On the geometry of asymptotically flat manifolds

TL;DR

This paper explores the geometry of asymptotically flat manifolds with controlled holonomy, revealing torus fibrations over ALE ends and establishing a Hitchin-Thorpe inequality for Ricci-flat 4-manifolds, with implications for their classification.

Contribution

It demonstrates that ends of such manifolds admit torus fibrations and proves a Hitchin-Thorpe inequality, advancing understanding of their geometric structure and classification.

Findings

Ends admit torus fibrations over ALE ends

Hitchin-Thorpe inequality holds for Ricci-flat 4-manifolds

Complete asymptotically flat Ricci-flat metrics on  are Euclidean or Taub-NUT if tangent cone condition is met

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 $\mathbb R \times \mathbb R_+$.