On the geometry of asymptotically flat manifolds

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