For our later invocation of Hein’s package (cf. section 5.1), we need to know the existence of distance-like functions with gradient and complex Hessian control.

Define a smooth function

$$\tilde{\rho}=(x_{1}^{2}+x_{2}^{2}+1)^{\frac{n+2}{4n}}.$$

As discussed in section 2.6, the distance function to the origin is uniformly equivalent outside a compact set to $|x|^{\frac{n+2}{2n}},$ and $\tilde{\rho}$ can be viewed as a regularized version. An easy consequence of Lemma 4.7 is

In terms of the distance-like function $\tilde{\rho}$, we can rewrite Cor. 4.13 as

$$\left\lVert Err_{2}\right\rVert_{k,\alpha,loc}=O(\tilde{\rho}^{-\frac{2n(2n-1)}{(n-1)(n+2)}}).$$

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Crucially for our later purpose, this decay is faster than quadratic $O(\tilde{\rho}^{-2})$ to all orders of derivatives. Morever, the formula for the Ricci form

$$Ric=-\sqrt{-1}\partial\bar{\partial}\log\left(\frac{(dd^{c}\phi_{glue})^{n}}{K_{0}\sqrt{-1}^{n^{2}}\Omega\wedge\overline{\Omega}}\right)$$

implies $|Ric|=O(\tilde{\rho}^{-2})$ for the glued ansatz.