Finite Diffeomorphism Types of Four Dimensional Ricci Flow with Bounded Scalar Curvature

In this paper, we consider Ricci flow on four dimensional closed manifold with bounded scalar curvature, noncollasping volume and bounded diameter. Under such conditions, we can show that the manifold has finitely many diffeomorphism types, which generalizes Cheeger-Naber’s result to bounded scalar curvature along Ricci flow. In particular, this implies the manifold has uniform L2 Riemann curvature bound. As an application, we point out that four dimensional Ricci flow would not have uniform scalar curvature upper bound if the initial metric only satisfying lower Ricci curvature bound, lower volume bound and upper diameter bound.