Convergence of Ricci flow on $\mathbb{R}^2$ to flat space

TL;DR

This paper proves that the Ricci flow on the Euclidean plane with bounded initial curvature converges to a flat metric, demonstrating stability and convergence properties of the flow in this setting.

Contribution

It establishes the convergence of Ricci flow on  to a flat metric under bounded initial curvature and conformal factor conditions.

Findings

Ricci flow on  converges to flat space

Convergence holds for initial metrics with bounded scalar curvature

Flow stability under bounded initial conditions

Abstract

We prove that, starting at an initial metric $g(0)=e^{2u_0}(dx^2+dy^2)$ on $\mathbb{R}^2$ with bounded scalar curvature and bounded $u_0$, the Ricci flow $\partial_t g(t)=-R_{g(t)}g(t)$ converges to a flat metric on $\mathbb{R}^2$.