On the geometry of asymptotically flat manifolds

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 $ℝ\times {ℝ}_{+}$.