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In this article we study the metric property and the function the- ory of asymptotically locally Euclidean (ALE) Kahler manifolds. In particular, we prove the Ricci atness under the assumption that the Ricci curvature of such manifolds is either nonnegative or nonpositive. The result provides a gen- eralization of previous gap type theorems established by Greene and Wu, Mok, Siu and Yau, etc. It can also be thought of as a general positive mass type result. The method also proves the Liouville properties of plurisubharmonic functions on such manifolds. We also give a characterization of Ricci atness of an ALE Kahler manifold with nonnegative Ricci curvature in terms of the structure of its cone at innity.

Ricci flatness of asymptotically locally Euclidean metrics