Obstruction-flat asymptotically locally Euclidean metrics

TL;DR

This paper proves that obstruction-flat ALE metrics are of a specific optimal order and extends the technique to general elliptic systems, also establishing a singularity removal theorem for such metrics with isolated orbifold singularities.

Contribution

It demonstrates that obstruction-flat ALE metrics must be of an optimal order and generalizes the proof technique to broader elliptic systems in any dimension.

Findings

Obstruction-flat ALE metrics are of a certain optimal order.

The proof technique applies to general elliptic systems in dimensions n ≥ 3.

A singularity removal theorem for obstruction-flat metrics with orbifold singularities.

Abstract

We show that any asymptotically locally Euclidean (ALE) metric which is obstruction-flat or extended obstruction-flat must be ALE of a certain optimal order. Moreover, our proof applies to very general elliptic systems and in any dimension $n \geq 3$. The proof is based on the technique of Cheeger-Tian for Ricci-flat metrics. We also apply this method to obtain a singularity removal theorem for (extended) obstruction-flat metrics with isolated $C^0$-orbifold singular points.