Manifolds with PIC1 pinched curvature

TL;DR

This paper extends recent results on three-dimensional manifolds with non-negative pinched Ricci curvature to higher dimensions, proving that PIC1 pinched manifolds are either flat or compact through Ricci flow analysis.

Contribution

It generalizes the existence of Ricci flows from 3D to higher-dimensional PIC1 pinched manifolds and establishes their flatness or compactness.

Findings

Constructed Ricci flow solutions for all time on PIC1 pinched manifolds.

Proved PIC1 pinched manifolds with non-negative complex sectional curvature are flat or compact.

Abstract

Recently it has been proved (Lee-Topping 2022, Deruelle-Schulze-Simon 2022, Lott 2019) that three-dimensional complete manifolds with non-negatively pinched Ricci curvature must be flat or compact, thus confirming a conjecture of Hamilton. In this paper we generalise our work on the existence of Ricci flows from non-compact 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 non-compact manifold that is PIC1 pinched. As an application we prove that any complete manifold of non-negative complex sectional curvature that is PIC1 pinched must be flat or compact.