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 ≥ 0 with scalar curvature converging to constant in L² norm should satisfy

where 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 ∞ in sup norm) on 4-manifolds.