Heat kernel and curvature bounds in Ricci flows with bounded scalar curvature

TLDR: In this article, the authors analyzed Ricci flows on which the scalar curvature is globally or locally bounded from above by a uniform or time-dependent constant, and established a new time-derivative bound for solutions to the heat equation.

Abstract: In this paper we analyze Ricci flows on which the scalar curvature is globally 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 to the 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 bound at 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 4 converges to an orbifold at a singularity. We also obtain a stronger $\varepsilon$-regularity theorem for Ricci flows. This result is particularly useful in the study of Kahler Ricci flows on Fano manifolds, where it can be used to derive certain convergence results.