PIC1 pinched manifolds are flat or compact
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.
Cite
@article{arxiv.2603.22086,
title = {PIC1 pinched manifolds are flat or compact},
author = {Alix Deruelle and Man-Chun Lee and Felix Schulze and Miles Simon and Peter M. Topping},
journal= {arXiv preprint arXiv:2603.22086},
year = {2026}
}