A general convergence result for the Ricci flow in higher dimensions

TL;DR

This paper proves that under certain curvature conditions, the normalized Ricci flow on higher-dimensional compact manifolds converges to a constant curvature metric, extending understanding of geometric evolution in higher dimensions.

Contribution

It establishes a new convergence result for the Ricci flow in higher dimensions under positive isotropic curvature conditions, which are weaker than some previously known curvature constraints.

Findings

Normalized Ricci flow deforms metrics to constant curvature under positive isotropic curvature.

The curvature condition used is stronger than 2-positive flag curvature but weaker than 2-positive curvature operator.

Convergence holds for compact manifolds of dimension n ≥ 4.

Abstract

Let (M,g_0) be a compact Riemannian manifold of dimension n \geq 4. We show that the normalized Ricci flow deforms g_0 to a constant curvature metric provided that (M,g_0) x R has positive isotropic curvature. This condition is stronger than 2-positive flag curvature but weaker than 2-positive curvature operator.