Manifolds with PIC1 pinched curvature

Lee, Man-Chun; Topping, Peter M

doi:10.2140/gt.2025.29.4767

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Manifolds with PIC1 pinched curvature

Man-Chun Lee and Peter M Topping

Geometry & Topology 29 (2025) 4767–4798

DOI: 10.2140/gt.2025.29.4767

Abstract

Recently it has been proved that three-dimensional complete manifolds with nonnegatively pinched Ricci curvature must be flat or compact, thus confirming a conjecture of Hamilton. We generalise our work on the existence of Ricci flows from noncompact pinched three-manifolds in order to prove a higher-dimensional analogue. We construct a solution to Ricci flow, for all time, starting with an arbitrary complete noncompact manifold that is PIC1 pinched. As an application we prove that any complete manifold of nonnegative complex sectional curvature that is PIC1 pinched must be flat or compact.

Keywords

Ricci flow, curvature pinching

Mathematical Subject Classification

Primary: 53C20, 53C21, 53E20

References

Publication

Received: 21 November 2023

Revised: 19 March 2025

Accepted: 10 May 2025

Published: 31 December 2025

Proposed: John Lott

Seconded: Tobias H Colding, Gang Tian

Authors

Man-Chun Lee
Department of Mathematics The Chinese University of Hong Kong Hong Kong Hong Kong

Peter M Topping
Department of Mathematics University of Warwick Coventry United Kingdom

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Volume 29, issue 9 (2025)