# Obstruction-Flat Asymptotically Locally Euclidean Metrics

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Antonio Ache, University of Wisconsin
Fine Hall 314

Given an even dimensional Riemannian manifold $(M^{n},g)$ with $n\ge 4$, it was shown in the work of Charles Fefferman and Robin Graham on conformal invariants the existence of a non-trivial 2-tensor which involves $n$ derivatives of the metric, arises as the first variation of a conformally invariant functional and vanishes for metrics that are conformally Einstein. This tensor is called the Ambient Obstruction tensor and is a higher dimensional generalization of the Bach tensor in dimension 4. We show that any asymptotically locally Euclidean (ALE) metric which is obstruction flat and scalar-flat must be ALE of a certain optimal order using a technique developed by Cheeger and Tian for Ricci-flat metrics. We also prove a singularity removal theorem for obstruction-flat metrics with isolated $C^{0}$-orbifold singularities. In addition, we show that our methods apply to more general systems. This is joint work with Jeff Viaclovsky.