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 –manifolds with curvature decay and controlled holonomy. As an application, we show that any complete, asymptotically flat, Ricci-flat metric on a –manifold which is homeomorphic to must be isometric to the Euclidean or the Taub–NUT metric, provided that the tangent cone at infinity is not .
"On the geometry of asymptotically flat manifolds." Geom. Topol. 25 (5) 2469 - 2572, 2021. https://doi.org/10.2140/gt.2021.25.2469