  Typical existence result on Ricci-flat metrics is in manifolds of finite
geometry, that is, on $F=\bar F-D$ where $\bar F$ is a compact K\"ahler
manifold and $D$ is a smooth divisor.
  We view this existence problem from a different perspective. For a given
complex manifold $X$, we take a suitable exhaustion $\{X_r\}_{r>0}$ admitting
complete \ke s of negative Ricci. Taking a positive decreasing sequence
$\{\lambda_r\}_{r>0}, \lim_{r\to\infty}\lambda_r=0$, we rescale the metric so
that $g_r$ is the complete \ke\ in $X_r$ of Ricci curvature $-\lambda_r$. The
idea is to show the limiting metric $\lim_{r\to\infty} g_r$ does exist. If so,
it is a Ricci-flat metric in $X$. Several examples: $X=\mathbb C^n$ and $X=TM$
where $M$ is a compact rank-one symmetric space have been studied in this
article.
  The existence of complete \ke s of negative Ricci in bounded domains of
holomorphy is well-known. Nevertheless, there is very few known for unbounded
cases. In the last section we show the existence, through exhaustion, of such
kind of metric in the unbounded domain $T^{\pi}H^n$.
