arXiv:2611.03736v1 Announce Type: new Abstract: We study uniqueness of scalar-flat K\"ahler metrics in a fixed K\"ahler class on an asymptotically locally Euclidean (ALE) K\"ahler 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 $u=O(r^{-\gamma})$, $\gamma>2n-4$, 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 K\"ahler forms. The argument also extends to asymptotically conical K\"ahler metrics approaching a fixed Ricci-flat K\"ahler cone under prescribed decay assumptions.
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