Obstruction-flat asymptotically locally Euclidean metrics

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 ≥ 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.

Calabi-Yau manifolds with isolated conical singularities

“…(3) Assuming only that Ric 0, (2) shows that the 0-homogeneous holomorphic vector fields on the cone contain p ⊕ Jp and that Ric = 0 on this space. Also, Jp clearly consists of Killing fields.…”

“…By scaling invariance, we can assume that k = 0. Let P denote the L 2 -orthogonal projection onto H 2 over A 1,2 . Define an open neighborhood U of K in G by the condition that g ∈ G lies in U if and only if g(A 1,2 ) ⊂ A 0,3 .…”

Critical metrics for Riemannian curvature functionals

“…The method in [AV12a] applies to much more general systems than just the obstruction tensors, and works in any dimension n ≥ 3. Given two tensor fields A, B, the notation A * B will mean a linear combination of contractions of A ⊗ B yielding a symmetric 2-tensor.…”

Local Gauge Conditions for Ellipticity in Conformal Geometry

“…These tensors have the following behavior under conformal scaling in various dimensions: W (cg) = cW (g) for n ≥ 4, C(cg) = C(g) for n = 3, and O(cg) = c − n−2 2 O(g) for n ≥ 4 even. See [1], [4], [7], [9], [12], [13], [27] for additional information on these tensors.…”