Uniqueness of scalar-flat ALE Kähler metrics

We study uniqueness of scalar-flat Kähler metrics in a fixed Kähler class on an asymptotically locally Euclidean (ALE) Kähler manifold, assuming only differentiated $o(1)$ decay to a fixed Euclidean metric on the end. In complex dimension $n\geq2$, under suitable polynomial decay of the reference metric, we prove that the relative Mabuchi functional is finite, continuous, and convex along weak geodesics whose endpoint potentials satisfy $u=O(r^{-γ})$, $γ>2n-4$, together with the corresponding derivative estimates. Combining this convexity with the analysis of the Lichnerowicz operator in weighted spaces, we obtain uniqueness by a perturbation argument. Decay improvement and comparison of the asymptotic charts produce a global potential with the required decay for the difference of the two cohomologous scalar-flat Kähler forms. The argument also extends to asymptotically conical Kähler metrics approaching a fixed Ricci-flat Kähler cone under prescribed decay assumptions.