Asymptotical flatness and cone structure at infinity

TL;DR

This paper studies the structure at infinity of asymptotically flat manifolds with cone structures, classifying possible cones and revealing the universal cover's form for nonnegative curvature cases.

Contribution

It classifies cones at infinity for simply connected ends and provides a complete classification of asymptotically flat manifolds with nonnegative curvature.

Findings

Finite number of ends in such manifolds

Classification of cones at infinity for simply connected ends

Universal cover is a product of Euclidean space and an asymptotically flat surface

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.