Hein [8, Chapter 3, 4] sets out a framework for solving the complex Monge-Ampère equation on complete noncompact manifolds, building on the seminal paper by Tian and Yau [16]. 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 metric is uniformly equivalent to the Euclidean metric, and the complex structure and the metric has $C^{k,\alpha}$ bounds. Beware that in our applications the injectivity radius can degenerate, and the charts involve local universal covers.

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  There is a function $\tilde{\rho}$ uniformly equivalent to $\text{dist}(0,x)+1$, and satisfies $|\nabla\tilde{\rho}|+\tilde{\rho}|dd^{c}\tilde{\rho}|\leq C$. 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 $p^{\prime}>2$. For $1\leq p\leq\frac{\dim_{\mathbb{R}}M}{\dim_{\mathbb{R}}M-2}$ and functions $u$ with $L^{2}$-gradient,

  $$(\int|u|^{2p}\tilde{\rho}^{p(p^{\prime}-2)-p^{\prime}}dvol)^{1/p}\leq C\int|\nabla u|^{2}.$$

  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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  Denote $\omega_{0}$ as the ambient Kähler form. Let $f\in C^{2,\alpha}$ satisfy $|f|\leq C\tilde{\rho}^{-q}$ for $p^{\prime}>q>2$. Then there is some $0<\alpha^{\prime}\leq\alpha$ and $u\in C^{4,\alpha^{\prime}}$ which solves $(\omega_{0}+dd^{c}u)^{\dim_{\mathbb{C}}M}=e^{f}\omega_{0}^{\dim_{\mathbb{C}}M}$, with decay estimate $|u|\leq C\tilde{\rho}^{2-q+\epsilon}$, 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.