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It is known that, for compact Kahler manifolds with vanishing first Chern-class, there is a unique Ricci-flat metric in a given Kahler class. What is known in the non-compact case? Are there certain assumptions you can impose on the asymptotic behavior that guarantees a nice generalization of this theorem?

user6013
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    It's not easily accessible to me right now, but see Joyce's book on manifolds with special holonomy. I believe he has a version of Calabi's theorem on manifolds which are "asymptotically locally Euclidean". I remember versions of this in other contexts (eg $G_2$ holonomy) being useful in the novel construction of compact manifolds with special holonomy. –  Jun 18 '18 at 20:45

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