Asymptotical flatness and cone structure at infinity

Abstract

We investigate asymptotically flat manifolds with cone structure at infinity. We show that any such manifold M has a finite number of ends. For simply connected ends we classify all possible cones at infinity, except for the 4-dimensional case where it remains open if one of the theoretically possible cones can actually arise. This result yields in particular a complete classification of asymptotically flat manifolds with nonnegative curvature: The universal covering of an asymptotically flat manifold with nonnegative sectional curvature is isometric to a product of Euclidean space and an asymptotically flat surface.