Volume 25, issue 5 (2021)

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On the geometry of asymptotically flat manifolds

Xiuxiong Chen and Yu Li

Geometry & Topology 25 (2021) 2469–2572

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 4 must be isometric to the Euclidean or the Taub–NUT metric, provided that the tangent cone at infinity is not × +.

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asymptotically flat, collapsing, torus fiber bundle
Mathematical Subject Classification 2010
Primary: 53C20, 53C21, 53C23, 53C25, 53C29
Received: 4 December 2019
Revised: 27 July 2020
Accepted: 28 August 2020
Published: 3 September 2021
Proposed: John Lott
Seconded: Mark Gross, Simon Donaldson
Xiuxiong Chen
Institute of Geometry and Physics
University of Science and Technology of China
Mathematics Department
Stony Brook University
Stony Brook, NY
United States
Yu Li
Institute of Geometry and Physics
University of Science and Technology of China