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arXiv · 2606.16176

Curvature at infinity of scalar-flat ALE four-manifolds

Jiangcheng You
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Abstract

We study refined asymptotics of scalar-flat ALE four-manifolds in the Tian--Viaclovsky setting, namely for self-dual or anti-self-dual metrics and for metrics with harmonic curvature. Starting from the ALE coordinates obtained by Tian--Viaclovsky, we construct preferred coordinates at infinity and identify the homogeneous $|x|^{-2}$ term in the metric expansion. This term splits canonically into a scalar part determined by the ALE ADM mass and an algebraic Weyl tensor at infinity. As an application, we consider scalar-flat K\"ahler ALE metrics on minimal resolutions $\pi:X\to\mathbb C^2/\Gamma$ of quotient surface singularities. In this case, the leading Weyl tensor at infinity vanishes exactly when the minimal resolution is crepant.

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Jiangcheng You. 2026-06-15. Curvature at infinity of scalar-flat ALE four-manifolds. https://arxiv.org/abs/2606.16176

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