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 refined torus fibration over an ALE manifold. 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 × R + .

Ancient solutions to the Ricci flow with isotropic curvature conditions

“…Now, we will construct a transition map ϕ x,y between two local fibrations ϕ x and ϕ y defined in Proposition 6.10. This strategy originates in [19,20], see also [21]. In the following, the function δ(ǫ) → 0 as ǫ → 0 and δ(ǫ) may be different line by line.…”

“…By using Lemma 6.14 and Lemma 6.15. we can follow the standard technique in [19,20] to modify all local fibrations to be compatible. For details, the reader can refer to [21,Theorem 5.16].…”