Global existence and uniqueness of solutions of the Ricci flow equation

Abstract

In this paper we will prove the global existence and uniqueness of solutions of the Ricci flow equation on $R^2$ $u_t=\Delta \text{ log }u$, $u>0$, in $R^2\times (0,\infty)$, $u(x,0) =u_0(x)$ for $x\in R^2$, satisfying the inequality $u_t\le u/t$ in $R^2\times (0,\infty)$ and the condition $\liminf_{r\to\infty}$ log $u(x,t)/\text{log }r\ge -2$ uniformly on any compact subset of $(0,\infty)$ as $r=|x|\to\infty$ for any $u_0\not\in L^1(R^2)$, $u_0\ge 0$, satisfying $u_0\in L_{loc}^p(R^2)$ for some $p>1$.