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

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 $|Ric|\le K$ must satisfy $|Rm|\le C{t}^{-1}$ for all $t\in (0,T]$, where $C$ depends only on the dimension $n$ and $T$ depends on $K$ and the injectivity radius $in{j}_{g(t)}$. 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.

arXiv commentStatement of Theorem 7 and other typos are corrected; references are revised

Cited by in corpus (1)

• A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature