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:

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  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.

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  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.

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  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:

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  (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 $2-q+\epsilon<0$.

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  (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

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

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 $g_{\text{Taub}}$:

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