What is the definition of a Calabi-Yau metric on a non-compact manifold?

Question

There is a lot of important recent work on the construction of Calabi-Yau metrics on non-compact complex manifolds, such as $\mathbb{C}^n$. For example:

[1] Li, Y. A new complete Calabi-Yau metric on $\mathbb{C}^3$. Invent. Math. 217 (2019), no. 1, 1–34.

[2] Székelyhidi, G. Degenerations of $\mathbb{C}^n$ and Calabi-Yau metrics. Duke Math. J. 168 (2019), no. 14, 2651–2700.

Being a complete outsider, I have a very hard time understanding what exactly they mean by a Calabi-Yau metric on a non-compact manifold such as $\mathbb{C}^n$ (they don't seem to define it in their papers). For compact manifolds, one usually defines a Calabi-Yau metric to be a metric with holonomy in $\mathrm{SU}(n)$ (see, e.g., Joyce's book Riemannian holonomy groups and calibrated geometries on page 54). For simply connected manifolds, this is equivalent to a Ricci-flat Kähler metric. So is a Calabi-Yau metric on $\mathbb{C}^n$ the same as a Ricci-flat Kähler metric?

What is the definition of a Calabi-Yau metric on $\mathbb{C}^n$? Are there several equivalent definitions?

Answer 1

There are two slightly different definitions. The first is that it is a Kähler metric that is Ricci-flat, and the second is that it is a Kähler metric on a (usually connected) complex $n$-manifold with holonomy in $\mathrm{SU}(n)$. They are equivalent in the simply-connected case, but not always in the non-simply connected case.

Some people reserve the term Calabi-Yau to mean that the holonomy is exactly equal to $\mathrm{SU}(n)$, so that, for example, the flat metric on $\mathbb{C}^n$ would not be Calabi-Yau in their terminology. However, this is not universal.