Shi-type estimates of the Ricci flow based on Ricci curvature

We prove that the magnitude of the derivative of Ricci curvature can be uniformly controlled by the bounds of Ricci curvature and injectivity radius along the Ricci flow.As a consequence, a precise uniform local bound of curvature operator can be constructed from local bounds of Ricci curvature and injectivity radius among all n-dimensional Ricci flows.In particular, we show that every Ricci flow with |Ric| ≤ K must satisfy |Rm| ≤ Ct -1 for all t ∈ (0, T ], where C depends only on the dimension n, and T depends on K and the injectivity radius inj g(t) .In the second part of this paper, we discuss the behavior of Ricci curvature and its derivative when the injectivity radius is thoroughly unknown.In particular, another Shi-type estimate for Ricci curvature is derived when the derivative of Ricci curvature is controlled by the derivative of scalar curvature.