On the normalized Ricci flow with scalar curvature converging to constant

Abstract

We show that the normalized Ricci flow $g(t)$ on a smooth closed manifold $M$ existing for all $t \geq 0$ with scalar curvature converging to constant in $L^2$ norm should satisfy $$\liminf_{t\rightarrow \infty} \int_M|\stackrel{\circ}{r}|_{g(t)}^2d\mu_{g(t)} =0,$$ where $\stackrel{\circ}{r}$ is the trace-free part of Ricci tensor. Using this, we give topological obstructions to the existence of such a Ricci flow (even with positive scalar curvature tending to $\infty$ in sup norm) on 4-manifolds.