Manifolds with 1/4-pinched Curvature are Space Forms

TL;DR

This paper proves that compact manifolds with pointwise 1/4-pinched curvature evolve under Ricci flow to constant curvature metrics, extending understanding of curvature pinching and flow behavior.

Contribution

It demonstrates that 1/4-pinched curvature manifolds are space forms by analyzing Ricci flow and positive isotropic curvature preservation.

Findings

Ricci flow deforms 1/4-pinched manifolds to constant curvature.

Positive isotropic curvature is preserved under Ricci flow in all dimensions.

The result confirms manifolds with 1/4-pinched curvature are space forms.

Abstract

Let (M,g_0) be a compact Riemannian manifold with pointwise 1/4-pinched sectional curvatures. We show that the Ricci flow deforms g_0 to a constant curvature metric. The proof uses the fact, also established in this paper, that positive isotropic curvature is preserved by the Ricci flow in all dimensions. We also rely on earlier work of Hamilton and of Bohm and Wilking.