  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.
