I am reading Hamilton's An isoperimetric estimate for the Ricci Flow on the Two-Sphere. It is the 9th paper of Collected papers on Ricci flow.
In this paper, Hamilton state that "Since the scalar curvature is bounded below, after dilating the limit will have nonnegative curvature." as picture below. The dilating is the blowing up of Ricci flow which is described in 3th picture below. I don't know why the scalar curvature is bounded below means that the limit have nonnegative curvature.
What I try: By the Theorem 3.3.1 of Lectures on the Ricci flow, I have $$ |\nabla Rm|\le \frac{CM}{\sqrt{t}} $$ namely, when $t$ closes $T$, $|\nabla Rm|$ is bounded upper (although I have not prove, I think it means $|\nabla R|$ is bounded upper). On the other hand, the area of surface $A(t_j)\rightarrow 0$ and $R_{max}(t_j) \rightarrow +\infty$ as $j\rightarrow \infty$. Therefore, when $j$ is large enough, I have $R(t_j)>0$. After dilating, I have $$ R(g_j(0))>0 $$ Since when $n=2$, the scalar curvature under Ricci flow satisfy $$ \partial_t R=\Delta R +R^2. $$ Then by the weak maximum principle for scalar (Theorem 3.1.1 of Lectures on the Ricci flow), I have $$ R(g_j(t))> 0~~~~~~\forall t\ge 0 $$ But when $t<0$, I don't know how to deal. Therefore, I can't get $$ R(g_{\infty}(t)) >0 $$ for all $t$.
