Bach-flat asymptotically locally Euclidean metrics

Abstract

We obtain a volume growth and curvature decay result for various classes of complete, noncompact Riemannian metrics in dimension 4; in particular our method applies to anti-self-dual or Kahler metrics with zero scalar curvature, and metrics with harmonic curvature. Similar results are known for Einstein metrics, but our analysis differs from the Einstein case in that (1) we consider more generally a fourth order system in the metric, and (2) we do not assume any pointwise Ricci curvature bound.

54 pages; final version

References in corpus (2)

• Hamilton-Jacobi Equations and Distance Functions on Riemannian Manifolds
• Einstein metrics and complex singularities

Cited by in corpus (37)

• A compactness theorem for complete Ricci shrinkers
• Ricci flow on asymptotically Euclidean manifolds
• On Conformally Kaehler, Einstein Manifolds
• A mass for ALF manifolds
• Some Progress in Conformal Geometry
• Rigidity of noncompact complete Bach-flat manifolds
• Compactness results for the Kähler-Ricci flow
• On Closed Manifolds with Harmonic Weyl Curvature
• Optimal Curvature Decays on Asymptotically Locally Euclidean Manifolds
• Volume growth, curvature decay, and critical metrics
• Moduli spaces of critical Riemannian metrics with L^{n/2} norm curvature bounds
• Existence and compactness theory for ALE scalar-flat Kähler surfaces
• Convergence of the Ricci flow on asymptotically flat manifolds with integral curvature pinching
• Limits of manifolds with a Kato bound on the Ricci curvature
• Asymptotics of the self-dual deformation complex
• Conformal harmonic coordinates
• Rigidity for Bach-flat metrics on manifolds with boundary and applications
• Remarks on $L^p$-vanishing results in geometric analysis
• Local Volume Estimate for Manifolds with $L^2$-bounded Curvature
• Calabi flow on toric varieties with bounded Sobolev constant, I
• Gap phenomena and curvature estimates for Conformally Compact Einstein Manifolds
• Deformation theory of scalar-flat Kähler ALE surfaces
• Local gauge conditions for ellipticity in conformal geometry
• A concentration-collapse decomposition for $L^2$ flow singularities
• On compact manifolds with harmonic curvature and positive scalar curvature
• Obstruction-flat asymptotically locally Euclidean metrics
• Moduli spaces of critical Riemannian metrics in dimension four
• Regularity and convergence of 4-dimensional extremal Kahler metrics
• Smoothing metrics on closed Riemannian manifolds through the Ricci flow
• $\epsilon$-regularity for shrinking Ricci solitons and Ricci flows
• The sharp Li-Yau equality on Shrinking Ricci Solitons with applications
• Conformally invariant rigidity theorems on four-manifolds with boundary
• Global $\epsilon$-regularity for 4-dimensional Ricci flow with integral scalar curvature bound
• Calabi flow on projective bundles, I
• Some $L^p$ rigidity results for complete manifolds with harmonic curvature
• Asymptotic Behaviors of Non-variational Elliptic Systems
• Convergence of compact Ricci solitons