[Paper Review] Compactness and Shi-type estimate of the Ricci flow based on Ricci curvature

Chih‐Wei Chen|arXiv (Cornell University)|Feb 5, 2016

Geometric Analysis and Curvature Flows22 references3 citations

TL;DR

This paper establishes a uniform local bound for the curvature operator in n-dimensional Ricci flows using only local bounds on Ricci curvature and injectivity radius, enabling new compactness theorems for Ricci flow and solitons without curvature operator assumptions. It further derives a Shi-type estimate for Ricci curvature under control of the scalar curvature's derivative, even when injectivity radius is unknown.

ABSTRACT

We show that a uniform local bound for the curvature operator can be derived from local bounds of Ricci curvature and injectivity radius among all $n$-dimensional Ricci flows. As a consequence, we obtain new compactness theorems for the Ricci flow and Ricci soliton without assuming any bounds on the curvature operator. 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, a Shi-type estimate for Ricci curvature is derived when the derivative of Ricci curvature is controlled by the derivative of scalar curvature.

Motivation & Objective

• To derive a uniform local bound for the curvature operator from local Ricci curvature and injectivity radius bounds in Ricci flows.
• To establish compactness theorems for Ricci flow and Ricci solitons without requiring curvature operator bounds.
• To analyze Ricci curvature and its derivative when the injectivity radius is not controlled.
• To derive a Shi-type estimate for Ricci curvature under control of the scalar curvature's derivative.

Proposed method

• Utilize local bounds on Ricci curvature and injectivity radius to derive a uniform local bound on the curvature operator via geometric analysis.
• Apply maximum principle techniques to control curvature quantities under Ricci flow evolution.
• Employ a priori estimates to bound the Ricci curvature and its derivatives in terms of scalar curvature derivatives.
• Use the structure of the Ricci flow equation and curvature evolution to derive estimates independent of curvature operator bounds.
• Incorporate injectivity radius control to stabilize curvature estimates in the absence of curvature operator bounds.
• Establish a Shi-type estimate for Ricci curvature by relating its derivative to that of the scalar curvature.

Experimental results

Research questions

• RQ1 Can a uniform local bound on the curvature operator be derived from local Ricci curvature and injectivity radius bounds alone in Ricci flows?
• RQ2 What compactness theorems for Ricci flow and solitons can be established without assuming curvature operator bounds?
• RQ3 How can Ricci curvature and its derivative be estimated when the injectivity radius is not bounded?
• RQ4 Under what conditions does a Shi-type estimate for Ricci curvature hold when only the derivative of the scalar curvature is controlled?

Key findings

• A uniform local bound for the curvature operator is obtained solely from local bounds on Ricci curvature and injectivity radius in n-dimensional Ricci flows.
• New compactness theorems for Ricci flow and Ricci solitons are established without requiring curvature operator bounds.
• A Shi-type estimate for Ricci curvature is derived when the derivative of Ricci curvature is controlled by the derivative of the scalar curvature.
• The analysis remains valid even when the injectivity radius is not known or bounded, extending applicability to broader geometric settings.
• The results demonstrate that curvature operator bounds are not necessary for compactness and derivative estimates in Ricci flow under Ricci curvature and injectivity radius control.