In this paper we analyze Ricci flows on which the scalar curvature isglobally or locally bounded from above by a uniform or time-dependent constant.On such Ricci flows we establish a new time-derivative bound for solutions tothe heat equation. Based on this bound, we solve several open problems: 1.distance distortion estimates, 2. the existence of a cutoff function, 3.Gaussian bounds for heat kernels, and, 4. a backward pseudolocality theorem,which states that a curvature bound at a later time implies a curvature boundat a slightly earlier time. Using the backward pseudolocality theorem, we next establish a uniform $L^2$curvature bound in dimension 4 and we show that the flow in dimension 4converges to an orbifold at a singularity. We also obtain a stronger$\varepsilon$-regularity theorem for Ricci flows. This result is particularlyuseful in the study of K\"ahler Ricci flows on Fano manifolds, where it can beused to derive certain convergence results.
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