Chapter 2 Taub-NUT Type Metrics on ℂ 3

2.1. First order asymptotic metric near infinity

We plan to construct an approximate Calabi-Yau metric using the generalised Gibbons-Hawking ansatz, on a singular $T^{2}$ -bundle $M$ over (the complement of a compact subset of) the real 4-dimensional base $\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\mathbb{C}_{\eta}$ , whose discriminant locus is

(2.1)

\begin{split}\mathfrak{D}&=\mathfrak{D}_{1}\cup\mathfrak{D}_{2}\cup\mathfrak{D}_{3}\cup\{0\}=\{\mu_{1}=0,\mu_{2}>0\}\cup\{\mu_{2}=0,\mu_{1}>0\}\cup\{\mu_{1}=\mu_{2}<0\}\cup\{0\}\\ &\subset\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\{0\}\subset\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\mathbb{C}_{\eta}\simeq\mathbb{R}^{4}.\end{split}

The topological situation is the same as in the Harvey-Lawson Example 1.7 in complex dimension 3. Our primary concern is that this metric should be approximately Calabi-Yau near spatial infinity, while on a compact set this approximation is allowed to fail.

The basic heuristic idea is to perturb the constant solution (cf. Example 1.6) in a way which incorporates the topology. The information in the constant solution is contained in the base metric

g_{a}=a_{ij}d\mu_{i}\otimes d\mu_{j}+A|d\eta|^{2},

where $(a_{ij})$ is a real symmetric positive definite $2\times 2$ matrix with inverse matrix $(a^{ij})$ , and $A=\det a$ . The matrix $a^{ij}$ describes the asymptotic metric on the $T^{2}$ -fibres. The associated volume measure is

d\text{Vol}_{a}=A^{3/2}d\mu_{1}\wedge d\mu_{2}\wedge d\text{Re}\eta\wedge d\text{Im}\eta.

Now in terms of the local potential $\Phi$ the Calabi-Yau condition (1.9) reads

\det(\frac{\partial^{2}\Phi}{\partial\mu_{i}\partial\mu_{j}})=-4\frac{\partial^{2}\Phi}{\partial\eta\partial\bar{\eta}},

whose linearised equation at the constant solution is the Laplace equation

\Delta_{a}\phi=a^{ij}\frac{\partial^{2}\phi}{\partial\mu_{i}\partial\mu_{j}}+4A^{-1}\frac{\partial^{2}\phi}{\partial\eta\partial\bar{\eta}}=0.

Here $\Delta_{a}$ is unsurprisingly the Laplacian of $g_{a}$ . This suggests that at least away from the discriminant locus, the first order correction to $V^{ij}$ and $W$ from the constant solution

(2.2)

v^{ij}=\frac{\partial^{2}\phi}{\partial\mu_{i}\partial\mu_{j}},\quad w=-4\frac{\partial^{2}\phi}{\partial\eta\partial\bar{\eta}}

ought to be given by $\Delta_{a}$ -harmonic functions,

(2.3)

\Delta_{a}v^{ij}=0,\quad\Delta_{a}w=0,\quad Aa^{ij}v^{ij}=w.

To incorporate the topology we need to recall the distributional equation (1.16) on $V^{ij}$ and $W$ . Since $v^{ij}$ and $w$ are linearisations, it makes sense to require the equation on currents

(2.4)

\frac{\sqrt{-1}}{4\pi}\left(\frac{\partial^{2}w}{\partial\mu_{i}\partial\mu_{j}}+4\frac{\partial^{2}v^{ij}}{\partial\eta\partial\bar{\eta}}\right)d\mu_{i}\wedge d\eta\wedge d\bar{\eta}\otimes e_{j}=\mathfrak{D}_{1}\otimes e_{1}-\mathfrak{D}_{2}\otimes e_{2}+\mathfrak{D}_{3}\otimes(e_{2}-e_{1}).

The task is to find a compatible solution to (2.2)(2.3)(2.4). Notice that $v^{ij}$ and $w$ are global quantities while $\phi$ is only local. We view $v^{ij}$ and $w$ as the unknown functions in this system of equations, and the existence of a local $\phi$ solving (2.2) is equivalent to some integrability conditions on $v^{ij}$ and $w$ away from $\mathfrak{D}$ ,

(2.5)

\frac{\partial v^{11}}{\partial\mu_{2}}=\frac{\partial v^{12}}{\partial\mu_{1}},\quad\frac{\partial v^{22}}{\partial\mu_{1}}=\frac{\partial v^{21}}{\partial\mu_{2}},

and

\frac{\partial^{2}w}{\partial\mu_{i}\partial\mu_{j}}=-4\frac{\partial^{2}v^{ij}}{\partial\eta\partial\bar{\eta}}.

Remark 2.1.

(Informal discussion on singularity) The $v^{ij}$ and $w$ should have very specific singularities along $\mathfrak{D}_{1}$ , $\mathfrak{D}_{2}$ , $\mathfrak{D}_{3}$ . Let us focus on what happens around $\mathfrak{D}_{1}$ . The delta forcing term appears in the component of (2.4) as

\frac{\sqrt{-1}}{4\pi}\left(\frac{\partial^{2}w}{\partial\mu_{1}\partial\mu_{1}}+4\frac{\partial^{2}v^{11}}{\partial\eta\partial\bar{\eta}}\right)d\mu_{1}\wedge d\eta\wedge d\bar{\eta}=\mathfrak{D}_{1}.

If we denote the Lebesgue measure $f\mapsto\int_{\mathfrak{D}_{1}}fd\mu_{2}$ as $\int_{\mathfrak{D}_{1}}d\mu_{2}$ , we may rewrite this equation as

\frac{1}{2\pi}\left(\frac{\partial^{2}w}{\partial\mu_{1}\partial\mu_{1}}+4\frac{\partial^{2}v^{11}}{\partial\eta\partial\bar{\eta}}\right)d\mu_{1}\wedge d\mu_{2}\wedge d\text{Re}\eta\wedge d\text{Im}{\eta}=-\int_{\mathfrak{D}_{1}}d\mu_{2}.

Since $v^{12},v^{22}$ do not see the forcing term, our best guess is that they are smooth along $\mathfrak{D}_{1}$ . Then modulo smooth terms $w\sim Aa^{11}v^{11}=a_{22}v^{11}$ along $\mathfrak{D}_{1}$ , from which the distributional equation gives the singularity structure along $\mathfrak{D}_{1}$ :

v^{11}\sim\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}},\quad w\sim\frac{a_{22}}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}.

The following base metric encoding the Gibbons-Hawking data has the singularity structure along $\mathfrak{D}_{1}$

\begin{split}&g_{a}+v^{ij}d\mu_{i}d\mu_{j}+w|d\eta|^{2}\sim g_{a}+\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}(d\mu_{1}^{2}+a_{22}|d\eta|^{2})\\ =&(\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}+\frac{A}{a_{22}})(d\mu_{1}^{2}+a_{22}|d\eta|^{2})+a_{22}(d(\mu_{2}+\frac{a_{12}}{a_{22}}\mu_{1}))^{2}\end{split}

from which we recognize the Taub-NUT metric appearing in directions transverse to $\mathfrak{D}_{1}$ . See Section 2.3 for further details.

We now move on to a more formal construction.

Lemma 2.1.

The functions

(2.6)

\begin{cases}\alpha_{1}(\mu_{1},\mu_{2},\eta)=&\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}\{\frac{1}{2}+\frac{1}{\pi}\arctan(\frac{a_{22}\mu_{2}+a_{12}\mu_{1}}{\sqrt{A}\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}})\},\\ \alpha_{2}(\mu_{1},\mu_{2},\eta)=&\frac{1}{2\sqrt{\mu_{2}^{2}+a_{11}|\eta|^{2}}}\{\frac{1}{2}+\frac{1}{\pi}\arctan(\frac{a_{11}\mu_{1}+a_{12}\mu_{2}}{\sqrt{A}\sqrt{\mu_{2}^{2}+a_{11}|\eta|^{2}}})\},\\ \alpha_{3}(\mu_{1},\mu_{2},\eta)=&\frac{1}{2\sqrt{(\mu_{1}-\mu_{2})^{2}+(a_{11}+2a_{12}+a_{22})|\eta|^{2}}}\\ &\{\frac{1}{2}+\frac{1}{\pi}\arctan(\frac{-a_{11}\mu_{1}-a_{12}\mu_{2}-a_{21}\mu_{1}-a_{22}\mu_{2}}{\sqrt{A}\sqrt{(\mu_{1}-\mu_{2})^{2}+(a_{11}+a_{12}+a_{21}+a_{22})|\eta|^{2}}})\}\end{cases}

satisfy the equations on measures

(2.7)

\begin{cases}(\Delta_{a}\alpha_{1})d\text{Vol}_{a}=-2\pi\sqrt{A}\int_{\mathfrak{D}_{1}}d\mu_{2},\\ (\Delta_{a}\alpha_{2})d\text{Vol}_{a}=-2\pi\sqrt{A}\int_{\mathfrak{D}_{2}}d\mu_{1},\\ (\Delta_{a}\alpha_{3})d\text{Vol}_{a}=2\pi\sqrt{A}\int_{\mathfrak{D}_{3}}d\mu_{1}\end{cases}

where the RHS are signed measures supported on $\mathfrak{D}_{1},\mathfrak{D}_{2},\mathfrak{D}_{3}$ . Morever,

(2.8)

\frac{\partial\alpha_{1}}{\partial\mu_{2}}=\frac{\partial\alpha_{2}}{\partial\mu_{1}}=(-\frac{\partial}{\partial\mu_{1}}-\frac{\partial}{\partial\mu_{2}})\alpha_{3}=\frac{\sqrt{A}}{2\pi|\vec{\mu}|_{a}^{2}}.

The singularity of $\alpha_{i}$ occurs along $\mathfrak{D}_{i}$ and modulo smooth terms looks like

\begin{cases}\alpha_{1}\sim\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}},\\ \alpha_{2}\sim\frac{1}{2\sqrt{\mu_{2}^{2}+a_{11}|\eta|^{2}}},\\ \alpha_{3}\sim\frac{1}{2\sqrt{(\mu_{1}-\mu_{2})^{2}+(a_{11}+2a_{12}+a_{22})|\eta|^{2}}}.\end{cases}

Proof.

We denote $\vec{\mu}=(\mu_{1},\mu_{2},\eta)$ and $|\vec{\mu}|_{a}=\sqrt{a_{ij}\mu_{i}\mu_{j}+A|\eta|^{2}}$ . The Green representation

\begin{split}&\frac{-1}{4\pi^{2}}\int_{0}^{\infty}\frac{1}{|\vec{\mu}-(0,s,0)|_{a}^{2}}ds\\ =&\frac{-1}{4\pi^{2}}\int_{0}^{\infty}\frac{1}{a_{11}\mu_{1}^{2}-2a_{12}\mu_{1}(s-\mu_{2})+a_{22}(s-\mu_{2})^{2}+A|\eta|^{2}}ds\\ =&\frac{-1}{4\pi^{2}}\int_{-\mu_{2}}^{\infty}\frac{1}{a_{11}\mu_{1}^{2}-2a_{12}\mu_{1}s+a_{22}s^{2}+A|\eta|^{2}}ds\\ =&\frac{-1}{4\pi}\frac{1}{\sqrt{A}}\frac{1}{\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}\{\frac{1}{2}+\frac{1}{\pi}\arctan(\frac{a_{22}\mu_{2}+a_{12}\mu_{1}}{\sqrt{A}\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}})\}\\ =&\frac{-1}{2\pi\sqrt{A}}\alpha_{1}\end{split}

shows the equality of the two measures

(\Delta_{a}\alpha_{1})d\text{Vol}_{a}=-2\pi\sqrt{A}\int_{\mathfrak{D}_{1}}d\mu_{2},

and it is easy to check from this integral calculation

\frac{\partial\alpha_{1}}{\partial\mu_{2}}=\frac{\sqrt{A}}{2\pi|\vec{\mu}|_{a}^{2}}.

To see the singularity structure near $\mathfrak{D}_{1}=\{\mu_{1}=0,\eta=0,\mu_{2}>0\}$ explicitly, we can write

\alpha_{1}=\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}\{1-\frac{1}{\pi}\arctan(\frac{\sqrt{A}\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}{a_{22}\mu_{2}+a_{12}\mu_{1}})\},

and Taylor expand the arctan function.

The situations of $\alpha_{2},\alpha_{3}$ are similar. A fast way to derive them by analogy is to remember that $\frac{\partial}{\partial\mu_{2}},\frac{\partial}{\partial\mu_{1}},-\frac{\partial}{\partial\mu_{1}}-\frac{\partial}{\partial\mu_{2}}$ are the directional vectors along $\mathfrak{D}_{1},\mathfrak{D}_{2},\mathfrak{D}_{3}$ , and notice

\iota_{\frac{\partial}{\partial\mu_{2}}}g_{a}=d(a_{12}\mu_{1}+a_{22}\mu_{2}),\quad g_{a}(\frac{\partial}{\partial\mu_{2}},\frac{\partial}{\partial\mu_{2}})=a_{22}.

∎

Proposition 2.2.

(First order linearised solution) The equations defining $v^{ij}$ and $w$

(2.9)

v^{11}=\alpha_{1}+\alpha_{3},\quad v^{12}=v^{21}=-\alpha_{3},\quad v^{22}=\alpha_{2}+\alpha_{3},\quad w=Aa^{ij}v^{ij}

or equivalently

v^{ij}d\mu_{i}\otimes d\mu_{j}=\alpha_{1}d\mu_{1}^{2}+\alpha_{2}d\mu_{2}^{2}+\alpha_{3}(d(\mu_{1}-\mu_{2}))^{2},\quad w=Aa^{ij}v^{ij}

provide a solution to the integrability condition (2.2) and the harmonicity condition (2.3) away from $\mathfrak{D}$ , which also satisfies the distributional equation (2.4) globally.

Proof.

The $\Delta_{a}$ -harmonicity away from $\mathfrak{D}$ follows from (2.7). To see the distributional equation (2.4), it suffices to notice that both sides are $\Delta_{a}$ -harmonic away from $\mathfrak{D}$ , and the singularities along $\mathfrak{D}$ match up by construction.

The integrability condition (2.5) is equivalent to (2.8). Together with the distributional equation this implies the local existence of the potential required by (2.2). ∎

Remark 2.2.

A Liouville theorem argument shows that the solution $v^{ij}$ and $w$ to the linear system of equation (2.2)(2.3)(2.4) is unique, in the sense that if another solution differs from it by functions with some power law decay at infinity, then the two solutions agree. The key point is to analyse the difference of the two solutions, and observe that now there is no forcing term in the distributional equation, so the $\Delta_{a}$ -harmonicity extend across the discriminant locus.

Now we define the Kähler ansatz $(g^{(1)},\omega^{(1)},J,\Omega)$ via the generalised Gibbons-Hawking construction, using the functions

V^{ij}_{(1)}=a_{ij}+v^{ij},\quad W_{(1)}=A+w.

Here the script $(1)$ signifies first order approximation. The complex structure and the holomorphic volume form are not scripted because they turn out to agree with the standard structures on $\mathbb{C}^{3}$ and will not be corrected in a later stage. Notice that the positive definite condition on $V^{ij}$ is implied by $\alpha_{i}\geq 0,i=1,2,3$ , which can be checked from the explicit formula. A grain of salt is that there is no a priori guarantee that the metric is smooth over the discriminant locus, a problem we shall take up in Section 2.3.

Let us examine the approximation to the Calabi-Yau condition. This is measured by the volume form error function

(2.10)

E^{(1)}=\frac{W_{(1)}}{\det(V^{ij}_{(1)})}-1=\frac{A+w}{A+Aa^{ij}v^{ij}+\det(v^{ij})}-1=-\frac{\det(v^{ij})}{A+w+\det(v^{ij})},

where $\det(v^{ij})=\alpha_{1}\alpha_{2}+\alpha_{1}\alpha_{3}+\alpha_{2}\alpha_{3}$ and $w=a_{22}\alpha_{1}+a_{11}\alpha_{2}+(a_{11}+a_{22}+2a_{12})\alpha_{3}$ . We denote $|\vec{\mu}|_{a}=\sqrt{a_{ij}\mu_{i}\mu_{j}+A|\eta|^{2}}$ . Near spatial infinity $E^{(1)}=O(\frac{1}{A^{1/2}|\mu|_{a}^{2}})$ sufficiently far away from the discriminant locus, and $E^{(1)}=O(\frac{1}{A^{1/4}|\mu|_{a}})$ near the discriminant locus. However in standard analytic packages [12] which construct Calabi-Yau metrics from asymptotic approximate solutions, it is essential to have faster than quadratic volume error decay rate, which is not satisfied by our ansatz, so this error must first be corrected. This issue will be explained more amply in Section 2.7.

Now we comment on the symmetry of the ansatz. Apart from the $T^{2}$ -symmetry from the construction, there is an additional $U(1)$ -symmetry for the Kähler metric commuting with the $T^{2}$ -action:

\mu_{i}\mapsto\mu_{i},\quad\eta\mapsto e^{i\theta}\eta.

However, the holomorphic volume form will be rotated by a phase angle under this action. This may be compared to the Taub-NUT metric, which has an $SO(3)$ -symmetry acting on the base. Our ansatz has less continuous symmetry because the base contains a distinguished trivalent graph, which is a new higher dimensional phenomenon. Another analogy to draw from this comparison is that when the Taub-NUT metric glues into the Ooguri-Vafa metric, these additional symmetries are broken, and the same phenomenon shall happen when we construct the Ooguri-Vafa type metrics on the positive vertex. This is because the Ooguri-Vafa type metrics involve an extra periodicity condition on $\eta$ which is not compatible with rotation; an alternative viewpoint is that the special Lagrangian fibration selects out a preferred phase angle.

In some special cases there can be some discrete symmetries from permuting the 3 edges of the trivalent graph. The most symmetric situation is where

a_{ij}d\mu_{i}d\mu_{j}\propto(d\mu_{1})^{2}+(d\mu_{2})^{2}+(d(\mu_{1}-\mu_{2}))^{2},

or equivalently

\begin{bmatrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{bmatrix}\propto\begin{bmatrix}2&-1\\ -1&2\end{bmatrix}

This choice of parameters has a special significance in the theory.

Morever, the family of ansatzs have a scaling symmetry which will be fundamental when we construct the Ooguri-Vafa type metrics later. This symmetry is prescribed by (1.17). In our concrete construction, this means replacing

a_{ij}\mapsto\Lambda a_{ij},\quad A\mapsto\Lambda^{2}A.

The region near $(\mu_{1},\mu_{2},\eta)$ in the $(a_{ij})$ -ansatz correspond to the region near the point $(\Lambda^{-1}\mu_{1},\Lambda^{-1}\mu_{2},\Lambda^{-1.5}\eta)$ in the $(\Lambda a_{ij})$ -ansatz. For example, a useful scaling-invariant quantity is $A^{1/4}|\vec{\mu}|_{a}$ :

(\Lambda^{2}A)^{1/4}\sqrt{(\Lambda a_{ij})(\Lambda^{-1}\mu_{i})(\Lambda^{-1}\mu_{j})+(\Lambda^{2}A)|\Lambda^{-1.5}\eta|^{2}}=A^{1/4}|\vec{\mu}|_{a}.

It defines the approximation scale $A^{1/4}|\vec{\mu}|_{a}\gtrsim 1$ , namely the region where the ansatz is approximately Calabi-Yau. Another scaling-invariant quantity is $A^{1/4}\ell$ where $\ell=A^{-1/4}+\text{dist}_{g_{a}}(\cdot,\mathfrak{D})$ . The scaling symmetry enables us to easily extract information about the $(\Lambda a_{ij})$ -ansatz by analysing the $(a_{ij})$ -ansatz, which is very useful for analytical questions.

Remark 2.3.

The constants appearing in this Chapter depend only on Hölder exponents and the following uniform ellipticity bound on $a_{ij}$ :

C^{-1}\delta_{ij}\leq a_{ij}\leq C\delta_{ij}.

The scaling argument can then be used to relax the uniform ellipticity to

(2.11)

C^{-1}A^{1/2}\delta_{ij}\leq a_{ij}\leq CA^{1/2}\delta_{ij}.

In strategic places we will in fact track down the $A$ -dependence as well.

2.2. Metric behaviour away from the discriminant locus

This Section uses weighted Hölder norms to quantify the idea that sufficiently away from the discriminant locus the metric $g^{(1)}$ is approximated by the constant solution.

Given a large number $C_{1}\gg 1$ , we consider $M$ over the base region

(2.12)

\begin{cases}A^{1/4}|\vec{\mu}|_{a}\geq C_{1},\\ |\vec{\mu}|_{a}\leq 2C_{1}\ell,\end{cases}

meaning that the region is far from the origin, and the $g_{a}$ -distance to $\mathfrak{D}$ is comparable to the $g_{a}$ -distance to the origin. Topologically the base region is obtained by removing the apex from a cone over a thrice-punctured 3-sphere. The $A$ -dependence is inserted for convenience when we analyse the scaling behaviours.

The flat model metric is simply constructed by applying the generalised Gibbons-Hawking ansatz to $V^{ij}_{\text{flat}}=a_{ij}$ and $W_{\text{flat}}=A$ :

g_{\text{flat}}=a_{ij}d\mu_{i}d\mu_{j}+A|d\eta|^{2}+a^{ij}\vartheta_{i}^{\text{flat}}\vartheta_{j}^{\text{flat}},

where $\vartheta_{i}^{\text{flat}}$ for $i=1,2$ are flat connections. Likewise we define $\omega_{\text{flat}}$ and $\Omega_{\text{flat}}$ . A subtlety is that $g_{\text{flat}}$ cannot model $g^{(1)}$ globally over the region defined by (2.12), because the Chern class of the $T^{2}$ -bundle for $g^{(1)}$ evaluates nontrivially on the $S^{2}$ cycles wrapping the 3 puncture points in $S^{3}$ , which obstructs the flat connection $\vartheta_{i}^{\text{flat}}$ . It is thence understood that we are comparing the model metric with $g^{(1)}$ over a finite number of contractible conical subregions which cover (2.12).

The deviation of $V^{ij}$ from $a_{ij}$ is measured by $\alpha_{1},\alpha_{2},\alpha_{3}$ . To estimate these quantities over these regions we introduce some weighted Hölder norms associated to the reference metrics $g_{\text{flat}}$ . For any $T^{2}$ -invariant tensor field $T$ defined over the region, we define the normalised Hölder seminorm

[T]_{\alpha}=\sup_{p}|\vec{\mu}|_{a}^{\alpha}\cdot\sup_{|p-p^{\prime}|_{a}<\frac{1}{10}\ell}\frac{|T(p)-T(p^{\prime})|}{d_{\text{flat}}(p,p^{\prime})^{\alpha}}

where we compare $T(p)$ and $T(p^{\prime})$ using parallel transport along minimal geodesics. The weighted norm of $T$ is then defined by

\left\lVert T\right\rVert_{C^{k,\alpha}_{\tau^{\prime}}}=A^{-\tau^{\prime}/4}\sum_{j=0}^{k}\left\lVert|\vec{\mu}|_{a}^{-\tau^{\prime}+j}\nabla^{j}T\right\rVert_{L^{\infty}}+A^{-\tau^{\prime}/4}[|\vec{\mu}|_{a}^{-\tau^{\prime}+k}\nabla^{k}T]_{\alpha}.

An estimate in this norm is thought as the higher order version of $|T|=O(A^{\tau^{\prime}/4}|\vec{\mu}|_{a}^{\tau^{\prime}})$ .

Lemma 2.3.

Over each of the finitely many contractible conical subregions $\left\lVert\alpha_{i}\right\rVert_{C^{k,\alpha}_{-1}}\leq{CA^{1/2}}$ .

Proof.

The absolute value estimate $|\alpha_{i}|\leq\frac{CA^{1/4}}{|\vec{\mu}|_{a}}$ is clear from the explicit defining formula. The higher order estimates use that $\Delta_{a}\alpha_{i}=0$ holds over a $g_{a}$ -ball of radius comparable to $|\vec{\mu}|_{a}$ . ∎

Next we estimate the deviation of $\vartheta_{i}$ from $\vartheta_{i}^{\text{flat}}$ . Since gauge equivalent choices of $\vartheta_{i}$ give rise to the same Kähler structure $(g^{(1)},\omega^{(1)},J,\Omega)$ up to holomorphic isometry, we may make any convenient gauge choice. The defining condition on $\vartheta_{i}$ is

d\vartheta_{i}=\sqrt{-1}\left(\frac{1}{2}\frac{\partial W_{(1)}}{\partial\mu_{i}}d\eta\wedge d\bar{\eta}+\frac{\partial V^{ij}_{(1)}}{\partial\eta}d\mu_{j}\wedge d\eta-\frac{\partial V^{ij}_{(1)}}{\partial\bar{\eta}}d\mu_{j}\wedge d\bar{\eta}\right),

and $d\vartheta_{i}^{\text{flat}}=0$ . Thus $\left\lVert d(\vartheta_{i}-\vartheta_{i}^{\text{flat}})\right\rVert_{C^{k,\alpha}_{-2}}\leq CA^{1/2}$ using the higher derivative estimates on $\alpha_{i}$ . Using the d-Poincaré lemma, we can find a gauge fixed choice of the 1-form $\vartheta_{i}-\vartheta_{i}^{\text{flat}}$ such that $\left\lVert\vartheta_{i}-\vartheta_{i}^{\text{flat}}\right\rVert_{C^{k,\alpha}_{-1}}\leq CA^{1/4}$ . Combining these discussions, and noticing $|d\mu_{i}|\leq CA^{-1/4},|d\eta|\leq CA^{-1/2}$ , we obtain

Corollary 2.4.

Over each of the finitely many contractible conical subregions, after suitable gauge fixing, we have the deviation estimates

\left\lVert g^{(1)}-g_{\text{flat}}\right\rVert_{C^{k,\alpha}_{-1}}\leq C,\quad\left\lVert\omega^{(1)}-\omega_{\text{flat}}\right\rVert_{C^{k,\alpha}_{-1}}\leq C,\quad\left\lVert\Omega-\Omega_{\text{flat}}\right\rVert_{C^{k,\alpha}_{-1}}\leq C.

Morever the volume form error function $E^{(1)}$ satisfies $\left\lVert E^{(1)}\right\rVert_{C^{k,\alpha}_{-2}}\leq C$ , namely the higher order version of quadratic decay estimate.

2.3. Structure near discriminant locus

We now study the metric near the discriminant locus but sufficiently far from the origin, which turns out to be locally modelled on a fibration by Taub-NUT metrics over a flat cylinder. A subtlety is that the smooth topology along $\mathfrak{D}_{i}$ is not a priori prescribed, and needs to be elucidated first.

We focus on the neighbourhood of $\mathfrak{D}_{1}$ far from the origin, where $\alpha_{1}\sim\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}$ and $\alpha_{2},\alpha_{3}$ are smooth. To leading order

V_{(1)}\sim V_{\text{Taub}}=\begin{bmatrix}\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}+a_{11}&a_{12}\\ a_{21}&a_{22}\end{bmatrix},W_{(1)}\sim W_{\text{Taub}}=A+\frac{a_{22}}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}.

The inverse matrix is

V_{(1)}^{-1}\sim V_{\text{Taub}}^{-1}=({A+\frac{a_{22}}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}})^{-1}\begin{bmatrix}a_{22}&-a_{21}\\ -a_{12}&\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}+a_{11}\end{bmatrix}.

Now if we apply the generalised Gibbons-Hawking ansatz to $V_{\text{Taub}}$ and $W_{\text{Taub}}$ , we obtain a model metric

g_{\text{Taub}}=V^{ij}_{\text{Taub}}d\mu_{i}d\mu_{j}+W_{\text{Taub}}|d\eta|^{2}+(V^{-1}_{\text{Taub}})^{ij}\vartheta_{i}^{\prime}\vartheta_{j}^{\prime}

where $\vartheta_{1}^{\prime},\vartheta_{2}^{\prime}$ are the connections. As $d\vartheta_{2}^{\prime}=0$ we may write $\vartheta_{2}^{\prime}=d\theta_{2}^{\prime}$ . Rewriting the model metric,

(2.13)

\begin{split}g_{\text{Taub}}=(\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}+\frac{A}{a_{22}})(d\mu_{1}^{2}+a_{22}|d\eta|^{2})+a_{22}(d(\mu_{2}+\frac{a_{12}}{a_{22}}\mu_{1}))^{2}\\ +(\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}+\frac{A}{a_{22}})^{-1}(\vartheta_{1}^{\prime}-\frac{a_{12}}{a_{22}}d\theta_{2}^{\prime})^{2}+\frac{1}{a_{22}}(d\theta_{2}^{\prime})^{2}.\end{split}

Notice the dual basis for $\{\vartheta_{1}^{\prime}-\frac{a_{12}}{a_{22}}d\theta_{2}^{\prime},\vartheta_{2}^{\prime}\}$ is given by $\{\frac{\partial}{\partial\theta_{1}},\frac{\partial}{\partial\theta_{2}}+\frac{a_{12}}{a_{22}}\frac{\partial}{\partial\theta_{1}}\}$ , which corresponds to the moment coordinates $\mu_{1}$ and $\mu_{2}+\frac{a_{12}}{a_{22}}\mu_{1}$ .

The variables $\mu_{2}+\frac{a_{12}}{a_{22}}\mu_{1}$ and $\theta_{2}^{\prime}$ define a cylinder $\mathbb{R}\times S^{1}$ . Translations in these variables are isometries of the model space. The model space fibres over this cylinder, and restricted to each fibre the metric is recognized as the Taub-NUT metric with parameter $\frac{A}{a_{22}}$ . The fibration is not always a metric product, because for the generators $\frac{\partial}{\partial\theta_{1}}$ and $\frac{\partial}{\partial\theta_{2}}+\frac{a_{12}}{a_{22}}\frac{\partial}{\partial\theta_{1}}$ to give rise to an integral basis of $H_{1}(T^{2})$ we need $\frac{a_{12}}{a_{22}}$ to be an integer. On the universal cover the metric becomes the product of Taub-NUT metric with the flat $\mathbb{R}^{2}$ , as $\theta_{2}$ becomes a real variable instead of a circle variable. In particular the universal cover is topologically $\mathbb{C}^{2}\times\mathbb{R}^{2}$ , and the model space is a discrete $\mathbb{Z}$ -quotient of $\mathbb{C}^{2}\times\mathbb{R}^{2}$ , so inherits a smooth topology. The Riemannian curvature on the model metric is bounded but does not decay as we move to infinity along $\mathfrak{D}_{1}$ .

Remark 2.4.

We wish to amplify the idea that the smooth topology of the $S^{1}$ -fibration map is subtle. Given a $T^{2}$ -fibration $M\to\mathcal{B}$ say, the $T^{2}$ -invariant smooth functions on $M$ descend into a sheaf of functions on the base, sitting between the sheaf of smooth functions on $\mathcal{B}$ and the sheaf of continuous functions on $\mathcal{B}$ . An example of such a function on our model space is $\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}$ . Had we chosen a different $a_{22}$ to begin with, this sheaf would be different. This means assigning a smooth topology on the compactification of a torus bundle across the discriminant locus, is a problem which involves extra data. In general this sheaf depends on functions along $\mathfrak{D}_{i}$ , so carries an infinite amount of information, and is therefore expected to be unstable under deformation. This subtlety is related to Joyce’s observation that special Lagrangian fibrations can fail to be given by smooth maps (cf. review Section 1.1.5 and Section 4.12).

Our next goal is to quantify the idea that the model $g_{\text{Taub}}$ is a good approximation to the metric ansatz $g^{(1)}$ . We view both metrics as defined on the same smooth manifold, fibred over the region

(2.14)

|\vec{\mu}|_{a}\geq C_{1}A^{-1/4},\quad|\vec{\mu}|_{a}\geq C_{1}\text{dist}_{g_{a}}(\cdot,\mathfrak{D}_{1})

where $C_{1}$ is a large number as in Section 2.2. In this region the $g_{a}$ -distance to $\mathfrak{D}_{2},\mathfrak{D}_{3}$ are both $O(|\vec{\mu}|_{a})$ , and $\text{dist}_{g_{a}}(\cdot,\mathfrak{D}_{1})$ is comparable to $A^{1/4}\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}$ , so

\ell=A^{-1/4}+\text{dist}_{g_{a}}(\cdot,\mathfrak{D})\sim A^{-1/4}+A^{1/4}\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}.

We introduce some weighted Hölder norms associated to the reference metric $g_{\text{Taub}}$ . The regularity scale of $g_{\text{Taub}}$ is comparable to $\ell$ . For any $T^{2}$ -invariant tensor field $T$ over the region (2.14) , define the normalised Hölder seminorm

[T]_{\alpha}=\sup_{p}\ell(p)^{\alpha}\cdot\sup_{p^{\prime}\in B_{g_{\text{Taub}}(p,\ell/10)}}\frac{|T(p)-T(p^{\prime})|}{d_{\text{Taub}}(p,p^{\prime})^{\alpha}}

where we compare $T(p)$ and $T(p^{\prime})$ using parallel transport along minimal geodesics. The weighted norm of $T$ is then defined by

\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta,\tau}}=A^{-\delta/4-\tau/4}\sum_{j=0}^{k}\left\lVert\ell^{-\delta+j}|\vec{\mu}|_{a}^{-\tau}\nabla^{j}T\right\rVert_{L^{\infty}}+A^{-\delta/4-\tau/4}[\ell^{-\delta+k}|\vec{\mu}|_{a}^{-\tau}\nabla^{k}T]_{\alpha}.

An estimate in this norm can be thought as the higher order version of $|T|=O(A^{\tau/4+\delta/4}\ell^{\delta}|\vec{\mu}|_{a}^{\tau})$ . Similar weighted Hölder norms are defined in the neighbourhood of $\mathfrak{D}_{2}$ and $\mathfrak{D}_{3}$ .

The deviation between $V^{ij}_{\text{Taub}}$ and $V^{ij}_{(1)}$ near $\mathfrak{D}_{1}$ is measured by the functions $\alpha_{1}-\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}$ , $\alpha_{2}$ and $\alpha_{3}$ .

Lemma 2.5.

In the above region (2.14) near $\mathfrak{D}_{1}$ ,

\begin{cases}\left\lVert\alpha_{2}\right\rVert_{C^{k,\alpha}_{0,-1}}\leq CA^{1/2},\\ \left\lVert\alpha_{3}\right\rVert_{C^{k,\alpha}_{0,-1}}\leq CA^{1/2},\\ \left\lVert\alpha_{1}-\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}\right\rVert_{C^{k,\alpha}_{0,-1}}\leq CA^{1/2}.\end{cases}

Consequently, if $C_{1}$ is chosen to be large enough, then

|V^{ij}_{(1)}-V^{ij}_{\text{Taub}}|\leq\frac{CA^{1/4}}{|\vec{\mu}|_{a}}\ll a_{ij}\leq V^{ij}_{\text{Taub}},\quad|W_{(1)}-W_{\text{Taub}}|\leq\frac{CA^{3/4}}{|\vec{\mu}|_{a}}\ll A\leq W_{\text{Taub}}.

Proof.

The $\Delta_{a}$ -harmonic function $\alpha_{2},\alpha_{3}$ are both of order $O(\frac{A^{1/4}}{|\vec{\mu}|_{a}})$ . The function

\alpha_{1}-\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}=-\frac{1}{2\pi\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}\arctan(\frac{\sqrt{A}\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}{a_{22}\mu_{2}+a_{12}\mu_{1}})

is also $\Delta_{a}$ -harmonic, and by the Taylor expansion of $\arctan$ is seen to be $O(\frac{A^{1/4}}{|\vec{\mu}|_{a}})$ as well. These functions are smooth on the base in the region (2.14) with regularity scale $O(|\vec{\mu}|_{a})$ . The $\Delta_{a}$ -harmonicity takes care of all higher order estimates. ∎

Next we analyse the deviation between the connections $\vartheta_{i}$ and $\vartheta_{i}^{\prime}$ for $i=1,2$ , corresponding to the ansatz $g^{(1)}$ and the model $g_{\text{Taub}}$ respectively. This involves the same gauge fixing issue as in Section 2.2. The defining condition on $\vartheta_{i}$ is

d\vartheta_{i}=\sqrt{-1}\left(\frac{1}{2}\frac{\partial W_{(1)}}{\partial\mu_{j}}d\eta\wedge d\bar{\eta}+\frac{\partial V^{ij}_{(1)}}{\partial\eta}d\mu_{i}\wedge d\eta-\frac{\partial V^{ij}_{(1)}}{\partial\bar{\eta}}d\mu_{i}\wedge d\bar{\eta}\right),

and similarly for $\vartheta_{i}^{\prime}$ . Thus $\left\lVert d(\vartheta_{i}-\vartheta_{i}^{\prime})\right\rVert_{C^{0,-2}}\leq CA^{1/2}$ using the higher derivative estimates on $V^{ij}_{(1)}-V^{ij}_{\text{Taub}}$ etc. Using the d-Poincaré lemma, we can find a gauge fixed choice of the smooth 1-form $\vartheta_{i}-\vartheta_{i}^{\prime}$ such that $\left\lVert\vartheta_{i}-\vartheta_{i}^{\prime}\right\rVert_{C^{k,\alpha}_{0,-1}}\leq CA^{1/4}.$ Combining the above, and recalling $|d\mu_{i}|\leq CA^{-1/4}$ , $|d\eta|\leq CA^{-1/2}$ , we obtain

Lemma 2.6.

The Kähler structure $(g^{(1)},\omega^{(1)},J,\Omega)$ extends smoothly over the region (2.14). The deviation from the model metric admits the estimates

\begin{cases}\left\lVert g^{(1)}-g_{\text{Taub}}\right\rVert_{C^{k,\alpha}_{0,-1}}\leq C,\quad&\left\lVert\omega^{(1)}-\omega_{\text{Taub}}\right\rVert_{C^{k,\alpha}_{0,-1}}\leq C,\\ \left\lVert J-J_{\text{Taub}}\right\rVert_{C^{k,\alpha}_{0,-1}}\leq C,\quad&\left\lVert\Omega-\Omega_{\text{Taub}}\right\rVert_{C^{k,\alpha}_{0,-1}}\leq C.\end{cases}

In particular, if $C_{1}$ is chosen large enough, then the magnitudes of the deviation

|g^{(1)}-g_{\text{Taub}}|\ll 1,\quad|\omega^{(1)}-\omega_{\text{Taub}}|\ll 1,\quad|\Omega-\Omega_{\text{Taub}}|\ll 1,\quad|J-J_{\text{Taub}}|\ll 1.

The volume form error function $E^{(1)}$ satisfies

\left\lVert E^{(1)}\right\rVert_{C^{k,\alpha}_{-1,-1}}\leq C.

Remark 2.5.

The same arguments show that the Kähler ansatz is smooth along the entire $\mathfrak{D}_{1}$ , although the smooth topology is not yet defined at the origin; this difficulty will later be resolved by shifting to the complex geometric viewpoint and doing a surgery to the Kähler ansatz.

Remark 2.6.

The metric deviation estimate and the volume form error estimate require $A^{1/4}|\vec{\mu}|_{a}\gtrsim 1$ . Heuristically we may think of the discriminant locus $\mathfrak{D}$ as the source of gravitating force, and for $A^{1/4}|\vec{\mu}|\lesssim 1$ the mutual interactions of $\mathfrak{D}_{1},\mathfrak{D}_{2},\mathfrak{D}_{3}$ become too strong, so the perturbative description breaks down.

Remark 2.7.

Over the subregion of (2.14) where $\ell\geq 2A^{-1/4}$ , namely outside the curvature scale along $\mathfrak{D}_{1}$ , the model metric $g_{\text{Taub}}$ is itself locally approximated by the flat model $g_{\text{flat}}$ (cf. Section 2.2) over $g_{a}$ -balls of radius $\sim\ell(x)$ :

\left\lVert g_{\text{Taub}}-g_{\text{flat}}\right\rVert_{C^{k,\alpha}_{-1,0}}\leq C.

2.4. Complex geometric perspective

We now identify the complex structure on $M$ with $\mathbb{C}^{3}$ . Recall $\zeta_{i}=V^{ij}_{(1)}d\mu_{j}+\sqrt{-1}\vartheta_{i}$ and formula (1.14) for their differentials. The main idea is to produce holomorphic differentials by adjusting $\zeta_{i}$ . The reader can refer to the Taub-NUT Example 1.8 for the warm up.

We define the functions $\beta_{i}(\mu_{1},\mu_{2},\eta)$ for $i=0,1,2$ ,

\begin{cases}\beta_{1}=2\displaystyle\lim_{(\mu_{1}^{\prime},\mu_{1}^{\prime}-\mu_{2}^{\prime})\to(+\infty,+\infty)}\int_{(\mu_{1}^{\prime},\mu_{2}^{\prime})}^{(\mu_{1},\mu_{2})}\frac{\partial\alpha_{1}}{\partial\eta}(s_{1},s_{2},\eta)ds_{1}+\frac{\partial\alpha_{3}}{\partial\eta}(s_{1},s_{2},\eta)d(s_{1}-s_{2})\\ \beta_{2}=2\displaystyle\lim_{(\mu_{2}^{\prime},\mu_{2}^{\prime}-\mu_{1}^{\prime})\to(+\infty,+\infty)}\int_{(\mu_{1}^{\prime},\mu_{2}^{\prime})}^{(\mu_{1},\mu_{2})}\frac{\partial\alpha_{2}}{\partial\eta}(s_{1},s_{2},\eta)ds_{2}+\frac{\partial\alpha_{3}}{\partial\eta}(s_{1},s_{2},\eta)d(s_{2}-s_{1})\\ \beta_{0}=2\displaystyle\lim_{(\mu_{1}^{\prime},\mu_{2}^{\prime})\to(-\infty,-\infty)}\int_{(\mu_{1}^{\prime},\mu_{2}^{\prime})}^{(\mu_{1},\mu_{2})}-\frac{\partial\alpha_{1}}{\partial\eta}(s_{1},s_{2},\eta)ds_{1}-\frac{\partial\alpha_{2}}{\partial\eta}(s_{1},s_{2},\eta)ds_{2}.\end{cases}

In these improper integrals $\eta$ is held fixed. Here the integrability condition (2.8) ensures the integrands are closed differentials, so the integral is path independent. We can take the limit in the definition of $\beta_{1}$ , because

\begin{cases}\frac{\partial\alpha_{1}}{\partial\eta}=O(\frac{a_{22}\bar{\eta}}{(\mu_{1}^{2}+a_{22}|\eta|^{2})^{3/2}}),\quad&\mu_{1}\to\infty,\\ \frac{\partial\alpha_{3}}{\partial\eta}=O(\frac{(a_{11}+2a_{12}+a_{22})\bar{\eta}}{((\mu_{1}-\mu_{2})^{2}+(a_{11}+2a_{12}+a_{22})|\eta|^{2})^{3/2}}),&\mu_{1}-\mu_{2}\to\infty.\end{cases}

Likewise with $\beta_{2},\beta_{0}$ . The domain of definition of $\beta_{1},\beta_{2},\beta_{0}$ are respectively $\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\mathbb{C}_{\eta}\setminus\{\eta=0,\mu_{1}\leq 0,\mu_{1}\leq\mu_{2}\}$ , $\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\mathbb{C}_{\eta}\setminus\{\eta=0,\mu_{2}\leq 0,\mu_{2}\leq\mu_{1}\}$ , and $\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\mathbb{C}_{\eta}\setminus\{\eta=0,\mu_{1}\geq 0,\mu_{2}\geq 0\}$ ; the singularities in the integrands prevent us from defining $\beta_{0},\beta_{1},\beta_{2}$ globally.

Lemma 2.7.

By construction

\begin{cases}\frac{\partial\beta_{1}}{\partial\mu_{1}}=2\frac{\partial}{\partial\eta}(\alpha_{1}+\alpha_{3})=2\frac{\partial v^{11}}{\partial\eta},\quad&\frac{\partial\beta_{1}}{\partial\mu_{2}}=2\frac{\partial v^{12}}{\partial\eta},\\ \frac{\partial\beta_{2}}{\partial\mu_{1}}=2\frac{\partial v^{21}}{\partial\eta},\quad&\frac{\partial\beta_{2}}{\partial\mu_{2}}=2\frac{\partial v^{22}}{\partial\eta}\\ \frac{\partial\beta_{0}}{\partial\mu_{1}}=-2\frac{\partial}{\partial\eta}(v^{11}+v^{21}),\quad&\frac{\partial\beta_{0}}{\partial\mu_{2}}=-2\frac{\partial}{\partial\eta}(v^{12}+v^{22}).\end{cases}\quad

Morever,

\frac{\partial\beta_{1}}{\partial\bar{\eta}}=-\frac{1}{2}\frac{\partial w}{\partial\mu_{1}},\quad\frac{\partial\beta_{2}}{\partial\bar{\eta}}=-\frac{1}{2}\frac{\partial w}{\partial\mu_{2}},\quad\frac{\partial\beta_{0}}{\partial\bar{\eta}}=\frac{1}{2}(\frac{\partial w}{\partial\mu_{1}}+\frac{\partial w}{\partial\mu_{2}}).

Therefore the type (1,0) forms

(2.15)

\zeta_{1}^{\prime}=\zeta_{1}+\beta_{1}d\eta,\quad\zeta_{2}^{\prime}=\zeta_{2}+\beta_{2}d\eta,\quad\zeta_{0}^{\prime}=-\zeta_{1}-\zeta_{2}+\beta_{0}d\eta

are closed, namely they are holomorphic differentials.

Proof.

The $\mu_{1},\mu_{2}$ derivatives are clear. For the $\bar{\eta}$ derivative, we can apply the component form of the distributional equation (2.4) away from $\mathfrak{D}$ , to see

\begin{split}\frac{\partial\beta_{1}}{\partial\bar{\eta}}&=2\lim\int\frac{\partial^{2}v^{11}}{\partial\eta\partial\bar{\eta}}d\mu_{1}+\frac{\partial^{2}v^{12}}{\partial\eta\partial\bar{\eta}}d\mu_{2}\\ &=-\frac{1}{2}\lim\int\frac{\partial^{2}w}{\partial\mu_{1}\partial\mu_{1}}d\mu_{1}+\frac{\partial^{2}w}{\partial\mu_{1}\partial\mu_{2}}d\mu_{2}=-\frac{1}{2}\frac{\partial w}{\partial\mu_{1}},\end{split}

where in the last equality we compare the asymptotic values at infinity to show there is no constant term depending on $\eta$ . Likewise with $\beta_{2},\beta_{0}$ . ∎

Lemma 2.8.

The sum $\beta_{1}+\beta_{2}+\beta_{0}=\frac{1}{\eta}.$ Equivalently,

\zeta_{1}^{\prime}+\zeta_{2}^{\prime}+\zeta_{0}^{\prime}=d\log\eta.

Proof.

By Lemma 2.7 the sum $\beta_{1}+\beta_{2}+\beta_{0}$ is independent of $\mu_{1},\mu_{2}$ . Given $\eta\neq 0$ , we shall evaluate this sum at the limit point $(\mu_{1}\to-\infty,\mu_{2}\to-\infty,\mu_{1}-\mu_{2}\to+\infty)$ . Then $\beta_{0}$ has no contribution, while $\beta_{1}$ contributes

2\lim_{\mu_{1}-\mu_{2}\to+\infty}\int_{\mu_{1}=+\infty,\text{fix $\mu_{1}-\mu_{2}$}}^{\mu_{1}=-\infty}\frac{\partial\alpha_{1}}{\partial{\eta}}d\mu_{1},

and $\beta_{2}$ contributes

2\lim_{\mu_{1}\to-\infty}\int_{\mu_{2}=+\infty,\text{fix $\mu_{1}$}}^{\mu_{2}=-\infty}\frac{\partial\alpha_{2}}{\partial{\eta}}d\mu_{2}

plus

-2\lim_{\mu_{1}\to-\infty}\int_{\mu_{1}-\mu_{2}=-\infty,\text{fix $\mu_{1}$}}^{\mu_{1}-\mu_{2}=+\infty}\frac{\partial\alpha_{3}}{\partial{\eta}}d(\mu_{1}-\mu_{2}).

Observe

\lim_{\mu_{1}\to-\infty,\text{fix $\mu_{1}-\mu_{2}$}}\alpha_{3}(\mu_{1},\mu_{2},\eta)=\frac{1}{2\sqrt{(\mu_{1}-\mu_{2})^{2}+(a_{11}+2a_{12}+a_{22})|\eta|^{2}}},

\lim_{\mu_{1}\to-\infty,\text{fix $\mu_{1}-\mu_{2}$}}\frac{\partial\alpha_{3}}{\partial\eta}=\frac{-(a_{11}+2a_{12}+a_{22})\bar{\eta}}{4((\mu_{1}-\mu_{2})^{2}+(a_{11}+2a_{12}+a_{22})|\eta|^{2})^{3/2}}

Using Lebesgue dominated convergence theorem, the third integral contribution is equal to

\frac{(a_{11}+2a_{12}+a_{22})\bar{\eta}}{2}\int_{-\infty}^{+\infty}\frac{1}{(x^{2}+(a_{11}+2a_{12}+a_{22})|\eta|^{2})^{3/2}}dx=\frac{1}{\eta}.

The other two contributions are zero by similar arguments. ∎

Now we notice that the multivalued holomorphic functions

\log z_{i}=\int\zeta_{i}^{\prime},\quad i=0,1,2

have periods in $2\pi\sqrt{-1}\mathbb{Z}$ , so the holomorphic functions $z_{0},z_{1},z_{2}$ are well defined on the domain of definition of $\beta_{0},\beta_{1},\beta_{2}$ respectively. Appropriate choices of multiplicative constants ensure the functional equation

(2.16)

z_{0}z_{1}z_{2}=\eta,

which enable us to extend $z_{0},z_{1},z_{2}$ over the complement of $\mathfrak{D}$ when $\eta=0$ .

Lemma 2.9.

The holomorphic functions $z_{0},z_{1},z_{2}$ extend smoothly over $\mathfrak{D}_{i}\subset\mathfrak{D}\subset\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\mathbb{C}_{\eta}$ . The function $z_{0},z_{1},z_{2}$ vanish over $\mathfrak{D}_{1}\cup\mathfrak{D}_{2}$ , $\mathfrak{D}_{1}\cup\mathfrak{D}_{3}$ and $\mathfrak{D}_{2}\cup\mathfrak{D}_{3}$ respectively.

Proof.

We focus on the neighbourhood of $\mathfrak{D}_{1}$ . Modulo smooth terms

\alpha_{1}\sim\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}},\quad\frac{\partial\alpha_{1}}{\partial\eta}\sim-\frac{a_{22}\bar{\eta}}{4({\mu_{1}^{2}+a_{22}|\eta|^{2}})^{3/2}},

so by the integral definitions, along $\mathfrak{D}_{1}$ the function $\beta_{2}$ is non-singular, and

\beta_{1}\sim\frac{-1}{2\eta}(\frac{\mu_{1}}{\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}-1),\quad\beta_{0}\sim\frac{1}{\eta}-\beta_{1}=\frac{1}{2\eta}(\frac{\mu_{1}}{\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}+1).

Now

\begin{split}&d\log|z_{1}|=(a_{11}+v^{11})d\mu_{1}+(a_{12}+v^{12})d\mu_{2}+\text{Re}(\beta_{1}d\eta)\\ &\sim\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}d\mu_{1}+\frac{1}{2}(1-\frac{\mu_{1}}{\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}})d\log|\eta|+a_{11}d\mu_{1}+a_{12}d\mu_{2}\\ &=\frac{1}{2}d\log|\eta|+\frac{1}{2}d\sinh^{-1}(\frac{\mu_{1}}{\sqrt{a_{22}}|\eta|})+d(a_{11}\mu_{1}+a_{12}\mu_{2}),\end{split}

hence up to multiplying by a smooth function

|z_{1}|\sim\text{const}\cdot(\frac{\mu_{1}}{\sqrt{a_{22}}}+\sqrt{\frac{\mu_{1}^{2}+a_{22}|\eta|^{2}}{{a_{22}}}})^{1/2}e^{a_{11}\mu_{1}+a_{12}\mu_{2}}\to 0

as the point moves to $\mathfrak{D}_{1}$ . Similarly

|z_{0}|\sim\text{const}\cdot(-\frac{\mu_{1}}{\sqrt{a_{22}}}+\sqrt{\frac{\mu_{1}^{2}+a_{22}|\eta|^{2}}{{a_{22}}}})^{1/2}e^{-a_{11}\mu_{1}-a_{12}\mu_{2}-a_{21}\mu_{1}-a_{22}\mu_{2}}\to 0.

The function $\log z_{2}$ encounters no singularity along $\mathfrak{D}_{1}$ . These calculations guarantee the continuous extension of the holomorphic functions $z_{0},z_{1},z_{2}$ over $\mathfrak{D}_{1}$ . Since the complex structure is compatible with the smooth topology by Section 2.3, these holomorphic functions in fact extend smoothly along $\mathfrak{D}_{1}$ . We remark that what happens in these calculations is essentially identical to the Taub-NUT metric near the origin. ∎

Next we show continuous extension of $z_{0},z_{1},z_{2}$ at the origin.

Lemma 2.10.

The functions $z_{0},z_{1},z_{2}$ tend to zero as $(\mu_{1},\mu_{2},\eta)\to 0$ .

Proof.

To see the main ideas, let us focus on $\eta=0$ and let $\mu_{1},\mu_{2}\to 0$ . By construction $d\log|z_{1}|=V^{1j}_{(1)}d\mu_{j}+\text{Re}(\beta_{1}d\eta).$ Restricted to $\eta=0$ ,

d\log|z_{1}|=(a_{11}+v^{11})d\mu_{1}+(a_{11}+v^{12})d\mu_{2}=d(a_{11}\mu_{1}+a_{12}\mu_{2})+\alpha_{1}d\mu_{1}+\alpha_{3}d(\mu_{1}-\mu_{2}),

hence

|z_{1}|=\text{const}\cdot e^{a_{11}\mu_{1}+a_{12}\mu_{2}}\exp\left(\int^{(\mu_{1},\mu_{2})}\alpha_{1}d\mu_{1}+\alpha_{3}d(\mu_{1}-\mu_{2})\right).

We need to show $|z_{1}|\to 0$ as $\mu_{1},\mu_{2}\to 0$ , namely $\int\alpha_{1}d\mu_{1}+\alpha_{3}d(\mu_{1}-\mu_{2})\to-\infty.$ Now because $\alpha_{1},\alpha_{3}$ are positive, this integral viewed as a function of $\mu_{1}$ and $\mu_{1}-\mu_{2}$ is an increasing functions of both variables, so it suffices to show this integral decreases to $-\infty$ as $(\mu_{1},\mu_{2})\to 0$ along the ray $\mathfrak{D}_{2}$ :

\begin{split}\int\alpha_{1}d\mu_{1}+\alpha_{3}d(\mu_{1}-\mu_{2})=&\log|\mu_{1}|\{\frac{1}{2}+\frac{1}{2\pi}\arctan(\frac{a_{12}}{\sqrt{A}})+\frac{1}{2\pi}\arctan(\frac{-a_{11}-a_{12}}{\sqrt{A}})\}\\ =&\log|\mu_{1}|\{\frac{1}{4}+\frac{1}{2\pi}\arctan(\frac{a_{12}+a_{22}}{\sqrt{A}})\}\to-\infty,\end{split}

where the first equality uses that the arctan functions are constant on the ray $\mathfrak{D}_{2}$ , and the second equality is an elementary trignometric identity.

In the more general case of $\eta\neq 0$ the $\arctan$ factor would no longer be exactly constant, but one can still make $|z_{1}|$ arbitrarily small for sufficiently small $|\eta|,\mu_{1},\mu_{2}$ . The cases of $z_{0}$ and $z_{2}$ are completely analogous. ∎

We have defined a holomorphic map from $M\setminus\{0\}$ to $\mathbb{C}^{3}_{z_{0},z_{1},z_{2}}\setminus\{0\}$ , which extends to a continuous map $M\to\mathbb{C}^{3}$ .

Proposition 2.11.

The map $M\setminus\{0\}\to\mathbb{C}^{3}_{z_{0},z_{1},z_{2}}\setminus\{0\}$ is a biholomorphism. The $T^{2}$ -action on the holomorphic functions is identified as

e^{i\theta_{1}}\cdot(z_{0},z_{1},z_{2})=(e^{-i\theta_{1}}z_{0},e^{i\theta_{1}}z_{1},z_{2}),\quad e^{i\theta_{2}}\cdot(z_{0},z_{1},z_{2})=(e^{-i\theta_{2}}z_{0},z_{1},e^{i\theta_{2}}z_{2}).

The holomorphic volume form $\Omega=-dz_{0}\wedge dz_{1}\wedge dz_{2}.$ Henceforth we identify $M$ with $\mathbb{C}^{3}$ .

Proof.

To identify the $T^{2}$ -action we examine the Hamiltonian vector field action. Recall that $\{\frac{\partial}{\partial\theta_{i}}\}_{i=1,2}$ is dual to the connection $\{\vartheta_{i}\}_{i=1,2}$ . We compute

\mathcal{L}_{\frac{\partial}{\partial\theta_{1}}}z_{i}=dz_{i}(\frac{\partial}{\partial\theta_{1}})=z_{i}\zeta_{i}^{\prime}(\frac{\partial}{\partial\theta_{1}}),

in particular

\mathcal{L}_{\frac{\partial}{\partial\theta_{1}}}z_{1}=z_{1}\sqrt{-1}\vartheta_{1}(\frac{\partial}{\partial\theta_{1}})=\sqrt{-1}z_{1},\quad\mathcal{L}_{\frac{\partial}{\partial\theta_{1}}}z_{0}=-\sqrt{-1}z_{0},\quad\mathcal{L}_{\frac{\partial}{\partial\theta_{1}}}z_{2}=0,

from which the first circle action is clear. Likewise with the second circle action.

The holomorphic (3,0)-form $\Omega$ is uniquely determined by the condition that $\Omega(\frac{\partial}{\partial\theta_{1}},\frac{\partial}{\partial\theta_{2}},\cdot)=d\eta$ . But

-dz_{0}\wedge dz_{1}\wedge dz_{2}(\frac{\partial}{\partial\theta_{1}},\frac{\partial}{\partial\theta_{2}},\cdot)=z_{0}z_{1}dz_{2}+z_{1}z_{2}dz_{0}+z_{0}z_{2}dz_{1}=d\eta

where in the last step we used the functional equation $z_{0}z_{1}z_{2}=\eta$ . This shows $\Omega=-dz_{0}\wedge dz_{1}\wedge dz_{2}$ . In particular the map $M\setminus\{0\}\to\mathbb{C}^{3}\setminus\{0\}$ is locally invertible.

To show the map is a homeomorphism, we notice that it is compatible with the fibration structure $M\to\mathbb{C}_{\eta}$ and $\mathbb{C}^{3}\to\mathbb{C}_{\eta}$ , so it suffices to show the fibres are identified, which follows from looking at the complexification of the $T^{2}$ -action into a $(\mathbb{C}^{*})^{2}$ -action.

Combining the above proves the biholomorphism claim. ∎

Remark 2.8.

Recall from Section 2.1 the additional $U(1)$ -symmetry

\mu_{i}\mapsto\mu_{i},\quad\eta\mapsto e^{i\theta}\eta,

acting on the base, which lifts to some $T^{2}$ -equivariant action on $M$ preserving the metric and rotating $\Omega$ . Properly speaking, the continuous symmetry group fits naturally into an extension sequence

1\to T^{2}\to U(1)^{3}\to U(1)\to 1,

and we are making a non-unique choice to split the extension. A particular choice can be identified complex geometrically as

(z_{1},z_{2},z_{0})\mapsto(z_{1},z_{2},e^{i\theta}z_{0}).

It is instructive to understand the $T^{2}$ -invariant Kähler metric in the complex geometric picture near spatial infinity. The reader will not fail to notice the analogy with the Taub-NUT metric. Our $\mathbb{C}^{3}$ admits a holomorphic fibration $\eta=z_{0}z_{1}z_{2}$ . Far away from $\eta=0$ , we are in the constant solution regime, so to leading order

\begin{cases}d\log z_{1}\sim a_{11}d\mu_{1}+a_{12}d\mu_{2}+\sqrt{-1}\vartheta_{1},\\ d\log z_{2}\sim a_{21}d\mu_{1}+a_{22}d\mu_{2}+\sqrt{-1}\vartheta_{2},\\ d\log z_{0}\sim-d\log z_{1}-d\log z_{2},\\ V^{ij}_{(1)}\sim a_{ij},\quad W_{(1)}\sim A,\end{cases}

hence the Kähler form is to leading order

(2.17)

\omega^{(1)}\sim A\frac{\sqrt{-1}}{2}d\eta\wedge d\bar{\eta}+d\mu_{j}\wedge\vartheta_{j}\sim\frac{\sqrt{-1}}{2}(\sum_{i,j=1,2}a^{ij}d\log z_{i}\wedge d\overline{\log z_{j}}+Ad\eta\wedge d\bar{\eta}).

This means in the horizontal direction the dominant term of $\omega^{(1)}$ is the pullback of a Euclidean metric $\frac{\sqrt{-1}}{2}Ad\eta\wedge d\bar{\eta}$ on $\mathbb{C}_{\eta}$ , and in the vertical direction $\omega^{(1)}$ is an almost flat metric on the fibre $\{z_{1}z_{2}z_{0}=\eta\}\simeq(\mathbb{C}^{*})^{2}$ written in the log coordinates.

When $\eta$ becomes small, the fibre will gradually break up into the union of 3 coordinate planes. Suppose at least two of $|z_{0}|,|z_{1}|,|z_{2}|$ remain large, then we are still far from the discriminant locus $\mathfrak{D}$ , and the metric asymptote (2.17) still applies. In particular the central fibre $\{\eta=0\}$ has 3 asymptotic branches, exemplified by $\{z_{0}=0,|z_{1}|\gg 1,|z_{2}|\gg 1\}$ which is metrically asymptotic to flat $\mathbb{R}^{2}\times T^{2}$ .

Finally the neighbourhood of $\{z_{i}=z_{j}=0\}$ corresponds to the discriminant locus $\mathfrak{D}$ . We focus on $\{z_{1}=z_{0}=0\}$ corresponding to $\mathfrak{D}_{1}$ . Approximately $d\log z_{2}\sim a_{12}d\mu_{1}+a_{22}d\mu_{2}+\sqrt{-1}\vartheta_{2}$ , and the function $\log z_{2}\in\mathbb{R}\times S^{1}$ provides a fibration structure over the cylinder $\mathbb{C}^{*}\simeq\mathbb{R}\times S^{1}$ , where the fibres are approximately $\mathbb{C}^{2}$ with the Taub-NUT metric (cf. Section 2.3).

2.5. Algebraic geometric perspective

We now take a closer examination of the complex geometry on $\mathbb{C}^{3}$ . We first raise two conceptual puzzles, and then we propose two conceptual explanations which suggest different directions of future investigations.

•

A priori speaking $M\simeq\mathbb{C}^{3}$ is only equipped with a complex structure, but the assignment of holomorphic coordinates $z_{0},z_{1},z_{2}$ canonically induces an algebraic structure. What is the origin of this algebraicity?
•

It is well known that $(\mathbb{C}^{3},\Omega)$ viewed as a complex manifold or an algebraic variety has a huge automorphism group preserving the holomorphic volume form. But our construction of coordinate functions are canonical up to multiplying by constants. What is the conceptual explanation?

The first explanation is that $\mathbb{C}^{3}$ has a toric structure. This comes from the holomorphic isometric action of $T^{3}$ , acting diagonally on $z_{0},z_{1},z_{2}$ . This induces a $(\mathbb{C}^{*})^{3}$ -action with an open dense orbit in $\mathbb{C}^{3}$ , making $\mathbb{C}^{3}$ a toric manifold and in particular algebraic. The canonical coordinates come from the eigenfunctions of this algebraic torus action, and $z_{0},z_{1},z_{2}$ up to constant scale factors are special because they have minimal vanishing orders on the toric boundary.

This explanation is simpler, but there are two possible criticisms. First, the $T^{3}$ has a preferred subgroup $T^{2}$ whose action has very different nature from the additional $U(1)$ -action, so it seems unnatural to put them on the same conceptual footing. Second, the additional $U(1)$ -symmetry is accidental to this particular example, which may not survive for other examples generalising our construction. A conjectural example without this $U(1)$ -symmetry is described in subsection 2.11.2.

The second and deeper explanation is based on the principle that algebraic structures arise from the ring of holomorphic functions with controlled growth (cf. [5]).

Lemma 2.12.

Any algebraic function $f$ on $\mathbb{C}^{3}$ satisfies the growth estimate

(2.18)

|f|\leq K_{1}e^{K_{2}(|\mu_{1}|+|\mu_{2}|)}(|\eta|+1)^{K_{3}}.

for some constants $K_{1},K_{2},K_{3}$ depending on $f$ .

Proof.

It suffices to prove the growth estimate for $z_{1},z_{2},z_{0}$ . By elementary calculation $|\frac{\partial\alpha_{1}}{\partial\eta}|\leq\frac{Ca_{22}|\eta|}{(\mu_{1}^{2}+a_{22}|\eta|^{2})^{3/2}}$ , so upon integration

\int_{\mu_{1}}^{\infty}|\frac{\partial\alpha_{1}}{\partial\eta}|d\mu_{1}\leq\frac{C}{|\eta|}(\frac{\mu_{1}}{\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}-1)\leq\frac{C}{|\eta|},

hence $|\beta_{i}|\leq\frac{C}{|\eta|}$ by the integral definition of $\beta_{i}$ . From

d\log|z_{1}|=d(a_{11}\mu_{1}+a_{12}\mu_{2})+\alpha_{1}d\mu_{1}+\alpha_{3}d(\mu_{1}-\mu_{2})+\text{Re}(\beta_{1}d\eta),

we integrate to obtain the growth bound on $z_{1}$ for $A^{1/4}|\vec{\mu}|_{a}\geq 1$ . But $|z_{1}|$ is a continuous function, so the bound holds also near the origin. Similarly we can bound $|z_{0}|$ and $|z_{2}|$ . ∎

Proposition 2.13.

The ring of algebraic functions on $\mathbb{C}^{3}$ coincides with the holomorphic functions satisfying the growth estimate (2.18) for some $K_{1},K_{2},K_{3}$ .

Proof.

We need to prove the converse to Lemma 2.12. The $T^{2}$ -symmetry acts on functions via

(e^{i\theta_{1}},e^{i\theta_{2}})\cdot f=f(e^{i(\theta_{1}+\theta_{2})}z_{0},e^{-i\theta_{1}}z_{1},e^{-i\theta_{2}}z_{2}).

This action allows us to expand any holomorphic function $f$ as a Fourier series on every $T^{2}$ -fibre:

f=\sum_{n,m\in\mathbb{Z}^{2}}f_{n,m},

where $f_{n,m}$ has weight $(n,m)$ with respect to the $T^{2}$ action. Since the $T^{2}$ action is holomorphic, the Fourier components $f_{n,m}$ are also holomorphic. Furthermore, these $f_{n,m}$ satisfy the same growth condition as $f$ after perhaps increasing $K_{1}$ .

We claim every $f_{n,m}$ is algebraic. To see this, we can find a suitable monomial of $z_{0},z_{1},z_{2}$ which has the same weight as $f_{n,m}$ , such that $f_{n,m}$ divided by this monomial has no pole along $\mathfrak{D}_{i}$ . But this quotient function is $T^{2}$ -invariant and holomorphic, so depends only on $\eta$ , and in fact has to be a polynomial of $\eta$ by the growth condition.

By applying the Parseval identify to every $T^{2}$ -fibre, we obtain

\frac{1}{4\pi^{2}}\int_{T^{2}}|f|^{2}\vartheta_{1}\wedge\vartheta_{2}=\sum_{n,m}|f_{n,m}|^{2}.

Both LHS and RHS are functions of $\mu_{1},\mu_{2},\eta$ , and LHS has a bound of type (2.18) by assumption. But for any given $K_{2},K_{3}$ , only finitely many monomials of $z_{0},z_{1},z_{2}$ satisfy the growth bound (2.18) globally, so only finitely many $f_{n,m}$ can appear as summands. Hence $f$ is algebraic as required. ∎

The proof in fact gives a double-index increasing filtration structure on the ring of algebraic functions:

\mathcal{F}_{K_{2},K_{3}}=\{f:\text{There exists $K_{1}$ such that }|f|\leq K_{1}e^{K_{2}(|\mu_{1}|+|\mu_{2}|)}(|\eta|+1)^{K_{3}}\},

such that every filtered piece is finite. This is the deeper mechanism why the complex automorphism group is cut down to finite size.

The insight from this discussion is that on our $\mathbb{C}^{3}$ the algebraic structure has a transcendental origin. The growth of holomorphic functions naturally involve transcendental functions such as $\exp$ and $\log$ . The ultimate reason is that torus fibrations are inherently transcendental in nature; this exponential growth behaviour already happened on the flat $\mathbb{C}^{*}$ .

At this moment we still have the freedom to normalise

(z_{0},z_{1},z_{2})\mapsto(\lambda_{0}z_{0},\lambda_{1}z_{1},\lambda_{2}z_{2}),

where $\lambda_{i}$ are constants satisfying $\lambda_{0}\lambda_{1}\lambda_{2}=1$ . Fixing a normalisation is important for keeping track of how estimates depend on the scaling parameter $A$ . We now make a choice so that the region $\{A^{1/4}|\vec{\mu}|_{a}\lesssim 1\}$ resemble a complex ball. Pick a point such that $|\mu_{1}|,|\mu_{2}|,|\mu_{1}-\mu_{2}|,A^{1/4}|\eta|$ are all comparable to $A^{-1/2}$ , so $A^{1/4}\text{dist}_{g_{a}}(\cdot,\Delta)\sim A^{1/4}|\vec{\mu}|_{a}\sim 1$ , and we demand $|z_{0}|=|z_{1}|=|z_{2}|=|\eta|^{1/3}$ at this point. This convention is compatible with both the $A$ -scaling and the functional equation $z_{0}z_{1}z_{2}=\eta$ . We did not mention the phase of $z_{i}$ because $T^{2}$ -gauge symmetry renders different phase choices equivalent. Under this convention, on the annulus region $1\lesssim A^{1/4}|\vec{\mu}|_{a}\lesssim C_{1}$ , the holomorphic functions $A^{1/4}z_{0},A^{1/4}z_{1},A^{1/4}z_{2}$ are bounded independent of scaling factor, and the metric $\omega^{(1)}$ is $C^{\infty}$ -equivalent to $\sum_{i}\sqrt{-1}dz_{i}\wedge d\bar{z}_{i}$ .

2.6. Surgery on the ansatz

We begin with some explanations about our strategy. The generalised Gibbons-Hawking ansatz is convenient for producing the metric ansatz $g^{(1)}$ , but very difficult for proving nonlinear existence theorems, due to the singularity issues caused by the distributional equation. So instead we will shift to the complex geometric viewpoint on $\mathbb{C}^{3}$ and attempt to solve the complex Monge-Ampère equation.

One minor problem is that there is no guarantee for $g^{(1)}$ to be smooth at the origin. So we do a surgery at the scale $A^{1/4}|\vec{\mu}|_{a}\lesssim 1$ , namely the scale $|z_{1}|,|z_{2}|,|z_{0}|\lesssim A^{-1/4}$ . In the annulus $\{1\leq A^{1/4}\sqrt{|z_{1}|^{2}+|z_{2}|^{2}+|z_{0}|^{2}}\leq 2\}$ , we write $\omega^{(1)}=\sqrt{-1}\partial\bar{\partial}\phi^{(1)}$ which is $C^{\infty}$ -equivalent to $\sqrt{-1}\sum_{i}dz_{i}\wedge d\bar{z}_{i}$ . Using a cutoff function

\chi=\begin{cases}0\quad&A^{1/4}\sqrt{|z_{1}|^{2}+|z_{2}|^{2}+|z_{0}|^{2}}\leq 1,\\ 1&A^{1/4}\sqrt{|z_{1}|^{2}+|z_{2}|^{2}+|z_{0}|^{2}}\geq 1.5,\end{cases}

we replace $\omega^{(1)}$ in the complex ball $\{A^{1/4}\sqrt{|z_{1}|^{2}+|z_{2}|^{2}+|z_{0}|^{2}}\leq 2\}$ by $\sqrt{-1}\partial\bar{\partial}(\chi\phi^{(1)})$ , which is now smooth but loses positive definiteness. The remedy is to add to $\sqrt{-1}\partial\bar{\partial}(\chi\phi^{(1)})$ a smooth semipositive closed $(1,1)$ -form, which is compactly supported in $\{A^{1/4}\sqrt{|z_{1}|^{2}+|z_{2}|^{2}+|z_{0}|^{2}}\leq 2\}$ , and larger than $C\sqrt{-1}\sum_{i}dz_{i}\wedge d\bar{z}_{i}$ on $\{A^{1/4}\sqrt{|z_{1}|^{2}+|z_{2}|^{2}+|z_{0}|^{2}}\leq 1.5\}$ for some sufficiently large constant $C$ . Let us call the modified Kähler form $\omega^{(2)}$ , which clearly agrees with $\omega^{(1)}$ outside $\{A^{1/4}\sqrt{|z_{1}|^{2}+|z_{2}|^{2}+|z_{0}|^{2}}\leq 2\}$ , and is $C^{\infty}$ -equivalent to $\sqrt{-1}\sum_{i}dz_{i}\wedge d\bar{z}_{i}$ inside $\{A^{1/4}\sqrt{|z_{1}|^{2}+|z_{2}|^{2}+|z_{0}|^{2}}\leq 3\}$ . Morever, with a little care in the construction the symmetries of $\omega^{(1)}$ persist on $\omega^{(2)}$ . The associated Kähler metric is $g^{(2)}$ , and we shall refer to both $g^{(2)}$ and $\omega^{(2)}$ interchangably. This metric is clearly complete.

A caveat is that the functions $\mu_{1},\mu_{2}$ on $\mathbb{C}^{3}$ are no longer the moment coordinates for $\omega^{(2)}$ . Furthermore in the smooth structure induced by the complex structure on $\mathbb{C}^{3}$ , the functions $\mu_{1},\mu_{2}$ may not be smooth at the origin. Henceforth in this Chapter we will abuse notation to denote $\mu_{1},\mu_{2}$ as their mollified version. In other words, we perform a surgery to the fibration $\mathbb{C}^{3}\xrightarrow{(\mu_{1},\mu_{2},\eta)}\mathbb{R}^{4}$ inside the ball $\{|z_{1}|^{2}+|z_{2}|^{2}+|z_{0}|^{2}\leq A^{-1/2}\}$ to make it defined by a smooth map.

We can now introduce the global weighted Hölder norms $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}$ for $T^{2}$ -invariant tensors $T$ on $\mathbb{C}^{3}$ .

•

In the region (2.12) away from the discriminant locus, the norm $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}$ is equivalent to $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta+\tau}}$ defined in Section 2.2.
•

In the region (2.14) close to $\mathfrak{D}_{1}$ but far from the origin, the norm $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}$ is equivalent to $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta,\tau}}$ introduced in Section 2.3. Similarly with the regions close to $\mathfrak{D}_{2},\mathfrak{D}_{3}$ and far away from the origin.
•

In the region $A^{1/4}|\vec{\mu}|_{a}\lesssim C_{1}$ where the metric $\omega^{(2)}$ is $C^{\infty}$ -equivalent to $\sqrt{-1}\sum dz_{i}\wedge d\bar{z}_{i}$ and $|z_{i}|\lesssim A^{-1/4}$ , the norm $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}$ is equivalent to the normalised $C^{k,\alpha}$ -norm on the complex ball of radius $\sim A^{-1/4}$ . For example on this ball the mollified functions $\mu_{i}$ satisfy $\left\lVert\mu_{i}\right\rVert_{C^{k,\alpha}}\lesssim A^{-1/2}$ for $i=1,2$ .

The point is that these regions cover the entire $\mathbb{C}^{3}$ and the norms are equivalent on overlapping regions, where the equivalence factor is independent of $A$ . These norms define the corresponding Banach spaces of $T^{2}$ -invariant functions/tensors on $\mathbb{C}^{3}$ . The spaces which are most relevant for us are $C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})$ , $C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3},\Lambda^{1})$ , $C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3},\text{Sym}^{2})$ and $C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3},\Lambda^{1,1})$ , corresponding to functions, 1-forms, real symmetric 2-tensors and real (1,1)-forms.

The volume form error function $E^{(2)}$ is defined by

(2.19)

\frac{3}{4}(1+E^{(2)})\sqrt{-1}\Omega\wedge\overline{\Omega}=(\omega^{(2)})^{3}.

By construction $E^{(2)}=E^{(1)}$ outside of the compact region where the surgery takes place. It follows from the discussions of Section 2.2 and 2.3 that

Lemma 2.14.

The volume form error has the global estimate

\left\lVert E^{(2)}\right\rVert_{C^{k,\alpha}_{-1,-1}(\mathbb{C}^{3})}\leq C.

2.7. Hein’s package and weighted Sobolev inequality

The following few Sections address the analytic problems. For convenience we assume $C^{-1}\delta_{ij}\leq a_{ij}\leq C\delta_{ij}$ , although we will indicate $A$ -dependence in strategic places. We rely heavily on the work of Hein (cf. Chapter 3,4 in [12]) which sets out a framework for solving the complex Monge-Ampère equation and its linear cousin the Poisson equation on complete noncompact manifolds, building on the seminal paper by Tian and Yau [28]. We explain Hein’s results in a variant form which follows from his arguments. The ambient complete manifold $M$ needs to satisfy the following analytic properties:

•

There is a $C^{k,\alpha}$ quasi-atlas with $k\geq 3$ , meaning a collection of charts on which the complex structure and the metric have $C^{k,\alpha}$ bounds, and the injectivity radius/regularity scale in these charts are bounded below. Clearly this condition is satisfied on $(\mathbb{C}^{3},\omega^{(2)})$ . This assumption allows one to speak of (unweighted) Hölder spaces.
•

There is a function $\rho(x)$ uniformly equivalent to the distance function $\text{dist}(0,x)$ outside the unit ball, and satisfies $|\nabla\rho|+\rho|\nabla^{2}\rho|\leq C$ . It is easy to check $\rho=\sqrt{|\vec{\mu}|_{a}^{2}+A^{-1/2}}$ works for $\mathbb{C}^{3}$ . This assumption is useful in integration by part arguments.
•

We need the weighted Sobolev inequality on functions: assume the power law volume growth $\text{Vol}(B(r))\sim r^{p^{\prime}}$ with rate $p^{\prime}>2$ . (In our case of interest $\dim_{\mathbb{R}}M=6$ , $p^{\prime}=4$ .) For $1\leq p\leq\frac{\dim_{\mathbb{R}}M}{\dim_{\mathbb{R}}M-2}$ and functions $u$ with $L^{2}$ -gradient,

$(\int|u|^{2p}(1+\rho)^{p(p^{\prime}-2)-p^{\prime}}d\text{Vol})^{1/p}\leq C\int|\nabla u|^{2}.$

These inequalities differ from the standard Sobolev inequalities in the sense that they do not require the manifold to have Euclidean volume growth, which makes them remarkably flexible.

The output of this package is:

•

(Poisson equation case) Let $f\in C^{0,\alpha}$ satisfy $|f|\leq C\rho^{-q}$ for given $p^{\prime}>q>2$ . Then there is a unique $C^{2,\alpha}$ solution to $\Delta u=f$ with decay estimate $|u|\leq C\rho^{2-q+\epsilon}$ , where $\epsilon\ll 1$ is any fixed small number satisfying $2-q+\epsilon<0$ .
•

(Complex Monge-Ampère equation case) Denote $\omega_{0}$ as the ambient Kähler form. Let $f\in C^{2,\alpha}$ satisfy $|f|\leq C\rho^{-q}$ for $p^{\prime}>q>2$ . Then there is some $0<\alpha^{\prime}\leq\alpha$ and $u\in C^{4,\alpha^{\prime}}$ which solves $(\omega_{0}+\sqrt{-1}\partial\bar{\partial}u)^{\dim_{\mathbb{C}}M}=e^{f}\omega_{0}^{\dim_{\mathbb{C}}M}$ , with decay estimate $|u|\leq C\rho^{2-q+\epsilon}$ , where $\epsilon\ll 1$ is any fixed small number.

Here we have separated the assumptions on the ambient manifolds from the decay assumptions to emphasize that these are difficulties of distinct nature. The key idea in Hein’s package is to obtain a priori $L^{\infty}$ estimates and power law decay estimates on potentials via the method of weighted Moser iteration, which hinges on the weighted Sobolev inequalities. The estimates from Hein’s package are constructive. It is essential to assume faster than quadratic decay on the source function $f$ , because the method needs the potential $u=O(\rho^{2-q})$ to be bounded. Another important remark is that Hein’s method respects compact group actions.

We give an elementary proof for the following

Proposition 2.15.

For $1\leq p\leq\frac{3}{2}$ , the weighted Sobolev inequality

(2.20)

(\int_{M}|u|^{2p}(A^{-1/4}+|\vec{\mu}|_{a})^{2p-4}d\text{Vol})^{1/p}\leq C\int_{M}|\nabla u|^{2}d\text{Vol}

holds for $T^{2}$ -invariant functions on $(\mathbb{C}^{3},g^{(2)})$ . The constant here depends only on the scale invariant ellipticity bound (2.11)

Proof.

By scaling analysis we may assume $A\sim 1$ . Let $u$ be a $T^{2}$ -invariant function with $\int_{M}|\nabla u|^{2}=1$ , so descends to a function on the base $\mathbb{R}^{4}$ . Since the weighted Sobolev inequality holds on Euclidean $\mathbb{R}^{4}$ (by an interpolation of standard Sobolev inequality and Hardy inequality),

(\int_{\mathbb{R}^{4}}|u|^{2p}(1+|\vec{\mu}|_{a})^{2p-4}d\text{Vol}_{a})^{1/p}\leq C\int_{\mathbb{R}^{4}}|\nabla_{g_{a}}u|^{2}d\text{Vol}_{a}\leq C\int_{M}|\nabla_{g^{(2)}}u|^{2}d\text{Vol}\leq C,

where the second inequality is easily seen using the model metric in Section 2.3. The LHS in this inequality is uniformly equivalent to the LHS in (2.20) except in the region $\{\text{dist}_{g_{a}}(\cdot,\mathfrak{D})\leq 1\}$ . So we are left to prove

(\int_{\text{dist}(\cdot,\mathfrak{D})\lesssim 1}|u|^{2p}(1+|\vec{\mu}|_{a})^{2p-4}d\text{Vol})^{1/p}\leq C.

For $x\in\mathfrak{D}_{1},\mathfrak{D}_{2},\mathfrak{D}_{3}$ , Sobolev inequality on bounded balls imply

(\int_{B(x,1)}|u-\bar{u}(x)|^{2p})^{1/p}\leq C\int_{B(x,2)}|\nabla u|^{2},\quad\bar{u}(x)=\text{Vol}(B(x,1))^{-1}\int_{B(x,1)}u

Furthermore we can find a point $x^{\prime}$ with $\text{dist}(x,x^{\prime})\leq 3$ , $\text{dist}(x^{\prime},\mathfrak{D})\gtrsim 2$ , and by Sobolev inequality

(\int_{B(x^{\prime},1)}|u-\bar{u}(x^{\prime})|^{2p})^{1/p}\leq C\int_{B(x,5)}|\nabla u|^{2},\quad\bar{u}(x^{\prime})=\text{Vol}(B(x^{\prime},1))^{-1}\int_{B(x^{\prime},1)}u.

By Poincaré inequality

|\bar{u}(x)-\bar{u}(x^{\prime})|^{2}\leq\int_{B(x,5)}|\nabla u|^{2}.

Combining these,

(\int_{B(x,1)}|u|^{2p})^{1/p}\leq C(\int_{B(x^{\prime},1)}|u|^{2p})^{1/p}+C\int_{B(x,5)}|\nabla u|^{2}.

Multiplying this inequality by $(1+|\vec{\mu}|_{a}(x))^{(2p-4)/p}$ , and summing over $x\in\mathfrak{D}$ , we obtain

\begin{split}&(\int_{\text{dist}(\cdot,\mathfrak{D})\lesssim 1}|u|^{2p}(1+|\vec{\mu}|_{a})^{2p-4}d\text{Vol})^{1/p}\\ \leq&C(\int_{\text{dist}(\cdot,\mathfrak{D})\gtrsim 1}|u|^{2p}(1+|\vec{\mu}|_{a})^{2p-4}d\text{Vol})^{1/p}+C\int|\nabla u|^{2}\leq C\end{split}

as required. ∎

Now applying the $T^{2}$ -equivariant version of Hein’s result on the Poisson equation,

Corollary 2.16.

Let $2<q<4$ and $0<\epsilon<q-2$ . There is a bounded Green operator for $T^{2}$ -invariant functions

G_{g^{(2)}}:\{f\in C^{0,\alpha}|f=O(|\vec{\mu}|_{a}^{-q})\text{ for large $|\vec{\mu}|_{a}$}\}\to\{u\in C^{2,\alpha}|u=O(|\vec{\mu}|_{a}^{2-q+\epsilon})\}.

such that $u=G_{g^{(2)}}f$ satisfies $\Delta_{g^{(2)}}u=f$ .

This mapping property is rather crude and unsuited for functions with slow decay rates at infinity. Improving our understanding of the Green operator shall be the task of Section 2.8.

Recall from Section 2.3 the model metric $g_{\text{Taub}}$ on a $\mathbb{Z}$ -quotient of the space $\text{Taub-NUT}\times\mathbb{C}$ . We can view $T^{2}$ -invariant functions as pullbacks of functions on the metric product space $\text{Taub-NUT}\times\mathbb{R}$ . A variant of the above discussions leads to weighted Sobolev inequalities and Green’s function estimates for $g_{\text{Taub}}$ :

Corollary 2.17.

Let $2<q<4$ and $0<\epsilon<q-2$ . There is a bounded Green operator for $T^{2}$ -invariant functions on the model space with the metric $g_{\text{Taub}}$

G_{\text{Taub}}:\{f\in C^{0,\alpha}|f=O(|\vec{\mu}|_{a}^{-q})\text{ for large $|\vec{\mu}|_{a}$}\}\to\{u\in C^{2,\alpha}|u=O(|\vec{\mu}|_{a}^{2-q+\epsilon})\}.

such that $u=G_{\text{Taub}}f$ satisfies $\Delta_{\text{Taub}}u=f$ .

The gist is that the Green’s function for $g_{\text{Taub}}$ decays like $O(|\vec{\mu}|_{a}^{-2+\epsilon})$ at infinity.

2.8. Harmonic analysis

This Section develops more precise mapping properties for the Green operator $G_{g^{(2)}}$ . Since we are ultimately interested in Kähler metrics rather than potentials, we need to bound the zeroth order operator $\sqrt{-1}\partial\bar{\partial}G_{g^{(2)}}$ for input functions with slow decay such as $E^{(2)}$ , a task which requires rather intricate harmonic analysis. Our strategy is to construct a parametrix by divide and conquer. In this Section we shall assume $C^{-1}\delta_{ij}\leq a_{ij}\leq C\delta_{ij}$ , and indicate $A$ -dependence in strategic places. The main result is Proposition 2.23.

Recall $\Delta_{a}$ is the Laplacian for the Euclidean metric $g_{a}$ on the base $\mathbb{R}^{4}$ . We shall identify $T^{2}$ -invariant functions with functions on the base $\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\mathbb{C}_{\eta}=\mathbb{R}^{4}$ .

Lemma 2.18.

Let $-3<\delta<0$ and $\delta+\tau<0$ . Let $f$ be a $T^{2}$ -invariant function on $\mathbb{C}^{3}$ supported in $\{\text{dist}(\cdot,\mathfrak{D})\gtrsim 1\}$ with $\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}\leq 1$ . Then the second order derivatives of the Euclidean potential $\Delta_{a}^{-1}f$ satisfies

|\nabla^{2}_{g_{a}}\Delta_{a}^{-1}f|_{g_{a}}\leq C\ell^{\delta}(|\vec{\mu}|_{a}+1)^{\tau}.

Morever if $\delta<-1$ and $\delta+\tau<-1$ , then

\left\lVert\nabla^{2}_{g^{(2)}}\Delta_{a}^{-1}f\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}\leq C.

The constants depend only on $k,\alpha,\delta,\tau$ and the uniform ellipticity bound on $a_{ij}$ .

Proof.

The main task is to estimate the Calderon-Zygmund type operator

G_{ij}f(x)=\int_{\mathbb{R}^{4}}\frac{(x-y)_{i}(x-y)_{j}}{|x-y|_{a}^{6}}f(y)d\text{Vol}_{a}(y).

where $(x-y)_{i}$ denotes the components of $x-y$ viewed as a vector in $\mathbb{R}^{4}$ . We say $x\in\mathbb{R}^{4}$ belongs to the dyadic scale $|x|\sim 2^{n}$ where $n\in\mathbb{N}$ , if either $n=0$ and $|x|\leq 1$ , or $n>0$ and $2^{n}\leq|x|\leq 2^{n+1}$ . To ensure the Green operator is well defined, we will temporarily assume $f$ to be compactly supported, with no quantitative restriction on the measure of its support.

Since $\delta>-3$ and $|f(y)|\lesssim\ell(y)^{\delta}|\vec{y}|_{a}^{\tau}$ , we have $\left\lVert f\right\rVert_{L^{1}(|y|\sim 2^{m})}\lesssim 2^{m(\delta+\tau+4)}$ . Thus if $|x|\sim 2^{n}$ does not belong to scale $m$ , then the contribution of $|y|\sim 2^{m}$ to $G_{ij}(x)$ is bounded by $O(2^{m(\delta+\tau+4)}\min\{2^{-4n},2^{-4m}\})$ . Adding up all contributions from $m\neq n$ , we get

|G_{ij}f(x)-\int_{|y|\sim 2^{n}}\frac{(x-y)_{i}(x-y)_{j}}{|x-y|_{a}^{6}}f(y)d\text{Vol}_{a}(y)|\lesssim 2^{n(\delta+\tau)}\lesssim(1+|x|_{a})^{\delta+\tau},

using $\delta+\tau<0$ for the summability of the series. Since the source is far from the observer, elliptic bootstrap implies higher order Hölder regularity.

Now that we are left with only one scale, it is clear that the claimed bound for second derivatives holds where $\ell$ is comparable to $|\vec{\mu}|_{a}$ . We now focus on $x$ close to $\mathfrak{D}$ . The contribution of $|y-x|_{a}\gtrsim\ell(x),|y|\sim|x|$ is estimated by

\begin{split}&C\int_{|y-x|_{a}\gtrsim\ell(x),|y|\sim|x|}\frac{1}{|x-y|_{a}^{4}}\ell(y)^{\delta}|y|^{\tau}d\text{Vol}_{a}(y)\\ \leq&C(1+|x|_{a})^{\tau}\int_{|y-x|_{a}\gtrsim\ell(x)}\frac{1}{|x-y|^{4-\delta}}(\frac{\ell(y-x)}{|x-y|_{a}})^{\delta}d\text{Vol}_{a}(y)\\ \leq&C(1+|x|_{a})^{\tau}\int_{r>\ell(x)}r^{\delta-1}dr\int_{S^{3}}(\frac{\ell(y^{\prime})}{|y^{\prime}|})^{\delta}d\text{Area}_{S^{3}}(y^{\prime})\\ \leq&C(1+|x|_{a})^{\tau}\ell(x)^{\delta}\end{split}

where we use $-3<\delta<0$ in the convergence of the integrals. Since the contribution comes from sources at distance at least $\ell(x)$ away from the observer, the higher Hölder norms are controlled. Finally, the estimates for the contribution from $|y-x|_{a}\lesssim\ell(x)$ follows simply from standard Schauder theory.

At this stage we have proved the second derivative bound

|\nabla^{2}_{g_{a}}\Delta_{a}^{-1}f|_{g_{a}}\leq C\ell^{\delta}(|\vec{\mu}|_{a}+1)^{\tau}

together with an implicit weighted $C^{k,\alpha}$ -bound in the $g_{a}$ -metric. Since $f$ is compactly supported by our temporary assumption, qualitatively $\Delta^{-1}_{a}f$ has quadratic decay at infinity. By integrating the second order derivatives from infinity, we can bound first order derivatives $d\Delta_{a}^{-1}f$ :

|d\Delta_{a}^{-1}f|_{g_{a}}\leq C\ell^{\delta+1}(|\vec{\mu}|_{a}+1)^{\tau},

using $\delta+\tau<-1$ and $\delta<-1$ in the integration. Alternatively the first order derivative bounds can be proved using the same singular integral operator method.

Now the Hessian $\nabla^{2}_{g^{(2)}}\Delta_{a}^{-1}f$ can be expanded as a linear combination of second derivatives $\frac{\partial^{2}}{\partial\mu_{i}\partial\mu_{j}}\Delta_{a}^{-1}f$ etc and first derivatives $\frac{\partial}{\partial\mu_{i}}\Delta_{a}^{-1}f$ etc. Hence

|\nabla^{2}_{g^{(2)}}\Delta_{a}^{-1}f|\leq\sum|\frac{\partial^{2}}{\partial\mu_{i}\partial\mu_{j}}\Delta_{a}^{-1}f||\nabla\mu_{i}||\nabla\mu_{j}|+\sum|\frac{\partial}{\partial\mu_{i}}\Delta_{a}^{-1}f||\nabla_{g^{(2)}}^{2}\mu_{i}|,

where the sum includes also $\eta$ -derivatives. Higher order derivatives of the Hessian can be expanded by the Leibniz rule. Using $\left\lVert d\mu_{i}\right\rVert_{C^{k,\alpha}_{0,0}}\leq C$ and $\left\lVert d\eta\right\rVert_{C^{k,\alpha}_{0,0}}\leq C$ , we obtain the Hessian bound $\left\lVert\nabla^{2}_{g^{(2)}}\Delta_{a}^{-1}f\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}\leq C$ as claimed.

Finally, an approximation argument in the weak topology removes the compact support assumption on $f$ , so we conclude that $\nabla^{2}_{g^{(2)}}\Delta_{a}^{-1}$ extends canonically to a bounded linear operator between the weighted Hölder spaces.

As a delicate side remark, to bound the integral operator $\Delta_{a}^{-1}$ itself we would need to impose further $\delta+\tau<-2$ and $\delta<-2$ , which would not be adequate for our intended applications. It is crucial in the above argument that the integral kernel of $G_{ij}$ decays two orders faster than the Green kernel. ∎

The Laplacian $\Delta_{g^{(2)}}$ is the trace of the Hessian $\nabla^{2}_{g^{(2)}}$ . The idea of the next Lemma is that for $T^{2}$ -invariant functions $G_{g^{(2)}}$ should be well approximated by $\Delta_{a}^{-1}$ as long as we stay sufficiently away from the discriminant locus $\mathfrak{D}$ .

Lemma 2.19.

In the situation of Lemma 2.18,

\left\lVert\Delta_{g^{(2)}}\Delta_{a}^{-1}f-f\right\rVert_{C^{k,\alpha}_{\delta-1,\tau}(\mathbb{C}^{3})}\leq C.

In particular for a large enough constant $C_{2}$ ,

\left\lVert\Delta_{g^{(2)}}\Delta_{a}^{-1}f-f\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3}\cap\{\text{dist}(\cdot,\mathfrak{D})>C_{2}/2\})}\leq\frac{C}{C_{2}}\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}\ll\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}.

Proof.

This follows from Lemma 2.6, Remark 2.7, and the fact that for the flat model $g_{\text{flat}}$ the Laplacian on $T^{2}$ -invariant functions coincides with the base Laplacian $\Delta_{a}$ . ∎

Next we study the Green operator $G_{\text{Taub}}$ for the model metric $g_{\text{Taub}}$ .

Lemma 2.20.

Let $\tau<1$ and $0<\epsilon\ll 1$ . Let $f$ be a $T^{2}$ -invariant function supported on $\{\text{dist}_{g_{a}}(\cdot,\mathfrak{D}_{1})<C_{2}\}$ inside the model space, with norm $\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta,\tau}}=1$ , so that $\left\lVert f\right\rVert_{C^{k,\alpha}_{0,\tau}}\lesssim 1$ . Then

\begin{cases}\left\lVert G_{\text{Taub}}f\right\rVert_{C^{k+2,\alpha}_{-1+\epsilon,\tau}}\leq C,\quad-1<\tau<1-\epsilon,\\ \left\lVert G_{\text{Taub}}f\right\rVert_{C^{k+2,\alpha}_{0,-2+\epsilon}}\leq C,\quad\tau\leq-1.\end{cases}

where the constant only depends on $C_{2},\delta,\epsilon,\tau,k,\alpha$ and the uniform ellipticity bound on $a_{ij}$ . In particular if

\begin{cases}\text{Either }-1<\tau<1,\quad-3+2\epsilon<\delta\leq 0,\\ \text{or }-2+\epsilon<\tau\leq-1,\quad\delta+\tau>-4+2\epsilon,\end{cases}

then $\left\lVert\nabla_{\text{Taub}}^{2}G_{\text{Taub}}f\right\rVert_{C^{k,\alpha}_{\delta,\tau}}\leq C.$

Proof.

As in the proof of Lemma 2.18 we may assume $f$ has compact support to ensure a priori the well definition of $G_{\text{Taub}}f$ . We use cutoff functions to decompose $f$ into a sum of functions $f_{n}$ supported on $\{n\lesssim\mu_{2}\lesssim n+1,\text{dist}_{g_{a}}(\cdot,\mathfrak{D}_{1})<C_{2}\}$ centred around points $x_{n}\in\mathfrak{D}$ , with Hölder bound $\left\lVert f_{n}\right\rVert_{C^{k,\alpha}(B(x_{n},C_{2}))}\lesssim n^{\tau}$ . At a fixed point $x$ bounded away from $\text{supp}(f_{n})$ , the contribution $G_{\text{Taub}}f_{n}$ is estimated by $|G_{\text{Taub}}f_{n}|\lesssim n^{\tau}(|x-x_{n}|_{a}+1)^{\epsilon-2}$ , where $\epsilon>0$ is any given small number (cf. Corollary 2.17 and notice the translational symmetry of $g_{\text{Taub}}$ along $\mathfrak{D}_{1}$ ). Elliptic bootstrap gives

\left\lVert G_{\text{Taub}}f_{n}\right\rVert_{C^{k+2,\alpha}_{0,0}(B(x,\ell(x)/10))}\lesssim n^{\tau}(|x-x_{n}|_{a}+1)^{\epsilon-2}.

Summing over all $n\in\mathbb{N}$ ,

\begin{split}&\left\lVert G_{\text{Taub}}f\right\rVert_{C^{k+2,\alpha}_{0,0}(B(x,\ell(x)/10))}\lesssim\sum n^{\tau}(|x-x_{n}|_{a}+1)^{\epsilon-2}\\ \lesssim&\int_{1}^{\infty}y^{\tau}(\ell(x)^{2}+|\mu_{2}(x)-y|^{2})^{\epsilon/2-1}dy\\ \lesssim&\begin{cases}(|x|_{a}+1)^{\tau}\ell(x)^{\epsilon-1}\quad-1<\tau<1-\epsilon,\\ (|x|_{a}+1)^{\epsilon-2},\quad\tau\leq-1.\end{cases}\end{split}

Thus

\begin{cases}\left\lVert G_{\text{Taub}}f\right\rVert_{C^{k+2,\alpha}_{-1+\epsilon,\tau}}\leq C,\quad-1<\tau<1-\epsilon,\\ \left\lVert G_{\text{Taub}}f\right\rVert_{C^{k+2,\alpha}_{0,-2+\epsilon}}\leq C,\quad\tau\leq-1.\end{cases}

which controls $\left\lVert\nabla_{\text{Taub}}^{2}G_{\text{Taub}}f\right\rVert_{C^{k,\alpha}_{\delta,\tau}}$ under the numerical conditions on weight exponents. ∎

Let $C_{3}\gg\max(C_{1},C_{2})$ be a large constant to be determined, depending on $k,\alpha,\delta,\tau,C_{2}$ and the ellipticity constant for $a_{ij}$ . Let $\chi_{1}$ be a cutoff function with regularity scale $\sim C_{3}$ on $\mathbb{R}^{4}$ ,

\chi=\begin{cases}1\quad|\vec{\mu}|_{a}>2C_{3}>4C_{1}\text{dist}_{g_{a}}(\cdot,\mathfrak{D}_{1}),\\ 0\quad|\vec{\mu}|_{a}<C_{3}\text{ or }\text{dist}_{g_{a}}(\cdot,\mathfrak{D}_{1})>C_{3}C_{1}^{-1}.\end{cases}

Over the support of $\chi_{1}$ the model metric $g_{\text{Taub}}$ and $g^{(2)}$ coexist, so $\chi_{1}G_{\text{Taub}}f$ can be viewed as a function on $(\mathbb{C}^{3},g^{(2)})$ . The next Lemma says that outside a neighbourhood of the origin $\chi_{1}G_{\text{Taub}}f$ is a good approximate solution to the Poisson equation.

Lemma 2.21.

In the situation of Lemma 2.20, if $C_{3}$ is sufficiently large, then

\left\lVert\Delta_{g^{(2)}}(\chi_{1}G_{\text{Taub}}f)-f\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3}\cap\{|\vec{\mu}|_{a}>2C_{3}\})}\leq CC_{3}^{-\epsilon}\ll 1.

Proof.

The error $\Delta_{g^{(2)}}(\chi_{1}G_{\text{Taub}}f)-f$ comes from two sources: the deviation of the metric $g_{\text{Taub}}$ from $g^{(2)}$ , and the cutoff error. The metric deviation error is estimated in Lemma 2.6 which we recall as $\left\lVert g_{\text{Taub}}-g^{(2)}\right\rVert_{C^{k,\alpha}_{0,-1}}\leq C.$ In particular for $|\vec{\mu}|_{a}>C_{3}$ and on the support of $\chi_{1}$ , we have $\left\lVert g_{\text{Taub}}-g^{(2)}\right\rVert_{C^{k,\alpha}_{0,0}}\leq CC_{3}^{-1},$ so the metric deviation error is $O(C_{3}^{-1})$ .

We turn to the cutoff error. By Lemma 2.20

\begin{cases}\left\lVert\chi_{1}G_{\text{Taub}}f\right\rVert_{C^{k+2,\alpha}_{-1+\epsilon,\tau}}\leq C,\quad-1<\tau<1-\epsilon,\\ \left\lVert\chi_{1}G_{\text{Taub}}f\right\rVert_{C^{k+2,\alpha}_{0,-2+\epsilon}}\leq C,\quad\tau\leq-1.\end{cases}

which implies

\begin{cases}\left\lVert\nabla^{2}_{\text{Taub}}(\chi_{1}G_{\text{Taub}}f)\right\rVert_{C^{k+2,\alpha}_{-3+\epsilon,\tau}}\leq C,\quad-1<\tau<1-\epsilon,\\ \left\lVert\nabla^{2}_{\text{Taub}}(\chi_{1}G_{\text{Taub}}f)\right\rVert_{C^{k+2,\alpha}_{-2,-2+\epsilon}}\leq C,\quad\tau\leq-1.\end{cases}

so in particular on $\text{supp}(d\chi_{1})\cap\{|\vec{\mu}|_{a}>2C_{3}\}$ where $\ell\sim C_{3}C_{1}^{-1}$ , we have

\left\lVert\nabla^{2}_{\text{Taub}}(\chi_{1}G_{\text{Taub}}f)\right\rVert_{C^{k,\alpha}_{\delta,\tau}}\leq CC_{3}^{-\epsilon}.

By the support assumptions $f=0$ on $\text{supp}(d\chi_{1})\cap\{|\vec{\mu}|_{a}>2C_{3}\}$ , hence the cutoff error is $O(C_{3}^{-\epsilon})$ . Combining the two errors give the claim. ∎

Clearly completely analogous results apply to the neighbourhood of $\mathfrak{D}_{2},\mathfrak{D}_{3}$ .

The source supported in a bounded region is treated by

Lemma 2.22.

Assume

\begin{cases}\text{Either }&-2\leq\delta\leq 0,\quad\tau>-2,\\ \text{Or }&\delta\leq-2,\quad\delta+\tau>-4,\end{cases}

and let $0<\epsilon\ll 1$ depending on $\delta,\tau$ . If $f$ is supported in the ball $\{|\vec{\mu}|_{a}<4C_{3}\}$ , with bound $\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}\leq 1$ or equivalently $\left\lVert f\right\rVert_{C^{k,\alpha}_{0,0}(\mathbb{C}^{3})}\lesssim 1$ , then $\left\lVert G_{g^{(2)}}f\right\rVert_{C^{k,\alpha}_{0,-2+\epsilon}(\mathbb{C}^{3})}\leq C,$ so in particular

\left\lVert\nabla^{2}_{g^{(2)}}G_{g^{(2)}}f\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}\leq\left\lVert\nabla^{2}_{g^{(2)}}G_{g^{(2)}}f\right\rVert_{C^{k,\alpha}_{-2,-2+\epsilon}(\mathbb{C}^{3})}\leq C.

Proof.

The absolute value is estimated by Corollary 2.16:

|G_{g^{(2)}}f|\leq C(1+|\vec{\mu}|_{a})^{-2+\epsilon}.

The higher order estimate $\left\lVert\nabla^{2}_{g^{(2)}}G_{g^{(2)}}f\right\rVert_{C^{k,\alpha}_{-2,-2+\epsilon}(\mathbb{C}^{3})}\leq C$ follows by bootstrapping, which controls $C^{k,\alpha}_{\delta,\tau}$ norm for the given range of weight exponents $\delta,\tau$ . ∎

We call the polyhedral set

\{-2\leq\delta<-1,-2<\tau<1,\delta+\tau<-1\}\cup\{-3<\delta\leq-2,\tau<1,-4<\delta+\tau\}

the good range of weight exponents for $g^{(2)}$ , namely the set where all the above Lemmas apply. As long as $(\delta,\tau)$ stays within a compact subset, the estimates in the Lemmas are in fact uniform in $\delta,\tau$ . The following Proposition is the main result of this Section.

Proposition 2.23.

Suppose $(\delta,\tau)$ stays within a compact subset of the good range of weight exponents. Then the operator $\mathcal{R}^{\prime}=\nabla^{2}_{g^{(2)}}G_{g^{(2)}}$ extends to bounded linear operators between the weighted Hölder spaces

\mathcal{R}^{\prime}:C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})\to C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3},\text{Sym}^{2}),\quad\left\lVert\mathcal{R}^{\prime}\right\rVert\leq C

where the constant depends only on $k,\alpha$ , the compact region of exponents $(\delta,\tau)$ , and the scale invariant ellipticity bound (2.11). The composition with the natural projection

\mathcal{R}:C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})\xrightarrow{\mathcal{R}^{\prime}}C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3},\text{Sym}^{2})\to C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3},\Lambda^{1,1})

extends the operator $\mathcal{R}=\sqrt{-1}\partial\bar{\partial}G_{g^{(2)}},$ which takes value in closed real (1,1)-forms and is inverse to taking trace.

Proof.

The key technique is to construct a parametrix $P_{g^{(2)}}$ for the Green operator semi-explicitly, with precise control on its mapping properties.

Given a function $f$ with $\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}=1$ , temporarily assumed to have sufficient decay at infinity, we will construct an approximate solution $u=P_{g^{(2)}}f$ to the Poisson equation as follows. Take a smooth cutoff function $\chi^{\prime}$

\chi^{\prime}=\begin{cases}1\quad&\text{dist}(\cdot,\mathfrak{D})\geq 2,\\ 0\quad&\text{dist}(\cdot,\mathfrak{D})\leq 1,\end{cases}

then $\chi^{\prime}f$ has norm $\left\lVert\chi^{\prime}f\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}\lesssim 1$ and is supported in $\{\text{dist}(\cdot,\mathfrak{D})\geq 1\}$ . Applying Lemma 2.18, the function $u_{0}=\Delta_{a}^{-1}(\chi^{\prime}f)$ satisfies $\left\lVert\nabla^{2}_{g^{(2)}}u_{0}\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}\leq C$ . By Lemma 2.19 we can choose $C_{2}\gg 1$ large enough independent of $f$ to ensure

\left\lVert\Delta_{g^{(2)}}u_{0}-f\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3}\cap\{\text{dist}(\cdot,\mathfrak{D})>C_{2}/2\})}\ll 1.

Next we take smooth cutoff functions $\chi_{1}^{\prime},\chi_{2}^{\prime},\chi_{3}^{\prime}$ near $\mathfrak{D}_{1},\mathfrak{D}_{2},\mathfrak{D}_{3}$ , such that

\chi_{1}^{\prime}=\begin{cases}1\quad&\text{dist}_{g_{a}}(\cdot,\mathfrak{D}_{1})\leq C_{2}/2\text{ and }|\vec{\mu}|_{a}>2C_{2},\\ 0\quad&\text{dist}_{g_{a}}(\cdot,\mathfrak{D}_{1})\geq C_{2}\text{ or }|\vec{\mu}|_{a}<C_{2}\end{cases}

and similarly with $\chi_{2}^{\prime},\chi_{3}^{\prime}$ . The function

f_{1}=\chi_{1}^{\prime}(f-\Delta_{g^{(2)}}u_{0})=\chi_{1}^{\prime}(f-\Tr_{g^{(2)}}\nabla^{2}_{g^{(2)}}u_{0})

is supported in $\{\text{dist}_{g_{a}}(\cdot,\mathfrak{D}_{1})\leq C_{2},|\vec{\mu}|_{a}\geq C_{2}\}$ with bound $\left\lVert f_{1}\right\rVert_{C^{k,\alpha}_{\delta,\tau}}\leq C$ . So we can apply Lemma 2.20 and Lemma 2.21 to find $u_{1}=\chi_{1}G_{\text{Taub}}f_{1}$ with bounds

\left\lVert\nabla^{2}_{g^{(2)}}u_{1}\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}\leq C,\quad\left\lVert\Delta_{{g^{(2)}}}u_{1}-f_{1}\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3}\cap\{|\vec{\mu}|_{a}>2C_{3}\})}\leq CC_{3}^{-\epsilon}\ll 1.

Completely analogous constructions are made near $\mathfrak{D}_{2}$ and $\mathfrak{D}_{3}$ , where we obtain $u_{2},u_{3}$ with similar bounds.

Let $\chi_{4}^{\prime}$ be a smooth cutoff function

\chi_{4}^{\prime}=\begin{cases}1\quad&|\vec{\mu}|_{a}\leq 2C_{3},\\ 0\quad&|\vec{\mu}|_{a}\geq 4C_{3},\end{cases}

and define $f_{4}=\chi_{4}^{\prime}(f-\Delta_{g^{(2)}}(u_{0}+u_{1}+u_{2}+u_{3}))$ , which is supported in the ball $\{|\vec{\mu}|_{a}\leq 4C_{3}\}$ and admits the bound $\left\lVert f_{4}\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}\leq C$ . Then we can apply Lemma 2.22 to obtain $u_{4}=G_{g^{(2)}}f_{4}$ with bounds

\left\lVert\nabla^{2}_{g^{(2)}}u_{4}\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}\leq C,\quad\Delta_{{g^{(2)}}}u_{4}=f_{4}.

We set $P_{g^{(2)}}f=u=u_{0}+u_{1}+u_{2}+u_{3}+u_{4}$ . The key point is that by construction

\left\lVert\nabla^{2}_{g^{(2)}}u\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}\leq C,\quad\left\lVert\Delta_{{g^{(2)}}}u-f\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}\ll 1,

namely $u$ is an approximate solution to the Poisson equation with bounds. A subtlety is that $\nabla^{2}_{g^{(2)}}u$ is fully controlled while $u$ is only controlled up to an additive constant. In any event, after extending the definition of the operator by removing the fast decay hypotheses on $f$ , we have defined a bounded linear operator between weighted Hölder spaces of $T^{2}$ -invariant functions and symmetric 2-tensors on $\mathbb{C}^{3}$

\nabla^{2}_{g^{(2)}}P_{g^{(2)}}:C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})\to C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3},\text{Sym}^{2}),

such that the operator $\Tr_{g^{(2)}}\nabla^{2}_{g^{(2)}}P_{g^{(2)}}$ is an approximation to the identity. Thus

\mathcal{R}^{\prime}=\nabla^{2}_{g^{(2)}}P_{g^{(2)}}(\Tr_{g^{(2)}}\nabla^{2}P_{g^{(2)}})^{-1}

is a bounded right inverse to $\Tr_{g^{(2)}}$ . Composing with the projection to the type (1,1)-forms defines the operator

\mathcal{R}=\sqrt{-1}\partial\bar{\partial}P_{g^{(2)}}(\Tr_{g^{(2)}}\nabla^{2}P_{g^{(2)}})^{-1},

which takes value in closed (1,1)-forms and is a bounded inverse to $\Tr_{g^{(2)}}$ . It is worth commenting that the same operators work for different exponents $\delta,\tau$ .

It remains to relate $\mathcal{R}$ and $\mathcal{R}^{\prime}$ to the Green operator $G_{g^{(2)}}$ when $f$ has sufficient decay at infinity. The point is that for fast decay weights $\delta+\tau<-2$ and $\delta<-2$ , the Hessian control $\left\lVert\nabla^{2}_{g^{(2)}}u\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}\leq C$ together with the a priori qualitative decay $u\to 0$ at infinity, imply the quantitative bound $\left\lVert u\right\rVert_{C^{k+2,\alpha}_{\delta+2,\tau}(\mathbb{C}^{3})}\leq C.$ This enables us to extend ${P}_{g^{(2)}}$ to a bounded linear operator

{P}_{g^{(2)}}:C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})\to C^{k+2,\alpha}_{\delta+2,\tau}(\mathbb{C}^{3})

and the operator

P_{g^{(2)}}(\Tr_{g^{(2)}}\nabla^{2}P_{g^{(2)}})^{-1}:C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})\to C^{k+2,\alpha}_{\delta+2,\tau}(\mathbb{C}^{3})

defines an inverse to the Laplacian $\Delta_{g^{(2)}}$ . By the uniqueness of decaying solution to the Poisson equation $G_{g^{(2)}}=P_{g^{(2)}}(\Tr_{g^{(2)}}\nabla^{2}P_{g^{(2)}})^{-1}$ . Hence

\mathcal{R}^{\prime}=\nabla^{2}_{g^{(2)}}G_{g^{(2)}},\quad\mathcal{R}=\sqrt{-1}\partial\bar{\partial}G_{g^{(2)}}

as required. ∎

Remark 2.9.

The construction of $\mathcal{P}_{g^{(2)}}$ , $\mathcal{R}$ , $\mathcal{R}^{\prime}$ can be made compatible with the symmetries of the ansatz.

Remark 2.10.

The moral of this proof is that for slowly decaying sources, it is easier to bound the Hessian of the Green operator than the Green operator itself.

Corollary 2.24.

(Solution to the Poisson equation) Let $(\delta,\tau)$ fall within the good range of weight exponents. Then given $f\in C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})$ , there exists a function $u$ solving $\Delta_{g^{(2)}}u=f$ with gradient bound

\left\lVert du\right\rVert_{C^{k+1,\alpha}_{\delta+1,\tau}(\mathbb{C}^{3},\Lambda^{1})}\leq CA^{-1/4}\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}.

Proof.

For $f$ with sufficient decay at infinity, we can find $u=P_{g^{(2)}}f$ with estimate $\left\lVert\nabla^{2}u\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3},\text{ Sym}^{2})}\leq C\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}.$ Using $\delta+\tau<-1$ and $\delta<-1$ , we can integrate from spatial infinity to obtain the required gradient bound.

For a general $f$ without fast decay assumption, take a weakly convergent sequence of fast decaying functions $f_{k}\to f$ bounded in $C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})$ , and find $u_{k}=P_{g^{(2)}}f_{k}$ with gradient bounds. After adjusting $u_{k}$ by additive constants to make $u_{k}(0)=0$ , we can extract the subsequential limit $u$ of $u_{k}$ , which solves $\Delta_{g^{(2)}}u=f$ with the gradient bound. ∎

2.9. Perturbation into a Calabi-Yau metric

In this Section we complete the construction of the promised Taub-NUT type Calabi-Yau metric on $\mathbb{C}^{3}$ .

Lemma 2.25.

Given $0<\epsilon\ll 1$ , there is a Kähler metric $\omega^{(3)}=\omega^{(2)}+\sqrt{-1}\partial\bar{\partial}\phi^{(3)}$ with estimate

\left\lVert d\phi^{(3)}\right\rVert_{C^{k+1,\alpha}_{-\epsilon,-1+\epsilon}(\mathbb{C}^{3},\Lambda^{1})}\leq CA^{-1/4},

such that the volume form error $E^{(3)}$ defined by

\frac{3}{4}(E^{(3)}+1)\sqrt{-1}\Omega\wedge\overline{\Omega}=(\omega^{(3)})^{3}

satisfies the fast decay estimate $\left\lVert E^{(3)}\right\rVert_{C^{k,\alpha}_{-4-\epsilon,-4+4\epsilon}}\leq C.$ Here the constants only depend on $k,\alpha,\epsilon,\kappa$ and the scale invariant uniform ellipticity bound (2.11). In particular $\omega^{(3)}$ is close to $\omega^{(2)}$ in the $C^{k,\alpha}$ -topology outside a compact set, and the volume form error decay rate is faster than quadratic.

Proof.

By Lemma 2.14 the initial volume form error is

\left\lVert E^{(2)}\right\rVert_{C^{k,\alpha}_{-1-\epsilon,-1+\epsilon}}\leq\left\lVert E^{(2)}\right\rVert_{C^{k,\alpha}_{-1,-1}}\leq C.

Applying Corollary 2.24 we can solve the Poisson equation with estimate

\Delta_{g^{(2)}}u_{1}=-2E^{(2)},\quad\left\lVert du_{1}\right\rVert_{C^{k+1,\alpha}_{-\epsilon,-1+\epsilon}(\mathbb{C}^{3},\Lambda^{1})}\leq CA^{-1/4},

so in particular

\left\lVert\partial\bar{\partial}u_{1}\right\rVert_{C^{k,\alpha}_{-1-\epsilon,-1+\epsilon}}\leq C,\quad\left\lVert(\partial\bar{\partial}u_{1})^{2}\right\rVert_{C^{k,\alpha}_{-2-2\epsilon,-2+2\epsilon}}\leq C.

Now $(\omega^{(2)^{\prime}})^{3}=(\omega^{(2)}+\sqrt{-1}\partial\bar{\partial}u_{1})^{3}=(\omega^{(2)})^{3}(1+\frac{1}{2}\Delta_{g^{(2)}}u_{1}+O(|\partial\bar{\partial}u_{1}|^{2}))$ , so the new volume form error has improved decay:

\frac{3}{4}(E^{(2)^{\prime}}+1)\sqrt{-1}\Omega\wedge\overline{\Omega}=(\omega^{(2)^{\prime}})^{3},\quad\left\lVert E^{(2)^{\prime}}\right\rVert_{C^{k,\alpha}_{-2,-2+2\epsilon}}\leq\left\lVert E^{(2)^{\prime}}\right\rVert_{C^{k,\alpha}_{-2-2\epsilon,-2+2\epsilon}}\leq C.

We notice that the modification to $\omega^{(2)}$ is $C^{0}$ -small outside a compact region, where the positive definite condition for the Kähler metric is not affected. Inside the compact set we can add on a locally supported semipositive (1,1)-form to guarantee the Kähler condition, as we have done in Section 2.6. We abuse notation to write this Kähler metric after surgery as $\omega^{(2)^{\prime}}$ , which inherits all the analytic properties of $\omega^{(2)}$ .

Applying Corollary 2.24 again to solve the Poisson equation with background metric $g^{(2)^{\prime}}$ ,

\Delta_{g^{(2)^{\prime}}}u_{2}=-2E^{(2)^{\prime}},\quad\left\lVert du_{2}\right\rVert_{C^{k+1,\alpha}_{-1,-2+2\epsilon}(\mathbb{C}^{3},\Lambda^{1})}\leq CA^{-1/4},

and using $(\omega^{(2)^{\prime}}+\sqrt{-1}\partial\bar{\partial}u_{2})^{3}=(\omega^{(2)^{\prime}})^{3}(1+\frac{1}{2}\Delta_{g^{(2)^{\prime}}}u_{2}+O(|\partial\bar{\partial}u_{2}|^{2})),$ the new volume form error is now bounded in $C^{k,\alpha}_{-4,-4+4\epsilon}$ -norm. Another surgery in the compact region ensures the Kähler property. ∎

Now we can prove the main theorem of this Chapter.

Theorem 2.26.

(Taub-NUT type Calabi-Yau metric on $\mathbb{C}^{3}$ ) There exists a complete metric $\omega_{\mathbb{C}^{3}}=\omega^{(2)}+\sqrt{-1}\partial\bar{\partial}\phi^{\mathbb{C}^{3}}$ on $\mathbb{C}^{3}$ satisfying $\omega_{\mathbb{C}^{3}}^{3}=\frac{3}{4}\sqrt{-1}\Omega\wedge\overline{\Omega}$ , with metric deviation estimate

\left\lVert d\phi^{\mathbb{C}^{3}}\right\rVert_{C^{k+1,\alpha}_{-\epsilon,-1+\epsilon}(\mathbb{C}^{3},\Lambda^{1})}\leq CA^{-1/4},\quad\left\lVert\sqrt{-1}\partial\bar{\partial}\phi^{\mathbb{C}^{3}}\right\rVert_{C^{k,\alpha}_{-1-\epsilon,-1+\epsilon}(\mathbb{C}^{3},\Lambda^{1,1})}\leq C.

Here $0<\epsilon\ll 1$ is an arbitrarily small given number, and the constants depend only on $k,\alpha,\epsilon$ and the scale invariant uniform ellipticity bound (2.11). This metric inherits all the symmetries of $\omega^{(2)}$ .

Proof.

We assume $C^{-1}\delta_{ij}\leq a_{ij}\leq C\delta_{ij}$ which can be relaxed by scaling. It suffices to solve the complex Monge-Ampère equation

(\omega^{(3)}+\sqrt{-1}\partial\bar{\partial}\phi^{\prime})^{3}=\frac{3}{4}\sqrt{-1}\Omega\wedge\overline{\Omega}.

In Hein’s analytic package (cf. Section 2.7), the conditions on the ambient metric including $C^{k,\alpha}$ quasi-atlas, existence of a distance-like function with Hessian bounds, and the weighted Sobolev inequality, are robust conditions which are inherited by $\omega^{(3)}$ from $\omega^{(2)}$ . The volume form error $E^{(3)}$ has faster than quadratic decay by construction:

|E^{(3)}|\leq C(|\vec{\mu}|_{a}+1)^{-4+\epsilon}.

Thus Hein’s package provides a potential $\phi^{\prime}$ solving the complex Monge-Ampère equation with decay estimate $|\phi^{\prime}|\leq C(|\vec{\mu}|_{a}+1)^{-2+2\epsilon}$ . Elliptic bootstrap gives the bound $\left\lVert\phi^{\prime}\right\rVert_{C^{k+2,\alpha}_{0,-2+2\epsilon}}\leq C,$ so $\left\lVert d\phi^{\prime}\right\rVert_{C^{k+1,\alpha}_{-1,-2+2\epsilon}(\mathbb{C}^{3},\Lambda^{1})}\leq C$ , which combined with Lemma 2.25 implies the metric deviation estimate. ∎

Some immediate geometric consequences are

Corollary 2.27.

The Taub-NUT type Calabi-Yau metric $g_{\mathbb{C}^{3}}$ has volume growth rate

C^{-1}\leq\frac{\text{Vol}(B_{g_{\mathbb{C}^{3}}}(r))}{r^{4}}\leq C,\quad r\geq A^{-1/4}.

and the tangent cone at infinity is the Euclidean $\mathbb{R}^{4}$ .

Corollary 2.28.

The Riemannian curvature satisfies the decay estimate

|\text{Rm}|\leq CA^{-1/4}\ell^{-3}.

Proof.

Using the metric deviation estimate, the Riemannian curvature is bounded. Morever if $\ell>2A^{-1/4}$ , then we can find a flat model $g_{\text{flat}}$ over a $g_{a}$ -ball of radius $\sim\ell/10$ , where $\left\lVert g_{\text{flat}}-g_{\mathbb{C}^{3}}\right\rVert_{C^{k,\alpha}_{-1,0}}\leq C.$ Using this bound up to second order derivatives, the Christoffel symbols in the local flat coordinates are $O(A^{-1/4}\ell^{-2})$ and the Riemannian curvature is of order $O(A^{-1/4}\ell^{-3})$ . ∎

In particular, in the generic region where $|\vec{\mu}|_{a}$ is comparable to $\ell$ , the Riemannian curvature decays as $\text{Rm}(x)=O(\text{dist}(x,0)^{-3}),$ although the Riemannian curvature does not decay at infinity along $\mathfrak{D}_{i}$ .

Corollary 2.29.

There exist $T^{2}$ -moment coordinates $\tilde{\mu}_{1}^{\mathbb{C}^{3}}$ , $\tilde{\mu}_{2}^{\mathbb{C}^{3}}$ on $(\mathbb{C}^{3},\omega_{\mathbb{C}^{3}})$ with global estimate

|\mu_{i}-\tilde{\mu}_{i}^{\mathbb{C}^{3}}|\leq CA^{-3/4}\ell^{-\epsilon}|\vec{\mu}|_{a}^{-1+\epsilon}.

The map $\mathbb{C}^{3}\xrightarrow{(\tilde{\mu}_{1}^{\mathbb{C}^{3}},\tilde{\mu}_{2}^{\mathbb{C}^{3}},\text{Im}(\eta))}\mathbb{R}^{3}$ is a special Lagrangian fibration with phase angle zero, whose critical point set is $\bigcup_{i,j}\{z_{i}=z_{j}=0\}$ and whose discriminant locus agrees with (1.2).

Remark 2.11.

Moment coordinates for the Taub-NUT type metric $\omega_{\mathbb{C}^{3}}$ should not be confused with the moment coordinates $\mu_{i}$ for the Kähler ansatz.

Proof.

The existence of moment coordinates follows from $H^{1}(\mathbb{C}^{3})=0$ , but for the purpose of estimation we wish to relate $\tilde{\mu}_{i}^{\mathbb{C}^{3}}$ to $\mu_{i}$ outside the ball $\{|\vec{\mu}|_{a}\lesssim A^{-1/4}\}$ where the surgery was performed. In this exterior region

\omega_{\mathbb{C}^{3}}=\omega^{(2)}+\sqrt{-1}\partial\bar{\partial}\phi^{\mathbb{C}^{3}}=\omega^{(1)}+dd^{c}\phi^{\mathbb{C}^{3}},\quad d^{c}=\frac{\sqrt{-1}}{2}(\bar{\partial}-\partial).

The 1-form $d^{c}\phi^{\mathbb{C}^{3}}$ is $T^{2}$ -invariant, so by Cartan’s formula

\iota_{\frac{\partial}{\partial\theta_{i}}}dd^{c}\phi^{\mathbb{C}^{3}}=-d\iota_{\frac{\partial}{\partial\theta_{i}}}d^{c}\phi^{\mathbb{C}^{3}},

which combined with $d\mu_{i}=-\iota_{\frac{\partial}{\partial\theta_{i}}}\omega^{(1)}$ allow us to find the moment coordinates:

\tilde{\mu}_{i}^{\mathbb{C}^{3}}=\mu_{i}+\iota_{\frac{\partial}{\partial\theta_{i}}}d^{c}\phi^{\mathbb{C}^{3}},\quad d\tilde{\mu}_{i}^{\mathbb{C}^{3}}=-\iota_{\frac{\partial}{\partial\theta_{i}}}\omega_{\mathbb{C}^{3}},\quad i=1,2.

Using the estimates $|d\phi^{\mathbb{C}^{3}}|\leq CA^{-1/2}\ell^{-\epsilon}|\vec{\mu}|_{a}^{-1+\epsilon}$ and $|\frac{\partial}{\partial\theta_{i}}|\leq CA^{-1/4},$ we see

|\mu_{i}-\tilde{\mu}_{i}^{\mathbb{C}^{3}}|\leq CA^{-3/4}\ell^{-\epsilon}|\vec{\mu}|_{a}^{-1+\epsilon}.

Now inside $\{|\vec{\mu}|_{a}\lesssim A^{-1/4}\}$ , we have $|\mu_{i}|\leq CA^{-1/2}$ , and $|d\tilde{\mu}_{i}|\leq CA^{-1/4}$ integrates to give $|\tilde{\mu}_{i}|\leq CA^{-1/2}$ . Thus globally on $\mathbb{C}^{3}$

|\mu_{i}-\tilde{\mu}_{i}^{\mathbb{C}^{3}}|\leq CA^{-3/4}\ell^{-\epsilon}(A^{-1/4}+|\vec{\mu}|_{a})^{-1+\epsilon}

as required. Morever $\tilde{\mu}_{1}^{\mathbb{C}^{3}},\tilde{\mu}_{2}^{\mathbb{C}^{3}},\tilde{\mu}_{1}^{\mathbb{C}^{3}}-\tilde{\mu}_{2}^{\mathbb{C}^{3}}$ vanish respectively along $\mathfrak{D}_{1},\mathfrak{D}_{2},\mathfrak{D}_{3}$ , due to the respective vanishing of the circle generators $\frac{\partial}{\partial\theta_{1}},\frac{\partial}{\partial\theta_{2}},\frac{\partial}{\partial\theta_{1}}-\frac{\partial}{\partial\theta_{2}}$ .

Now consider the map $\mathbb{C}^{3}\xrightarrow{(\tilde{\mu}_{1}^{\mathbb{C}^{3}},\tilde{\mu}_{2}^{\mathbb{C}^{3}},\text{Im}(\eta))}\mathbb{R}^{3}$ . It is a special Lagrangian fibration by Remark 1.6.

At a critical point $p\in\mathbb{C}^{3}$ the Zariski tangent space of the fibre, namely the annihilator of $\text{span}(d\tilde{\mu}_{1}^{\mathbb{C}^{3}},d\tilde{\mu}_{2}^{\mathbb{C}^{3}},d\text{Im}\eta)$ , is a linear subspace of $T_{p}\mathbb{C}^{3}$ of real dimension at least 4. It contains $\frac{\partial}{\partial\theta_{1}},\frac{\partial}{\partial\theta_{2}}$ and is $g_{\mathbb{C}^{3}}$ -orthogonal to $I\frac{\partial}{\partial\theta_{1}},I\frac{\partial}{\partial\theta_{2}}$ . If $d\eta\neq 0$ at $p$ , then $\frac{\partial}{\partial\theta_{1}},\frac{\partial}{\partial\theta_{2}}$ are linearly independent, so the Zariski tangent space is the orthogonal complement of $\text{span}(I\frac{\partial}{\partial\theta_{1}},I\frac{\partial}{\partial\theta_{2}})$ by dimension counting. Since $d\text{Im}\eta$ vanishes on the Zariski tangent space, and $d\eta$ vanishes on $\text{span}_{\mathbb{C}}(\frac{\partial}{\partial\theta_{1}},\frac{\partial}{\partial\theta_{2}})$ , we deduce $d\eta=0$ on $T_{p}\mathbb{C}^{3}$ , contradiction. Thus the critical points must satisfy $d\eta=0$ , or equivalently $p\in\bigcup_{i,j}\{z_{i}=z_{j}=0\}$ . Conversely all points in $\bigcup_{i,j}\{z_{i}=z_{j}=0\}$ are critical. Having identified the critical point set, the discriminant locus follows from the argument in Lemma 1.6. ∎

2.10. Uniqueness and moduli

In this Section we show that under $T^{2}$ symmetry, there is only one complete Calabi-Yau metric $g_{\mathbb{C}^{3}}$ on $\mathbb{C}^{3}$ within a suitably restrictive asymptotic class prescribed by the metric deviation estimate in Theorem 2.26. The strategy is similar to the one used by Conlon and Hein [2].

Lemma 2.30.

Let $\delta<-1$ and $\tau<0$ . If a function $u$ satisfies $\Delta_{g_{\mathbb{C}^{3}}}u=0$ with bound $\left\lVert du\right\rVert_{C^{k+1,\alpha}_{\delta+1,\tau}(\mathbb{C}^{3},\Lambda^{1})}<\infty$ , then $du=0$ .

Proof.

Since $g_{\mathbb{C}^{3}}$ is a Ricci-flat metric, the Bochner formula implies

\Delta(\frac{1}{2}|\nabla u|^{2})=|\nabla^{2}u|^{2}+\langle\Delta\nabla u,\nabla u\rangle=|\nabla^{2}u|^{2}+\langle\nabla\Delta u,\nabla u\rangle=|\nabla^{2}u|^{2},

so $|du|^{2}$ is a non-negative subharmonic function. The decay condition implies it converges to zero at infinity, so maximum principle gives $du=0$ . ∎

Proposition 2.31.

Let $\delta<-1$ and $\tau<0$ . If a $T^{2}$ -invariant potential $\phi^{\prime}$ satisfies $(\omega_{\mathbb{C}^{3}}+\sqrt{-1}\partial\bar{\partial}\phi^{\prime})^{3}=\omega_{\mathbb{C}^{3}}^{3}$ with bound $\left\lVert d\phi^{\prime}\right\rVert_{C^{k+1,\alpha}_{\delta+1,\tau}(\mathbb{C}^{3},\Lambda^{1})}<\infty$ , then $d\phi^{\prime}=0$ .

Proof.

The strategy is to improve the decay rate of $d\phi^{\prime}$ iteratively, until it becomes sufficiently fast. We rewrite the equation as a Poisson equation

\frac{1}{2}(\Delta_{g_{\mathbb{C}^{3}}}\phi^{\prime})\omega_{\mathbb{C}^{3}}^{3}=-3(\sqrt{-1}\partial\bar{\partial}\phi^{\prime})^{2}\wedge\omega_{\mathbb{C}^{3}}-(\sqrt{-1}\partial\bar{\partial}\phi^{\prime})^{3}.

Notice that $\sqrt{-1}\partial\bar{\partial}\phi^{\prime}$ lives in $C^{k,\alpha}_{\delta,\tau}$ , so its square lives in $C^{k,\alpha}_{2\delta,2\tau}$ . As long as $(\delta,\tau)$ stays in the good range of weight exponents, Corollary 2.24 and the above vanishing lemma imply that the solution $\phi^{\prime}$ to this Poisson equation must satisfy $\left\lVert d\phi^{\prime}\right\rVert_{C^{k+1,\alpha}_{2\delta+1,2\tau}(\mathbb{C}^{3},\Lambda^{1})}<\infty$ . This is an improved decay estimate because $2\delta+1<\delta$ and $2\tau<\tau$ . Since each iteration improves the decay rate by a definite amount, within a finite number of steps we can assume $\delta<-2$ and $\tau<0$ . Then $\left\lVert d\phi^{\prime}\right\rVert_{C^{k+1,\alpha}_{\delta+1,\tau}(\mathbb{C}^{3},\Lambda^{1})}<\infty$ implies that $\left\lVert\phi^{\prime}\right\rVert_{C^{k+2,\alpha}_{\delta+2,\tau}(\mathbb{C}^{3})}<\infty$ after adjusting $\phi^{\prime}$ by a constant. Then we can use the standard integration by part argument for the complex Monge-Ampère equation to see

\int_{\mathbb{C}^{3}}|\nabla\phi^{\prime}|^{2}\phi^{\prime p}\omega_{\mathbb{C}^{3}}^{3}=0,\quad p\gg 1.

Hence $\phi^{\prime}$ is a constant, and the metric is unique. ∎

It follows from the uniqueness result that the natural parameter space of our Taub-NUT type metrics is the space of positive definite rank 2 matrices $(a_{ij})$ , which involves 3 parameters. The discrete group $S_{3}$ acts on the parameter space by permuting the edges $\mathfrak{D}_{1},\mathfrak{D}_{2},\mathfrak{D}_{3}$ , or equivalently interchanging the 3 positive numbers $a_{11},a_{22},a_{11}+a_{12}+a_{21}+a_{22}.$ This permutation does not change the holomorphic isometry type of the Taub-NUT type metrics, so the moduli space of our construction is the $S_{3}$ -quotient of the parameter space. The scaling transformations act on the parameter space by

a_{ij}\mapsto\Lambda a_{ij},\quad A\mapsto\Lambda^{2}A.

The size of $A$ is inversely related to the area of the asymptotic $T^{2}$ in the generic region near infinity, and the inverse matrix $(a^{ij})$ up to scale describes the shape of the asymptotic $T^{2}$ . If we restrict attention to $C^{-1}\delta_{ij}\leq a_{ij}\leq C\delta_{ij}$ , then the Taub-NUT type metrics on $\mathbb{C}^{3}$ are uniformly equivalent.

We mention two interesting problems:

Question.

What kind of degenerations would happen if the scale invariant uniform ellipticity bound (2.11) fails?

Question.

Can we prove uniqueness under a weaker hypothesis? For instance, if a complete Calabi-Yau metric on $\mathbb{C}^{3}$ is uniformly equivalent to $g_{\mathbb{C}^{3}}$ , then does it need to be a member of our family of Taub-NUT type metrics? If we are only given the topology of $\mathbb{C}^{3}$ , then is it possible to characterise our Taub-NUT type metrics in terms of its tangent cone at infinity and some extra curvature decay conditions?

The author feels this uniqueness question would be the beginning of a classification program of higher dimensional gravitational instantons (cf. Section 2.11.2 for more discussions).

2.11. Exotic metrics: past and future

This informal Section aims to connect the new Taub-NUT type metric on $\mathbb{C}^{3}$ to a circle of ideas in the literature, and sketch the directions for plausible generalisations and the scope for future research.

2.11.1. Exotic metrics on $\mathbb{C}^{n}$

We begin with some historical remarks about the fundamental problem:

Question.

Given $n\geq 2$ , what are the complete Calabi-Yau metrics on $\mathbb{C}^{n}$ equipped with the standard holomorphic volume form?

The initial guess was that the only solution is the flat metric. The rationale is that the moduli of compact Calabi-Yau manifolds depends on the cohomology class of the Kähler form and the holomorphic volume form, and since $\mathbb{C}^{n}$ has trivial topology, it seemed that there was no room to admit nontrivial Calabi-Yau metrics. The situation changed when LeBrun first observed that the Taub-NUT metric gives a counterexample on $\mathbb{C}^{2}$ (cf. Section 1.8). Hindsight shows that the necessary amount of nontrivial topology comes from an additional fibration structure. In fact the Taub-NUT metric admits two kinds of fibration structures: a holomorphic fibration $\mathbb{C}^{2}_{z_{0},z_{1}}\xrightarrow{z_{0}z_{1}}\mathbb{C}$ which gives an algebraic perspective, and a circle fibration coming from the Gibbons-Hawking ansatz which gives a transcendental perspective.

In [18] the author realised that if we take the holomorphic fibration one step further, namely if we start from the standard Lefschetz fibration $\mathbb{C}^{3}\xrightarrow{f=z_{1}^{2}+z_{2}^{2}+z_{3}^{2}}\mathbb{C}$ on $\mathbb{C}^{3}$ , then we can construct a nontrivial complete Calabi-Yau metric on $\mathbb{C}^{3}$ , such that near spatial infinity, the restricted metric on the affine quadric fibres are approximately the Eguchi-Hanson metrics on the fibres, and the horizontal part of the metric is approximately the pullback of the Euclidean metric on the base. This work was soon generalised independently by Conlon-Rochon [3] and Székelyhidi [27], who developed more substantial linear analysis to treat more complicated holomorphic fibrations. In the most general known version, we start from a weighted homogeneous polynomial $f:\mathbb{C}^{n}\to\mathbb{C}$ where $n\geq 3$ , such that the only singularities in the fibration $f$ are isolated singularities on the central fibre $f^{-1}(0)$ , and we require the weighted cone $f^{-1}(0)$ to admit a conical Calabi-Yau metric whose Reeb vector field action is compatible with the weights. Algebro-geometrically, the singular fibre must have klt singularity, and the requirement for the existence of a conical Calabi-Yau metric imposes a stability condition on the singular fibre. Then by standard results the smoothing fibres $f^{-1}(c)$ are equipped with asymptotically conical Calabi-Yau metrics, which now play the same role as the Eguchi-Hanson metrics played in the $\mathbb{C}^{3}$ example setting. The final output of their theory is a complete Calabi-Yau metric on $\mathbb{C}^{n}$ associated to the fibration $f:\mathbb{C}^{n}\to\mathbb{C}$ , equipped with the standard holomorphic volume form.

The most important Riemannian geometric aspect of this infinite class of complete Calabi-Yau metrics is that the volume of metric balls have Euclidean volume growth rate

C^{-1}\leq\frac{\text{Vol}(B(r))}{\text{Vol}(B_{\text{Euclid}}^{2n}(r))}\leq 1,\quad r>0.

Since these manifolds are Ricci-flat, it makes sense to take the tangent cone at infinity, which is identified as the singular variety $f^{-1}(0)\times\mathbb{C}$ with the product metric, and in particular has the same dimension as $\mathbb{C}^{n}$ . This aspect is contrasted with the Taub-NUT metric in complex dimension 2, whose volume growth rate is $\text{Vol}(B(r)\sim r^{3}$ which is not Euclidean. This failure can be traced back to the fact that the singular fibre $z_{0}z_{1}=0$ for the Taub-NUT $\mathbb{C}^{2}$ is not even irreducible, let alone having a Calabi-Yau cone metric.

Furthermore, the metric distance to the origin for these examples on $\mathbb{C}^{n}$ are bi-Hölder equivalent to the standard Euclidean distance, but not uniformly equivalent. This has the consequence that the ring of algebraic functions on these exotic $\mathbb{C}^{n}$ coincides with the ring of holomorphic functions with polynomial growth, but the filtration structure on these functions induced by the growth rate is not the standard filtration.

Now we turn to the new Taub-NUT type Calabi-Yau metric on $\mathbb{C}^{3}$ . Like the Taub-NUT $\mathbb{C}^{2}$ , it is associated to both a holomorphic fibration structure and a torus fibration structure. The holomorphic fibration is given by $\mathbb{C}^{3}\xrightarrow{z_{0}z_{1}z_{2}}\mathbb{C}$ , which may be viewed as a degenerate case where the fibration is allowed to have more severe singularities: here $z_{0}z_{1}z_{2}=0$ is reducible into 3 pieces, and morever its singularity is non-isolated, stretching all the way into spatial infinity. This explains why the Riemannian curvature does not decay at infinity along the locus $\{z_{i}=z_{j}=0\}$ , a phenomenon similar to Joyce’s examples of quasi-ALE Calabi-Yau metrics [15]. Another viewpoint is that the generic fibre is stable while the central singular fibre is unstable. Their delicate balance produces a global metric on $\mathbb{C}^{3}$ , but the instability near the singular fibre produces large quantum fluctuation effects.

However, the principal novalty of our Taub-NUT type $\mathbb{C}^{3}$ metric comes from the $T^{2}$ -fibration structure. An immediate consequence of the fact that 2 spatial dimensions are ‘compactified’, is that the volume growth rate is sub-Euclidean: in fact $\text{Vol}(B(r)\sim r^{4}$ and the tangent cone at infinity is the flat $\mathbb{R}^{4}$ . This sub-Euclidean growth is otherwise known as collapsing in Riemannian geometry.

An important conceptual feature of real tori is that they are inherently transcendental objects, tied up intimately with the fundamental functions $\log$ and $\exp$ ; we saw the pervasive presence of such transcendental functions in Section 2.4 in the metric asymptote. Another manifestation of this is that the ring of algebraic functions on the Taub-NUT type $\mathbb{C}^{3}$ is defined by holomorphic functions with an exponential type growth condition, rather than the more familiar polynomial growth which is the expected feature in the Euclidean volume growth situation.

The evidence suggests that the full mystery of complete Calabi-Yau metrics on $\mathbb{C}^{n}$ involves at least 3 fundamental phenomena:

•

holomorphic fibrations with a suitable notion of stability, which is associated with Euclidean volume growth rate and polynomial growth rate on holomorphic functions.
•

torus fibrations, which is associated with collapsing phenomenon and exponential growth rate on holomorphic functions.
•

an additional layer of combinatorial complexity involving iterative fibrations (cf. subsection 2.11.3 for the flavour).

This picture seems to fit well with Kontsevich and Soibelman’s conjectural picture for collapsing compact Calabi-Yau manifolds (cf. Chapter 2, 3 in [16]). The relation between the two situations will be further explained in subsection 2.11.4.

2.11.2. Gravitational instantons

A gravitational instanton is a complete non-compact hyperKähler 4-manifold with $\int|\text{Rm}|^{2}d\text{Vol}<\infty$ . The theory of gravitational instantons is very rich, with important contributions from Kronheimer, Atiyah, Hitchin, Hein, and many others. Recent breakthrough made by Chen and Chen [1] is a decisive step towards a complete classification. A conspicuous feature of this classification program is the crucial role played by the volume growth rate. In the Euclidean volume growth rate case, these are the ALE metrics (‘asymptotically locally Euclidean’) classified by Kronheimer. In the sub-Euclidean volume growth case, in all known situations the asymptotic geometry near infinity is approximately a flat torus fibration over a flat base.

We will not attempt to review this extensive literature, but limit ourselves to examine a simple class of examples known as multi-Taub-NUT metrics. In the Gibbons-Hawking coordinates (cf. Section 1.2), this is given by the potential

V=A+\sum_{i=1}^{k}\frac{1}{2\sqrt{|\mu-\mu_{i}|^{2}+|\eta-\eta_{i}|^{2}}},

where $(\mu_{i},\eta_{i})$ are disjoint given points on the base $\mathbb{R}^{3}=\mathbb{R}_{\mu}\oplus\mathbb{C}_{\eta}$ , and $k\geq 1$ . The asymptotic geometry is given by a degree $k$ circle bundle over the complement of a compact region in $\mathbb{R}^{3}$ , whose circle fibres have approximate length $2\pi A^{-1/2}$ . This behaviour is known as asymptotically locally flat, or ALF for short. The $k=1$ case is the usual Taub-NUT metric.

Let’s assume for convenience that the $\eta_{i}$ are all distinct, which is the generic situation. From the holomorphic perspective, the multi-Taub-NUT metrics lives on the smooth algebraic varieties

\{z_{0}z_{1}=F(\eta)=(\eta-\eta_{1})(\eta-\eta_{2})\ldots(\eta-\eta_{k})\}\subset\mathbb{C}^{3}_{z_{0},z_{1},\eta},

with nowhere vanishing holomorphic volume form $\Omega=\frac{1}{F^{\prime}(\eta)}\sqrt{-1}dz_{0}\wedge dz_{1}$ , and the circle action is

e^{i\theta}\cdot(z_{0},z_{1})=(e^{-i\theta}z_{0},e^{i\theta}z_{1}),

which ensures $\iota_{\frac{\partial}{\partial\theta}}\Omega=d\eta$ .

A crucial aspect of multi-Taub-NUT metrics is that they come in a moduli space, determined by the positions of $(\mu_{i},\eta_{i})$ . In general, the fact that a family of geometric objects has natural moduli indicates the possibility that in some degenerate limit they decompose into more primary objects, and the parameters in the moduli comes from the parameters in these building blocks and the combinatorics of the gluing construction. This is the case when the spatial separation distance of the monopole points $(\mu_{i},\eta_{i})\in\mathbb{R}^{3}$ is far larger than the circle length parameter $A^{-1/2}$ . Then we can view the multi-Taub-NUT metric as obtained from gluing $k$ copies of the Taub-NUT metrics, whose curvature centres are far separated and therefore whose mutual interaction is weak.

Now we can try to push this story to higher dimensions. The natural generalisation of complete hyperKähler 4-folds is complete Calabi-Yau manifolds. Since in our Taub-NUT type $\mathbb{C}^{3}$ example the Riemannian curvature does not decay at infinity along $\mathfrak{D}_{i}$ , the total $L^{2}$ -curvature integral is infinite. Finding the correct generalised notion of finite curvature condition is clearly fundamental to any classification program. We do not fully understand what this notion is. A tentative idea compatible with Chen and Chen’s work [1] and our Taub-NUT type $\mathbb{C}^{3}$ example is to require that $|\text{Rm}|\leq C$ globally and $|\text{Rm}|(x)\leq C\text{dist}(x,0)^{-2-\epsilon}$ for some $\epsilon>0$ in the generic region.

In the direction of constructing more examples, we comment that Hein’s existence package is by no means limited to the case of $\mathbb{C}^{n}$ . Focusing on complex dimension 3, the distinguished role of our Taub-NUT type metrics on $\mathbb{C}^{3}$ is instead that they are more primary objects, and in particular ought to have a more rigid moduli space, than most of the other 3-dimensional complete Calabi-Yau metrics with similar behaviours. It is perhaps best to illustrate this by a conjectural example which generalises the multi-Taub-NUT metrics.

Let $\eta_{i}$ be all distinct and take the smooth algebraic varieties

\{z_{0}z_{1}z_{2}=F(\eta)=(\eta-\eta_{1})(\eta-\eta_{2})\ldots(\eta-\eta_{k})\}\subset\mathbb{C}^{4}_{z_{0},z_{1},z_{2},\eta},

with nowhere vanishing holomorphic volume form $\Omega=\frac{-1}{F^{\prime}(\eta)}dz_{0}\wedge dz_{1}\wedge dz_{2}$ . These admit a $T^{2}$ -action

e^{i\theta_{1}}\cdot(z_{0},z_{1},z_{2})=(e^{-i\theta_{1}}z_{0},e^{i\theta_{1}}z_{1},z_{2}),\quad e^{i\theta_{2}}\cdot(z_{0},z_{1},z_{2})=(e^{-i\theta_{2}}z_{0},z_{1},e^{i\theta_{2}}z_{2})

which ensures $\Omega(\frac{\partial}{\partial\theta_{1}},\frac{\partial}{\partial\theta_{2}})=d\eta$ . It seems likely that Hein’s package can be made to provide a multi-parameter family of complete Calabi-Yau metrics on these varieties. Morever, when the $T^{2}$ -fibres have much smaller lengths compared to the spatial separation of $\eta_{i}$ , then the author expects such metrics to have a gluing description in terms of our Taub-NUT type metric on $\mathbb{C}^{3}$ . On the other extreme, if we allow $\eta_{i}$ to collide, then we may see new metric behaviours not yet understood in the literature.

Remark 2.12.

Another conjectural example of this flavour can be found in the final Section of the author’s paper [20].

2.11.3. Generalisation of ALF geometry

We now discuss the problems of generalising the Taub-NUT type $\mathbb{C}^{3}$ to higher dimensional exotic metrics on $\mathbb{C}^{n}$ . The key issue seems to be an extra layer of combinatorial complexity of recursive nature. This calls for a theory which deals with linear analysis on quasi-ALF geometry. Roughly put, a quasi-ALF geometry of complexity 1 asymptotically looks like a flat torus fibration over a flat base. A quasi-ALF geometry of complexity $k$ is a singular torus fibration, whose asymptotic behaviour away from the neighbourhood of a lower dimensional stratified singular set looks ALF, and whose behaviour transverse to the singular locus is modelled on a quasi-ALF geometry of complexity $k-1$ . We shall not attempt to make a formal definition, but merely point out that theories of a very similar flavour are much studied, such as QALE spaces by Joyce [15], and QAC spaces by Degeratu and Mazzeo [4].

A conjectural example which illustrates the main ideas is the direct generalisation of our Taub-NUT type metric to $\mathbb{C}^{n}$ with $n\geq 4$ . We take the holomorphic fibration

\mathbb{C}^{n}\xrightarrow{\eta=z_{0}z_{1}\ldots z_{n-1}}\mathbb{C}_{\eta},\quad\Omega=\sqrt{-1}^{n-1}dz_{0}\wedge dz_{1}\ldots\wedge dz_{n-1}.

which admits the action by the diagonal torus $T^{n-1}\subset\text{SL}(n,\mathbb{C})$ . The asymptotic geometry is as follows:

•

Far away from $\cup\{z_{i}=z_{j}=0\}$ , the metric looks like a flat $T^{n-1}$ -fibration over a flat base. In the holomorphic persepcitive, the fibres of $\eta=z_{0}\ldots z_{n-1}$ have a almost flat cylindrical metric on $(\mathbb{C}^{*})^{n-1}$ , and the horizontal part of the metric looks like the pullback of a Euclidean metric on $\mathbb{C}_{\eta}$ .
•

Near $\cup\{z_{i}=z_{j}=0\}$ but far from $\cup\{z_{i}=z_{j}=z_{k}\}$ , we see the Taub-NUT metric appearing in the transverse direction to $\cup\{z_{i}=z_{j}=0\}$ .
•

Near $\cup\{z_{i}=z_{j}=z_{k}\}$ but far from the intersection of 4 coordinate hyperplanes, we see the Taub-NUT type $\mathbb{C}^{3}$ appearing in the transverse direction to $\cup\{z_{i}=z_{j}=z_{k}\}$ .

…
•

Near $\{z_{1}=\ldots z_{n-1}=0\}$ but far from $\{z_{0}=0\}$ , we see the conjectural metric on $\mathbb{C}^{n-1}$ appearing in the transverse direction.

The point is that if one has a sufficiently powerful linear theory which could correct the initial volume form errors to have faster than quadratic decay near infinity, then one can invoke Hein’s package to produce a global Calabi-Yau metric. The whole construction follows a clearly inductive pattern.

2.11.4. Connection to collapsing compact Calabi-Yau metrics

A family of Calabi-Yau metrics $(X_{t},\omega_{t})$ living on a flat family of compact Calabi-Yau manifolds is said to be collapsing if there is no uniform estimate

\text{Vol}_{\omega_{t}}(B(x_{t},r))\geq\kappa r^{\dim_{\mathbb{R}}X_{t}},\quad\forall x_{t}\in X_{t},\forall 0<r<\text{diam}(X_{t}),\quad\kappa>0.

Two well-studied basic mechanisms for collapsing are:

•

Fix the complex structure of $X_{t}=X$ and a reference Kähler class $[\omega_{X}]$ on $X$ . Assume there is a holomorphic fibration $f:X\to Y$ to a lower dimensional Kähler manifold $Y$ with Kähler class $[\omega_{Y}]$ . Then we take $\omega_{t}$ to be the Calabi-Yau metric in the class $[t\omega_{X}+f^{*}\omega_{Y}]$ , where $t\ll 1$ . Crucially the fibre volume is cohomologically determined, and the fibre length scale is much smaller compared to the diameter of the base (cf. [29]).
•

Fix a polarisation on a 1-parameter flat family $X_{t}$ , which prescribes the Kähler class, and assume there is a holomorphic volume form $\Omega_{\mathcal{X}}$ on the total space, so there are induced holomorphic volume forms $\Omega_{t}$ on $X_{t}$ depending on $t$ in a holomorphic way. Then we study the Calabi-Yau metrics $\omega_{t}$ as we allow the complex structure to degenerate, in such a way that the central fibre $X_{0}$ has worse than klt singularities.

Kontsevich and Soibelman observe that in the polarised collapsing situation, the resolution of singularity implies

\int_{X_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}=C(\log|t|)^{m}|t|^{k}(1+o(1)),

where $C$ is some constant, $k$ is an integer which can be taken as zero by adjusting $\Omega_{t}$ , and $0<m\leq\dim_{\mathbb{C}}X_{t}$ if the central fibre has worse than klt singularities . The integer $m$ is determined by Hodge theory for the degeneration. The curious presence of the transcendental factor $(\log|t|)^{m}$ is interpreted by Kontsevich and Soilbelman as indicating the presence of an $m$ -dimensional torus fibration; in the special case of the large complex structure limit $\dim_{\mathbb{C}}X_{t}=m$ they predict a $T^{m}$ -fibration, which is compatible with the SYZ proposal (cf. Section 3.1 [16]). Transcendental phenomenon is captured by non-archimdean analysis. They also suggest that collapsing phenomenon in general involves an iterative fibration structure, based on motivations from conformal field theory (cf. Section 2.3 in [16]).

There is a simple conceptual relation between collapsing families of Calabi-Yau metrics $(X_{t},\omega_{t})$ on compact manifolds, and non-compact complete Calabi-Yau metrics. If we scale the metrics such that $\sup|\text{Rm}|=1$ inside a region of interest, then there is a dichotomy:

•

If the injectivity radius is bounded below, then the pointed Gromov-Hausdorff limit is a smooth complete Calabi-Yau manifold (a ‘complete bubble’).
•

If the injectivity is not bounded below, then we are in the situation of collapsing with bounded curvature, and we should instead look at the covering geometry.

It often happens that the original $X_{t}$ has a natural fibration structure, which would strongly motivate a complete Calabi-Yau manifold with the same kind of fibration structure.

To explain the role of the Euclidean volume growth condition for the complete Calabi-Yau manifolds, we recall a basic fact in Riemannian geometry called Bishop-Gromov monotonicity, which implies that for Ricci-flat manifolds of real dimension $N$ , the normalised volume

\frac{\text{Vol}(B(x,r))}{\text{Vol}(B^{N}_{\text{ Euclid }}(0,r))}

is a decreasing function of the radius $r$ . Thus if one has a geometric reason for the non-collapsing bound $\text{Vol}(B(x,R))\geq\kappa R^{N}$ at a particular distance scale $R$ , then in all smaller scales $r$ we have also $\text{Vol}(B(x,r))\geq\kappa r^{N}$ . In particular, even though a family of Calabi-Yau metric is collapsing globally, it can happen that in a local region of interest the non-collapsing bound holds, so the complete bubble inherits the Euclidean volume growth condition. The reader is referred to the author’s papers [19][20] for concrete examples where this phenomenon happens.

Finally, focusing on complex dimension 3, recall from subsection 2.11.2 that the Taub-NUT type metrics on $\mathbb{C}^{3}$ are expected to be primary objects, while the conjectural multi-Taub-NUT type metrics are composite objects which naturally arise in high dimensional families. We suggest that this means the Taub-NUT type metric on $\mathbb{C}^{3}$ typically occurs as a complete bubble in a suitably generic 1-parameter collapsing family of compact Calabi-Yau metrics when the Euclidean volume growth condition fails, while most other complete bubbles are relevant for multi-parameter degenerations.

Chapter 2 Taub-NUT Type Metrics on ℂ 3 [03ZF]