Complete Ricci-flat metrics through a rescaled exhaustion

Abstract

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 $T^{\pi}H^n$.