#### Volume 25, issue 5 (2021)

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On the geometry of asymptotically flat manifolds

### Xiuxiong Chen and Yu Li

Geometry & Topology 25 (2021) 2469–2572
##### Abstract

We investigate the geometry of asymptotically flat manifolds with controlled holonomy. We show that any end of such manifold admits a torus fibration over an ALE end. In addition, we prove a Hitchin–Thorpe inequality for oriented Ricci-flat $4$–manifolds with curvature decay and controlled holonomy. As an application, we show that any complete, asymptotically flat, Ricci-flat metric on a $4$–manifold which is homeomorphic to ${ℝ}^{4}$ must be isometric to the Euclidean or the Taub–NUT metric, provided that the tangent cone at infinity is not $ℝ×{ℝ}_{+}$.

##### Keywords
asymptotically flat, collapsing, torus fiber bundle
##### Mathematical Subject Classification 2010
Primary: 53C20, 53C21, 53C23, 53C25, 53C29