Uniqueness of scalar-flat ALE Kähler metrics
Abstract
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≥2, 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.
Related
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BibTeX
@misc{yao2026uniqueness,
title = {{Uniqueness of scalar-flat ALE Kähler metrics}},
author = {Qi Yao},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03736},
eprint = {2611.03736},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03736},
}
RIS
TY - UNPB
TI - Uniqueness of scalar-flat ALE Kähler metrics
AU - Qi Yao
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.03736
UR - https://arxiv.org/abs/2611.03736
AB - 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≥2, 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.
ER -