Some aspects of Ricci flow on the 4-sphere

TL;DR

This paper investigates Ricci flow on 4-spheres, establishing conditions under which the flow converges exponentially to the standard sphere by analyzing integral conformal invariants and curvature decay.

Contribution

It introduces a conformal gap theorem and demonstrates exponential convergence of Yamabe metrics with small Weyl tensor norm under Ricci flow.

Findings

Conformal gap theorem for 4-spheres.

Exponential convergence of metrics to the standard sphere.

Monotonic decay of reduced curvature tensor norms.

Abstract

In this paper, on 4-spheres equipped with Riemannian metrics we study some integral conformal invariants, the sign and size of which under Ricci flow characterize the standard 4-sphere. We obtain a conformal gap theorem, and for Yamabe metrics of positive scalar curvature with $L^2$ norm of the Weyl tensor of the metric suitably small, we establish the monotonic decay of the $L^p$ norm for certain $p>2$ of the reduced curvature tensor along the normalized Ricci flow, with the metric converging exponentially to the standard 4-sphere.