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 Kähler 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.
From: Jiangcheng You [view email]
[v1]
Mon, 15 Jun 2026 03:44:15 UTC (43 KB)
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