Curvature at the infinity of asymptotically flat Einstein manifold

TL;DR

This paper develops coordinate systems at infinity for asymptotically flat Ricci-flat manifolds with finite curvature norm, enabling the definition of Weyl tensor and renormalized volume at infinity in any dimension.

Contribution

It introduces a method to construct coordinates at infinity for Ricci-flat manifolds with finite curvature norm, allowing new geometric invariants to be defined.

Findings

Coordinates at infinity can be constructed under finite $L^{m/2}$ curvature norm.

Weyl tensor can be defined at infinity for these manifolds.

A version of renormalized volume is established for ALE Ricci-flat manifolds.

Abstract

In this paper, we construct coordinates at the infinity of an asymptotically flat end of a Ricci-flat manifold $(M_m, g)$ as long as the $L^{m/2}$ norm of the curvature is finite in this end. As applications, we can define a Weyl tensor at the infinity of $M$ and a version of renormalized volume following Biquard and Hein to ALE Ricci-flat manifolds/orbifolds of any dimension.