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

TL;DR

This paper develops new bounds and estimates for Ricci flows with bounded scalar curvature, solving open problems and analyzing flow behavior near singularities, with applications to Kähler Ricci flows.

Contribution

It introduces a new time-derivative bound for heat solutions, solves key open problems, and establishes curvature bounds and convergence results in four-dimensional Ricci flows.

Findings

Established distance distortion estimates.

Proved existence of cutoff functions.

Derived Gaussian bounds for heat kernels.

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 K\"ahler Ricci flows on Fano manifolds, where it can be used to derive certain convergence results.