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

TL;DR

This paper establishes new Shi-type estimates for Ricci flow, linking Ricci curvature bounds to curvature operator bounds, and explores the behavior of Ricci curvature when injectivity radius is unknown, leading to improved compactness results.

Contribution

It introduces novel Shi-type estimates based on Ricci curvature and injectivity radius, and extends understanding of Ricci flow behavior without known injectivity radius.

Findings

Bounded Ricci curvature implies curvature operator bounds

Derived new compactness theorems for Ricci flow and solitons

Established Shi-type estimates when injectivity radius is unknown

Abstract

We construct a uniform local bound of curvature operator from local bounds of Ricci curvature and injectivity radius among all $n$-dimensional Ricci flows. Thus new compactness theorems for the Ricci flow and Ricci solitons are derived. In particular, we show that every Ricci flow with $|Ric|\leq K$ must satisfy $|Rm|\leq Ct^{-1}$ for all $t\in (0,T]$, 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.