The gradient flow of the $L^2$ curvature energy near the round sphere

TL;DR

This paper studies the gradient flow of the $L^2$ curvature energy on four-manifolds, proving long-term existence and exponential convergence to constant curvature metrics under certain initial conditions.

Contribution

It establishes the long-time existence and exponential convergence of the flow near the round sphere for metrics with positive Yamabe constant and small initial energy.

Findings

Flow exists for a long time under specified conditions.

Flow converges exponentially to a constant curvature metric.

Results apply to four-manifolds with positive Yamabe constant.

Abstract

We investigate the low-energy behavior of the gradient flow of the $L^2$ norm of the Riemannian curvature on four-manifolds. Specifically, we show long time existence and exponential convergence to a metric of constant sectional curvature when the initial metric has positive Yamabe constant and small initial energy.