PIC1 pinched manifolds are flat or compact

TL;DR

This paper extends Hamilton's pinching conjecture to all dimensions, proving that certain non-compact manifolds with pinched Ricci curvature are either flat or compact, using a novel lifting technique and Ricci flow estimates.

Contribution

It introduces a new lifting method for collapsed manifolds and provides a Ricci flow curvature estimate without strong positivity assumptions.

Findings

Pinched Ricci curvature implies flatness or compactness in all dimensions.

Developed a lifting technique for collapsed manifolds at infinity.

Established a Ricci flow curvature estimate without Harnack inequality.

Abstract

Hamilton's pinching conjecture, that three-dimensional complete non-compact manifolds with pinched Ricci curvature are flat, has recently been resolved using Ricci flow. In this paper we prove a direct analogue of that result in all dimensions. In order to do so we develop a lifting technique that allows us to handle manifolds that are collapsed at infinity. This new method also gives an alternative way of handling collapsed manifolds in the known three-dimensional case. As part of this approach, we prove a Ricci flow curvature estimate of a type that would normally be derived from the Harnack inequality, but without requiring the strong curvature positivity hypothesis demanded by Harnack. We give an improved gap theorem as a further application.