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In this note, we shall prove geodesic convexity of the space of K\"ahlerpotentials on an ALE K\"ahler manifold. This extends earlier results in thecompact case proved in the fundamental work of X-X. Chen. We further prove theboundedness from below of the Mabuchi energy, and give an alternative proof forthe uniqueness of scalar-flat metrics in the category of ALE K\"ahler manifold.

The space of K\"ahler potentials on an asymptotically locally euclidean K\"ahler manifold