode[STIX]{x1D719}\circ \unicode[STIX]{x1D6F7}_{Y})_{\ast }g_{\unicode[STIX]{x1D713}}\in C_{\unicode[STIX]{x1D6FF}}^{1,\unicode[STIX]{x1D6FC}}(X,\mathbf{f}^{\flat })$ .

Proof. Fix $\unicode[STIX]{x1D6FC}_{2}$ as in the statement, and consider the map

$$\begin{eqnarray}\displaystyle \begin{array}{@{}ccl@{}}\unicode[STIX]{x1D6EF}:\mathscr{V}_{\unicode[STIX]{x1D6FF}-1}^{2,\unicode[STIX]{x1D6FC}_{2}}\times \mathbf{Met}_{\unicode[STIX]{x1D6FF}}^{1,\unicode[STIX]{x1D6FC}_{2}}(\mathbf{f}^{\flat })\, & \longrightarrow \, & C_{\unicode[STIX]{x1D6FF}+1}^{0,\unicode[STIX]{x1D6FC}_{2}}(T^{\ast }\mathbb{C}^{2},\mathbf{f}^{\flat })\\ (Z,g)\quad \;\;\, & \longmapsto \, & B^{\unicode[STIX]{x1D719}_{Z}^{\ast }\mathbf{f}^{\flat }}(g).\end{array} & & \displaystyle \nonumber\end{eqnarray}$$

We would like to solve the equation

(25) $$\begin{eqnarray}B^{\unicode[STIX]{x1D719}_{Z}^{\ast }\mathbf{f}^{\flat }}(g_{\unicode[STIX]{x1D713}})=0,\quad \text{i.e}~\unicode[STIX]{x1D6EF}(Z,g_{\unicode[STIX]{x1D713}})=0,\end{eqnarray}$$

near infinity, and for this use the implicit function theorem near $(0,\mathbf{f}^{\flat })$ , since the differential of $\unicode[STIX]{x1D6EF}$ with respect to $Z$ is $(\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }})^{\ast }\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }}$ , which as we shall see enjoys surjectivity properties. We first solve a linearised version (25), to get in the $\unicode[STIX]{x1D708}\geqslant 0$ position of Lemma 1.19, better adapted to solve (a duly modified version of) this nonlinear equation.

Step 1. Let $Z\in C_{\unicode[STIX]{x1D6FF}-1,0}^{2,\unicode[STIX]{x1D6FC}}(\{R^{\flat }\geqslant R_{0}\},\mathbf{f}^{\flat })$ be a smooth vector field such that $(\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }})^{\ast }\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }}[\mathbf{f}^{\flat }(Z,\cdot )]=B^{\mathbf{f}^{\flat }}(g_{\unicode[STIX]{x1D713}})$ near infinity. As we shall see below, such an equation can be solved and, picking some $\unicode[STIX]{x1D6FF}^{\prime }\in (0,\unicode[STIX]{x1D6FF})$ , and up to using some cut-off function (which does not affect the equation verified by $Z$ near infinity), we can assume that $\Vert Z\Vert _{C_{\unicode[STIX]{x1D6FF}^{\prime }-1}^{2,\unicode[STIX]{x1D6FC}}(\mathbf{f}^{\flat })}$ is small enough so that $Z\in \mathscr{V}_{\unicode[STIX]{x1D6FF}^{\prime }-1}^{2,\unicode[STIX]{x1D6FC}}$ , i.e., $\unicode[STIX]{x1D719}_{Z}\in \mathbf{Diff}_{\unicode[STIX]{x1D6FF}^{\prime }-1,R_{0}}^{1,\unicode[STIX]{x1D6FC}}$ (in particular, it is a diffeomorphism); as $Z$ has regularity $C_{\unicode[STIX]{x1D6FF}-1}^{2,\unicode[STIX]{x1D6FC}}(\mathbf{f}^{\flat })$ , we moreover get $\unicode[STIX]{x1D719}_{Z}\in \mathbf{Diff}_{\unicode[STIX]{x1D6FF}-1,R_{0}}^{1,\unicode[STIX]{x1D6FC}}$ . Taking a mean along the action of ${\mathcal{D}}_{k}$ , we can also assume that $Z$ is ${\mathcal{D}}_{k}$ -invariant, and hence that $\unicode[STIX]{x1D719}_{Z}$ commutes to the ${\mathcal{D}}_{k}$ -action. The smoothness of $Z$ , purely local, comes at once from that of $\mathbf{f}^{\flat }$ and $B^{\mathbf{f}^{\flat }}(g)$ , and ellipticity of $(\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }})^{\ast }\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }}$ .

Now, given metrics $g$ , $g^{\prime }$ and $g+h$ (with $g^{\prime }$ and $g+h$ seen as perturbations of $g$ ), repeated use of equation (20) provides

$$\begin{eqnarray}B^{g+h}(g^{\prime })=B^{g}(g^{\prime })-B^{g}(h)+Q_{g}(h,g^{\prime }),\end{eqnarray}$$

where $Q_{g}(h,g^{\prime })$ admits a formal expansion

$$\begin{eqnarray}\displaystyle Q_{g}(h,g^{\prime }) & = & \displaystyle \unicode[STIX]{x1D6FB}^{g}h\ast \{(g+h)^{-1}-g^{-1}\}+(g+h)^{-2}\ast \unicode[STIX]{x1D6FB}^{g}h\ast h\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D6FB}^{g}g^{\prime }\ast \{(g+h)^{-1}-g^{-1}\}(g+h)^{-2}\ast \unicode[STIX]{x1D6FB}^{g}h\ast (g^{\prime }-g).\nonumber\end{eqnarray}$$

Taking $g=\mathbf{f}^{\flat }$ , $h=h_{Z}:=(\unicode[STIX]{x1D719}_{Z})^{\ast }\mathbf{f}^{\flat }-\mathbf{f}^{\flat }$ and $g^{\prime }=g_{\unicode[STIX]{x1D713}}$ , we get that $Q_{\mathbf{f}^{\flat }}(h_{Z},g_{\unicode[STIX]{x1D713}})$ factors through $(\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }}h_{Z})\ast h_{Z}$ , $(\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }}h_{Z})\ast (\mathbf{f}^{\flat }-g_{\unicode[STIX]{x1D713}})$ and $(\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }}g_{\unicode[STIX]{x1D713}})\ast h_{Z}$ . This way, by Lemma 1.20 and as $Z$ is smooth, $Q_{\mathbf{f}^{\flat }}(h_{Z},g_{\unicode[STIX]{x1D713}})\in C_{\text{loc}}^{\infty }\cap C_{1+2\unicode[STIX]{x1D6FF}}^{0,\unicode[STIX]{x1D6FC}}$ . Moreover, as $h_{Z}=-\unicode[STIX]{x1D6FF}^{\mathbf{f}^{\flat }}[\mathbf{f}^{\flat }(Z,\cdot )]+h_{Z}^{\prime }$ with $h_{Z}^{\prime }\in C_{\text{loc}}^{\infty }\cap C_{2\unicode[STIX]{x1D6FF}}^{1,\unicode[STIX]{x1D6FC}}$ ,

$$\begin{eqnarray}B^{\mathbf{f}^{\flat }}(h_{Z})=B^{g}(\unicode[STIX]{x1D6FF}^{\mathbf{f}^{\flat }}[\mathbf{f}^{\flat }(Z,\cdot )])+B^{\mathbf{f}^{\flat }}(h_{Z}^{\prime })=(\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }})^{\ast }\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }}[\mathbf{f}^{\flat }(Z,\cdot )]+B^{\mathbf{f}^{\flat }}(h_{Z}^{\prime }),\end{eqnarray}$$

with $B^{\mathbf{f}^{\flat }}(h_{Z}^{\prime })\in C_{\text{loc}}^{\infty }\cap C_{1+2\unicode[STIX]{x1D6FF}}^{0,\unicode[STIX]{x1D6FC}}$ . In conclusion, $B^{(\unicode[STIX]{x1D719}_{Z})^{\ast }\mathbf{f}^{\flat }}(g_{\unicode[STIX]{x1D713}})=B^{\mathbf{f}^{\flat }}(g_{\unicode[STIX]{x1D713}})-B^{\mathbf{f}^{\flat }}(h_{Z})+Q_{\mathbf{f}^{\flat }}(h_{Z},g_{\unicode[STIX]{x1D713}})=\underbrace{B^{\mathbf{f}^{\flat }}(g_{\unicode[STIX]{x1D713}})-(\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }})^{\ast }\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }}[\mathbf{f}^{\flat }(Z,\cdot )]}_{=\,0\text{ near infinity}}+B^{\mathbf{f}^{\flat }}(h_{Z}^{\prime })+Q_{\mathbf{f}^{\flat }}(h_{Z},g_{\unicode[STIX]{x1D713}})$ is now in $C_{\text{loc}}^{\infty }\cap C_{1+2\unicode[STIX]{x1D6FF}}^{0,\unicode[STIX]{x1D6FC}}$ . Pushing forward by $\unicode[STIX]{x1D719}_{Z}$ , we have $B^{\mathbf{f}^{\flat }}[(\unicode[STIX]{x1D719}_{Z})_{\ast }g_{\unicode[STIX]{x1D713}}]\in C_{\text{loc}}^{\infty }\cap C_{1+2\unicode[STIX]{x1D6FF}}^{0,\unicode[STIX]{x1D6FC}}(\mathbf{f}^{\flat })$ , where we had $B^{\mathbf{f}^{\flat }}(g_{\unicode[STIX]{x1D713}})\in C_{\text{loc}}^{\infty }\cap C_{1+\unicode[STIX]{x1D6FF}}^{0,\unicode[STIX]{x1D6FC}}(\mathbf{f}^{\flat })$ (and $1+2\unicode[STIX]{x1D6FF}>2>1+\unicode[STIX]{x1D6FF}$ ; this is where $\unicode[STIX]{x1D6FF}>\frac{1}{2}$ is used).

Step 2. Set $\bar{\unicode[STIX]{x1D6FF}}_{1}=2\unicode[STIX]{x1D6FF}-1\in (0,1)$ , and take $\unicode[STIX]{x1D6FF}_{1}\in (0,\bar{\unicode[STIX]{x1D6FF}}_{1})$ . As $B^{\mathbf{f}^{\flat }}[(\unicode[STIX]{x1D719}_{Z})_{\ast }g_{\unicode[STIX]{x1D713}}]\in C_{\text{loc}}^{\infty }\cap C_{2+\bar{\unicode[STIX]{x1D6FF}}_{1}}^{0,\unicode[STIX]{x1D6FC}}(\mathbf{f}^{\flat })$ , $\Vert \unicode[STIX]{x1D712}_{R_{2}}(R^{\flat })B^{\mathbf{f}^{\flat }}[(\unicode[STIX]{x1D719}_{Z})_{\ast }g_{\unicode[STIX]{x1D713}}]\Vert _{C_{2+\unicode[STIX]{x1D6FF}_{1}}^{0,\unicode[STIX]{x1D6FC}}(\mathbf{f}^{\flat })}$ goes to 0 as $R_{2}$ goes to infinity; here $\unicode[STIX]{x1D712}_{R_{2}}=\unicode[STIX]{x1D712}(\cdot -R_{2})$ , with $\unicode[STIX]{x1D712}$ a cut-off function as in Proposition 1.12. In other words, given any neighbourhood of $B^{\mathbf{f}^{\flat }}[(\unicode[STIX]{x1D719}_{Z})^{\ast }g_{\unicode[STIX]{x1D713}}]$ in $C_{2+\unicode[STIX]{x1D6FF}_{1}}^{0,\unicode[STIX]{x1D6FC}}(\mathbf{f}^{\flat })$ , the 1-form $(1-\unicode[STIX]{x1D712}_{R_{2}}(R^{\flat }))B^{\mathbf{f}^{\flat }}[(\unicode[STIX]{x1D719}_{Z})_{\ast }g_{\unicode[STIX]{x1D713}}]$ lies in this neighbourhood for $R_{2}$ large enough; by construction it moreover vanishes on  $\{R^{\flat }\geqslant R_{2}+1\}$ . This being said, we now work for simplicity with the operator $z\mapsto \unicode[STIX]{x1D6EF}[z,(\unicode[STIX]{x1D719}_{Z})_{\ast }g_{\unicode[STIX]{x1D713}}]=B^{\unicode[STIX]{x1D719}_{z}^{\ast }\mathbf{f}^{\flat }}[(\unicode[STIX]{x1D719}_{Z})_{\ast }g_{\unicode[STIX]{x1D713}}]$ seen as a map $\mathscr{V}_{\unicode[STIX]{x1D6FF}_{1}}^{2,\unicode[STIX]{x1D6FC}}\rightarrow C_{\unicode[STIX]{x1D6FF}_{1}+2}^{0,\unicode[STIX]{x1D6FC}}(T^{\ast }\mathbb{C}^{2},\mathbf{f}^{\flat })$ thanks to the formulas on $B^{g+h}(g^{\prime })$ and $Q_{g}(h,g^{\prime })$ of Step 1. Finding one solution $z$ to the equation $\unicode[STIX]{x1D6EF}[z,(\unicode[STIX]{x1D719}_{Z})_{\ast }g_{\unicode[STIX]{x1D713}}]=0$ near infinity, that is, finding a solution to the equation $\unicode[STIX]{x1D6EF}[z,(\unicode[STIX]{x1D719}_{Z})_{\ast }g_{\unicode[STIX]{x1D713}}]=(1-\unicode[STIX]{x1D712}(R^{\flat }-R_{2}))B^{\mathbf{f}^{\flat }}[(\unicode[STIX]{x1D719}_{Z})_{\ast }g_{\unicode[STIX]{x1D713}}]$ for some arbitrarily large $R_{2}$ , with $z\in C_{\unicode[STIX]{x1D6FF}_{1},0}^{2,\unicode[STIX]{x1D6FC}}(\{R^{\flat }\geqslant R_{0}\},\mathbf{f}^{\flat })$ , thus amounts by the implicit function theorem to establishing the surjectivity of the operator

(26) $$\begin{eqnarray}\displaystyle & & \displaystyle (\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }})^{\ast }\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }}+[(\unicode[STIX]{x1D719}_{Z})_{\ast }g_{\unicode[STIX]{x1D713}}-\mathbf{f}^{\flat }]\ast (\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }})^{2}\cdot +\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }}[(\unicode[STIX]{x1D719}_{Z})_{\ast }g_{\unicode[STIX]{x1D713}}]\ast \unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }}\cdot \nonumber\\ \displaystyle & & \displaystyle \quad \!\!=\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6EF}}{\unicode[STIX]{x2202}z}\bigg|_{(0,(\unicode[STIX]{x1D719}_{Z})_{\ast }g_{\unicode[STIX]{x1D713}})}\!\!:C_{\unicode[STIX]{x1D6FF}_{1},0}^{2,\unicode[STIX]{x1D6FC}}(\{R^{\flat }\geqslant R_{0}\},\mathbf{f}^{\flat })\longrightarrow C_{\unicode[STIX]{x1D6FF}_{1}+2}^{0,\unicode[STIX]{x1D6FC}}(\{R^{\flat }\geqslant R_{0}\},T^{\ast }\mathbb{C}^{2},\mathbf{f}^{\flat })\cong C_{\unicode[STIX]{x1D6FF}_{1}+2}^{0,\unicode[STIX]{x1D6FC}}(\{R^{\flat }\geqslant R_{0}\},\mathbf{f}^{\flat }).\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Now, as $[(\unicode[STIX]{x1D719}_{Z})_{\ast }g_{\unicode[STIX]{x1D713}}-\mathbf{f}^{\flat }]$ is $C_{\unicode[STIX]{x1D6FF}}^{1,\unicode[STIX]{x1D6FC}}(\mathbf{f}^{\flat })$ and $\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }}[(\unicode[STIX]{x1D719}_{Z})_{\ast }g_{\unicode[STIX]{x1D713}}]$ is $C_{\unicode[STIX]{x1D6FF}+1}^{0,\unicode[STIX]{x1D6FC}}(\mathbf{f}^{\flat })$ , the operator (26) differs from $(\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }})^{\ast }\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }}$ by some asymptotically vanishing term in the $C_{\unicode[STIX]{x1D6FF}_{1}}^{2,\unicode[STIX]{x1D6FC}}(\mathbf{f}^{\flat })$ -to- $C_{\unicode[STIX]{x1D6FF}_{1}+2}^{0,\unicode[STIX]{x1D6FC}}(\mathbf{f}^{\flat })$ norm. Hence (up to working in Step 2 with an $R_{0}$ possibly larger than in Step 1), it is enough to see that $(\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }})^{\ast }\unicode[STIX]{x1D6FB}^{\mathbf{f}^{\flat }}:C_{\unicode[STIX]{x1D717},0}^{2,\unicode[STIX]{x1D6FC}}(\{R^{\flat }\geqslant R_{0}\},\mathbf{f}^{\flat })\rightarrow C_{\unicode[STIX]{x1D717}+2}^{0,\unicode[STIX]{x1D6FC}}(\{R^{\flat }\geqslant R_{0}\},\mathbf{f}^{\flat })$ admits a bounded right inverse for $\unicode[STIX]{x1D717}\in (0,1)$ (playing the role of $\unicode[STIX]{x1D6FF}_{1}$ ), with norm independent of the domain as we let $R_{0}$ grow; we also do the case $\unicode[STIX]{x1D717}\in (-1,0)$ (playing the role of $\unicode[STIX]{x1D6FF}$ ) without the independence of the bounds, needed to get $Z$ in the beginning of the proof. We work with $\mathbf{f}$ , diffeomorphic to $\mathbf{f}^{\flat }$ (and replace $R^{\flat }$ by $R$ , and so on). As for $i=0,\ldots ,3$ , $(\unicode[STIX]{x1D6FB}^{\mathbf{f}})^{\ast }\unicode[STIX]{x1D6FB}^{\mathbf{f}}(ve_{i}^{\ast })$ equals $(\unicode[STIX]{x1D6E5}_{\mathbf{f}}v)e_{i}^{\ast }$ plus a linear combination of the $(e_{j}\cdot v)\unicode[STIX]{x1D6FB}_{e_{k}}^{\mathbf{f}}e_{\ell }$ and the $v(\unicode[STIX]{x1D6FB}^{\mathbf{f}})_{e_{j},e_{k}}^{2}e_{\ell }$ , up to increasing $R_{0}$ , by a standard perturbation argument, the existence of a bounded right inverse for

$$\begin{eqnarray}(\unicode[STIX]{x1D6FB}^{\mathbf{f}})^{\ast }\unicode[STIX]{x1D6FB}^{\mathbf{f}}:C_{\unicode[STIX]{x1D717},0}^{2,\unicode[STIX]{x1D6FC}}(\{R\geqslant R_{0}\},T^{\ast }\mathbb{C}^{2},\mathbf{f})\longrightarrow C_{\unicode[STIX]{x1D717}+2}^{0,\unicode[STIX]{x1D6FC}}(\{R\geqslant R_{0}\},T^{\ast }\mathbb{C}^{2},\mathbf{f})\end{eqnarray}$$

amounts to the analogue for the scalar Laplacian with Dirichlet condition

$$\begin{eqnarray}\unicode[STIX]{x1D6E5}_{\mathbf{f}}:C_{\unicode[STIX]{x1D717},0}^{2,\unicode[STIX]{x1D6FC}}(\{R\geqslant R_{0}\},\mathbb{R},\mathbf{f})\longrightarrow C_{\unicode[STIX]{x1D717}+2}^{0,\unicode[STIX]{x1D6FC}}(\{R\geqslant R_{0}\},\mathbb{R},\mathbf{f}).\end{eqnarray}$$

Using a similar procedure as in [Reference MinerbeMin09, § 2.1] (where the ‘ $\unicode[STIX]{x1D6FF}$ ’ there should be thought of as ‘ $\frac{3}{2}-\unicode[STIX]{x1D717}$ ’ with our $\unicode[STIX]{x1D717}$ ), we first get, for $\unicode[STIX]{x1D717}\in (-1,1)\setminus \{0\}$ , the existence of a bounded linear ${\mathcal{G}}_{\unicode[STIX]{x1D717}}:C_{\unicode[STIX]{x1D717}+2}^{0,\unicode[STIX]{x1D6FC}}(\{R\geqslant R_{0}\},\mathbf{f})\rightarrow C_{\unicode[STIX]{x1D717}}^{2,\unicode[STIX]{x1D6FC}}(\{R\geqslant R_{0}\},\mathbf{f})$ satisfying $\unicode[STIX]{x1D6E5}_{\mathbf{f}}\circ {\mathcal{G}}_{\unicode[STIX]{x1D717}}=\text{id}_{C_{\unicode[STIX]{x1D717}+2}^{0,\unicode[STIX]{x1D6FC}}(\{R\geqslant R_{0}\})}$ ; beware we momentarily drop the 0 index to the target space $C_{\unicode[STIX]{x1D717}}^{2,\unicode[STIX]{x1D6FC}}$ of ${\mathcal{G}}_{\unicode[STIX]{x1D717}}$ . Moreover, a careful reading of Minerbe’s construction shows that for $\unicode[STIX]{x1D717}<1$ (which prevents us to be in the case ‘ $\unicode[STIX]{x1D6FF}<\unicode[STIX]{x1D6FF}_{j}<2-\unicode[STIX]{x1D6FF}$ ’ of [Reference MinerbeMin09, bottom of p. 937]; remember $\unicode[STIX]{x1D6FF}$ plays the role of $\frac{3}{2}-\unicode[STIX]{x1D717}$ , and that $\unicode[STIX]{x1D6FF}_{j}$ has shape $\frac{3}{2}+j$ , $j\in \mathbb{N}$ [Reference MinerbeMin09, p. 931]) gives that the above ${\mathcal{G}}_{\unicode[STIX]{x1D717}}$ has image in $C_{\unicode[STIX]{x1D717},0}^{2,\unicode[STIX]{x1D6FC}}(\{R\geqslant R_{0}\},\mathbf{f})$ . Finally, when $\unicode[STIX]{x1D717}\in (0,1)$ , a bound on ${\mathcal{G}}_{\unicode[STIX]{x1D717}}$ (of shape $C/\unicode[STIX]{x1D717}(1-\unicode[STIX]{x1D717})$ and) uniform in $R_{0}$ can also be extracted from this construction; a similar bound, uniform in $R_{0}$ , is also valid on ${\mathcal{G}}_{\unicode[STIX]{x1D717}-1}$ . Consequently, for all $\unicode[STIX]{x1D717}\in (-1,1)\setminus \{0\}$ ,

$$\begin{eqnarray}(\unicode[STIX]{x1D6FB}^{\mathbf{f}})^{\ast }\unicode[STIX]{x1D6FB}^{\mathbf{f}}:C_{\unicode[STIX]{x1D717},0}^{2,\unicode[STIX]{x1D6FC}}(\{R\geqslant R_{0}\},T^{\ast }\mathbb{C}^{2},\mathbf{f})\longrightarrow C_{\unicode[STIX]{x1D717}+2}^{0,\unicode[STIX]{x1D6FC}}(\{R\geqslant R_{0}\},T^{\ast }\mathbb{C}^{2},\mathbf{f})\end{eqnarray}$$

is a surjective map, with a bounded right inverse ${\mathcal{G}}_{\unicode[STIX]{x1D717}}$ enjoying a bound independent of $R_{0}$ , as claimed.

We conclude Step 2 by taking a smooth solution $z\in C_{\unicode[STIX]{x1D6FF}_{1},0}^{2,\unicode[STIX]{x1D6FC}}(\{R^{\flat }\geqslant R_{0}^{\prime }\},\mathbf{f}^{\flat })$ to $\unicode[STIX]{x1D6EF}(z,(\unicode[STIX]{x1D719}_{Z})_{\ast }g_{\unicode[STIX]{x1D713}})=0$ near infinity (chosen ${\mathcal{D}}_{k}$ -invariant), producing a smooth $\unicode[STIX]{x1D719}_{z}\in \mathbf{Diff}_{\unicode[STIX]{x1D6FF}_{1},R_{0}^{\prime }}^{1,\unicode[STIX]{x1D6FC}}$ , and by putting $\unicode[STIX]{x1D719}=(\unicode[STIX]{x1D719}_{z}\circ \unicode[STIX]{x1D719}_{Z^{\prime }})^{-1}\in \mathbf{Diff}_{\unicode[STIX]{x1D6FF}-1,R_{0}^{\prime }}^{1,\unicode[STIX]{x1D6FC}}$ (which is smooth), with $Z^{\prime }=\unicode[STIX]{x1D712}(R^{\flat }-R_{0}^{\prime })Z$ .◻

Regularity of  $g_{\unicode[STIX]{x1D713}}$ . We conclude this paragraph by the following statement, which finally allows us to apply Theorem 1.16.

Proof. The assertion on the curvature of $g_{\unicode[STIX]{x1D713}}$ directly follows from the estimate stated on $\unicode[STIX]{x1D700}:=\unicode[STIX]{x1D719}^{\ast }\mathbf{f}^{\flat }-g_{\unicode[STIX]{x1D713}}$ (or $\unicode[STIX]{x1D719}_{\ast }\unicode[STIX]{x1D700}$ ), and the fact that $|\!\operatorname{Rm}^{\mathbf{f}^{\flat }}\!|_{\mathbf{f}^{\flat }}=O(R^{-3})$ . For the regularity statement on $\unicode[STIX]{x1D700}$ , proceed as follows: set $\mathbf{F}=\unicode[STIX]{x1D719}^{\ast }\mathbf{f}^{\flat }$ , and define the operator $\unicode[STIX]{x1D6F7}^{\mathbf{F}}$ by

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}^{\mathbf{F}}(h)=\operatorname{Ric}(h)+(\unicode[STIX]{x1D6FF}^{h})^{\ast _{h}}[h_{\mathbf{F}}\cdot B^{\mathbf{F}}h]\end{eqnarray}$$

on $C_{\text{loc}}^{2}$ metrics, where $h_{\mathbf{F}}$ is the endomorphism of $T^{\ast }X$ such that $(h_{\mathbf{F}}\cdot \unicode[STIX]{x1D6FC},\cdot )_{h}=(\unicode[STIX]{x1D6FC},\cdot )_{\mathbf{F}}$ on 1-forms. This way, $\unicode[STIX]{x1D6F7}^{\mathbf{F}}(\mathbf{F})=\unicode[STIX]{x1D6F7}^{\mathbf{F}}(g_{\unicode[STIX]{x1D713}})=0$ near infinity. Now $\unicode[STIX]{x1D6F7}^{\mathbf{F}}$ is of order 2, hence schematically,

$$\begin{eqnarray}0=\unicode[STIX]{x1D6F7}^{\mathbf{F}}(\mathbf{F})-\unicode[STIX]{x1D6F7}^{\mathbf{F}}(g_{\unicode[STIX]{x1D713}})=(d_{\boldsymbol{ F}}\unicode[STIX]{x1D6F7}^{\mathbf{F}})(\unicode[STIX]{x1D700})+P_{\boldsymbol{ F}}(\unicode[STIX]{x1D700}),\end{eqnarray}$$

with $P_{\mathbf{F}}(\unicode[STIX]{x1D700})$ an at-least-quadratic combination of $\unicode[STIX]{x1D700}$ , its first and its second derivatives, with coefficients depending on $\mathbf{F}$ . Quoting [Reference BamlerBam11, p. 16], one has more precisely:

(27) $$\begin{eqnarray}\displaystyle 2P_{\mathbf{F}}(\unicode[STIX]{x1D700}) & = & \displaystyle (g_{\unicode[STIX]{x1D713}})^{uv}(g_{\unicode[STIX]{x1D713}})^{pq}(\unicode[STIX]{x1D6FB}_{u}^{\unicode[STIX]{x1D713}}\unicode[STIX]{x1D700}_{pa}\unicode[STIX]{x1D6FB}_{v}^{\unicode[STIX]{x1D713}}\unicode[STIX]{x1D700}_{qb}-\unicode[STIX]{x1D6FB}_{p}^{\unicode[STIX]{x1D713}}\unicode[STIX]{x1D700}_{ua}\unicode[STIX]{x1D6FB}_{v}^{\unicode[STIX]{x1D713}}\unicode[STIX]{x1D700}_{qb}+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D6FB}_{a}^{\unicode[STIX]{x1D713}}\unicode[STIX]{x1D700}_{up}\unicode[STIX]{x1D6FB}_{b}^{\unicode[STIX]{x1D713}}\unicode[STIX]{x1D700}_{vq})\nonumber\\ \displaystyle & & \displaystyle +\,(g_{\unicode[STIX]{x1D713}})^{uv}(-\unicode[STIX]{x1D6FB}_{u}^{\unicode[STIX]{x1D713}}\unicode[STIX]{x1D700}_{vp}+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D6FB}_{p}^{\unicode[STIX]{x1D713}}\unicode[STIX]{x1D700}_{uv})(g_{\unicode[STIX]{x1D713}})^{pq}(\unicode[STIX]{x1D6FB}_{a}^{\unicode[STIX]{x1D713}}\unicode[STIX]{x1D700}_{qb}+\unicode[STIX]{x1D6FB}_{b}^{\unicode[STIX]{x1D713}}\unicode[STIX]{x1D700}_{qa}-\unicode[STIX]{x1D6FB}_{q}^{\unicode[STIX]{x1D713}}\unicode[STIX]{x1D700}_{ab})\nonumber\\ \displaystyle & & \displaystyle +\,\mathbf{F}^{uv}(-\unicode[STIX]{x1D6FB}_{u}^{\unicode[STIX]{x1D713}}\unicode[STIX]{x1D700}_{vp}+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D6FB}_{p}^{\unicode[STIX]{x1D713}}\unicode[STIX]{x1D700}_{uv})\mathbf{F}^{pq}\unicode[STIX]{x1D6FB}_{q}^{\unicode[STIX]{x1D713}}\unicode[STIX]{x1D700}_{ab}\nonumber\\ \displaystyle & & \displaystyle +\,\mathbf{F}^{uv}\mathbf{F}^{pq}(-\unicode[STIX]{x1D700}_{ap}(\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D713}})_{bu}^{2}\unicode[STIX]{x1D700}_{vq}-\unicode[STIX]{x1D700}_{bp}(\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D713}})_{au}^{2}\unicode[STIX]{x1D700}_{vq}+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D700}_{ap}(\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D713}})_{bq}^{2}\unicode[STIX]{x1D700}_{uv}+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D700}_{bp}(\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D713}})_{aq}^{2}\unicode[STIX]{x1D700}_{uv})\nonumber\\ \displaystyle & & \displaystyle +\,((g_{\unicode[STIX]{x1D713}})^{uv}-\mathbf{F}^{uv})((\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D713}})_{ua}^{2}\unicode[STIX]{x1D700}_{bv}+(\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D713}})_{ub}^{2}\unicode[STIX]{x1D700}_{av}-(\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D713}})_{uv}^{2}\unicode[STIX]{x1D700}_{ab}+(\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D713}})_{ab}^{2}\unicode[STIX]{x1D700}_{uv}).\end{eqnarray}$$

The interest of this formula lies in the following: in $P_{\mathbf{F}}(\unicode[STIX]{x1D700})$ ,

1. (i) the only occurrence of the second derivatives of $\unicode[STIX]{x1D700}=g_{\unicode[STIX]{x1D713}}-\mathbf{F}$ with respect to $g_{\unicode[STIX]{x1D713}}$ , denoted by $(\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D713}})^{2}\unicode[STIX]{x1D700}$ , in (27), is via tensors factoring through $\unicode[STIX]{x1D700}\ast (\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D713}})^{2}\unicode[STIX]{x1D700}$ ; using (20), we can moreover rewrite $(\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D713}})^{2}\unicode[STIX]{x1D700}$ as $(1+\mathbf{F}^{-1}\ast \unicode[STIX]{x1D700})\ast (\unicode[STIX]{x1D6FB}^{\mathbf{F}})^{2}\unicode[STIX]{x1D700}$ plus some terms factorising through $(\unicode[STIX]{x1D6FB}^{\mathbf{F}}\unicode[STIX]{x1D700})^{\ast 2}$ ;

2. (ii) all other terms factor through $(\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D713}})^{\ast 2}\unicode[STIX]{x1D700}$ ; according to (20), one can say these terms factor through $(\unicode[STIX]{x1D6FB}^{\mathbf{F}})^{\ast 2}\unicode[STIX]{x1D700}$ as well;

3. (iii) the algebraic coefficients are controlled (for $\mathbf{F}$ say) in $C^{1,\unicode[STIX]{x1D6FC}}$ .

We sum these three points up by writing

(28) $$\begin{eqnarray}{\textstyle \frac{1}{2}}\mathscr{L}_{\unicode[STIX]{x1D719}^{\ast }\mathbf{f}^{\flat }}\unicode[STIX]{x1D700}+\unicode[STIX]{x1D700}\star (\unicode[STIX]{x1D6FB}^{\mathbf{F}})^{2}\unicode[STIX]{x1D700}=(\unicode[STIX]{x1D6FB}^{\mathbf{F}}\unicode[STIX]{x1D700})^{\ast 2}\ast Q(\unicode[STIX]{x1D700},\unicode[STIX]{x1D6FB}^{\mathbf{F}}\unicode[STIX]{x1D700}),\end{eqnarray}$$

where $\mathscr{L}_{\unicode[STIX]{x1D719}^{\ast }\mathbf{f}^{\flat }}=d_{\mathbf{F}}\unicode[STIX]{x1D6F7}^{\mathbf{F}}$ is the Lichnerowicz Laplacian of $\mathbf{F}=\unicode[STIX]{x1D719}^{\ast }\mathbf{f}^{\flat }$ , the symbols $\star$ denote algebraic operations. Since $\unicode[STIX]{x1D700}\in C_{\unicode[STIX]{x1D6FF}}^{1,\unicode[STIX]{x1D6FC}}(\mathbb{R}^{4},\mathbf{F})$ the right-hand side of (28) is in $C_{2\unicode[STIX]{x1D6FF}+2}^{0,\unicode[STIX]{x1D6FC}}(\mathbb{R}^{4},\mathbf{F})$ . Again since $\unicode[STIX]{x1D700}\in C_{\unicode[STIX]{x1D6FF}}^{1,\unicode[STIX]{x1D6FC}}$ , the linear operator $\unicode[STIX]{x1D702}\,\mapsto \,\frac{1}{2}\mathscr{L}_{\unicode[STIX]{x1D719}^{\ast }\mathbf{f}^{\flat }}\unicode[STIX]{x1D702}+\unicode[STIX]{x1D700}\star (\unicode[STIX]{x1D6FB}^{\mathbf{F}})^{2}\unicode[STIX]{x1D702}$ is elliptic and one can draw for this operator weighted estimates similar to those for $\mathscr{L}_{\unicode[STIX]{x1D719}^{\ast }\mathbf{f}^{\flat }}$ . From this we deduce that $\unicode[STIX]{x1D700}\in C_{\unicode[STIX]{x1D6FF}}^{2,\unicode[STIX]{x1D6FC}}$ . Repeating this argument, we get $\unicode[STIX]{x1D700}\in C_{\unicode[STIX]{x1D6FF}}^{\infty }(\mathbf{F})$ , hence $\mathbf{f}^{\flat }-\unicode[STIX]{x1D719}_{\ast }g_{\unicode[STIX]{x1D713}}=\unicode[STIX]{x1D719}_{\ast }\unicode[STIX]{x1D700}\in C_{\unicode[STIX]{x1D6FF}}^{\infty }(\mathbf{f}^{\flat })$ . Notice that one could even get from this scheme $\unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}_{0}+\unicode[STIX]{x1D700}^{\prime }$ , with $\unicode[STIX]{x1D700}_{0}$ an $\mathscr{L}_{\unicode[STIX]{x1D719}^{\ast }\mathbf{f}^{\flat }}$ -harmonic tensor in $C_{\unicode[STIX]{x1D6FF}}^{\infty }(\mathbf{F})$ , hence in $C_{1}^{\infty }(\mathbf{F})$ , and $\unicode[STIX]{x1D700}^{\prime }\in C_{2\unicode[STIX]{x1D6FF}}^{\infty }(\mathbf{F})$ ; this yields in the end $\unicode[STIX]{x1D719}_{\ast }\unicode[STIX]{x1D700}\in C_{1}^{\infty }(\mathbf{f}^{\flat })$ , which is better than needed.◻