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.

Now recall [9, Def. 1.1] a complete manifold $(M,g)$ is called $\text{SOB}(p^{\prime})$, if there exist $x_{0}\in M$ and $C\geq 1$ such that

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  $B(x_{0},s)\setminus B(x_{0},t)$ is connected for all $s\geq t\geq C$,

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  $\text{Vol}(B(x_{0},s))\leq Cs^{p^{\prime}}$ for all $s\geq C$,

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  $\text{Vol}(B(x,(1-C^{-1})r(x)))\geq C^{-1}r(x)^{p^{\prime}}$ and $Ric(x)\geq-Cr(x)^{-2}$ if $r(x)=dist(x_{0},x)\geq C$.

Hein [9] shows that when $p^{\prime}>2$, then $SOB(p^{\prime})$ is a sufficient conditions for the weighted Sobolev inequality. As observed in [20, section 7], the connectivity of the annulus can be relaxed to the weaker requirement of relative connected annulus:

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  For sufficiently large $D$, any two points $x_{1},x_{2}\in M$ with $d(x_{0},x_{i})=D$ can be joined by a curve of length at most $CD$, lying in the annulus $B(x_{0},CD)\setminus B(x_{0},C^{-1}D)$, for a uniform constant $C>1$.

For our particular metric ansatz $dd^{c}\phi_{glue}$, the assumptions on the volume growth rate of balls are immediate consequences of the our much more refined description of the generalized Calabi ansatz, and the metric behaviour near the Tian-Yau region. The quadratic decay on the Ricci tensor is a consequence of the faster than quadratic decay on the volume form error to all derivatives (cf. section 4.7).

To verify the relative connnected annulus property, notice any point close to the Tian-Yau region can be first connected via a path of length $O(\tilde{\rho})$ contained inside the annulus, to a point in the generic region where $x_{1},x_{2}$ are comparable, and the statement is obvious for two points in the generic region.

The smallness of $dd^{c}\phi_{rel}$ near infinity means that the main features of the asymptotic geometry are preserved. In particular, the volume growth of the Calabi-Yau metric $dd^{c}\phi$ is $\text{Vol}(B(\tilde{\rho}))\sim\tilde{\rho}^{\frac{4n}{n+2}}$. The tangent cone at infinity refers to the pointed Gromov-Hausdorff limit of rescaled geodesic balls centred at a fixed reference point. In our case, the rescaling procedure obliviates the $T^{2}$ and $Y$-fibres . The tangent cone is topologically $\mathbb{R}_{\geq 0}^{2}$ with the variables $x_{1},x_{2}$, and metrically it is up to a constant the Hessian metric

where $u(x_{1},x_{2})$ is the solution to the non-archimedean Monge-Ampère equation. The renormalized measure on the tangent cone, which comes from pushing forward the complex Monge-Ampère measure, is up to constant factor $dx_{1}dx_{2}$.