On the geometry of asymptotically flat manifolds

Abstract

In this paper, 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 ${\mathbb{R}}^4$ must be isometric to the Euclidean or the Taub-NUT metric, provided that the tangent cone at infinity is not $\mathbb{R}\times {\mathbb{R}}_{+}$.

94 pages, revised version, to appear in Geometry & Topology

References in corpus (1)

• Asymptotical flatness and cone structure at infinity

Cited by in corpus (2)

• Topology of Toric Gravitational Instantons
• Ancient solutions to the Ricci flow with isotropic curvature conditions