The Ricci-DeTurck flow on complete manifolds

TL;DR

This paper establishes short-time existence, uniqueness, and continuous dependence of Ricci flows on complete manifolds with bounded curvature, using tensor heat kernel estimates and a novel continuous dependence estimate.

Contribution

It provides a unified proof of Ricci flow existence and uniqueness without requiring injectivity radius bounds, advancing the understanding of Ricci flows on complete manifolds.

Findings

Proves short-time existence and uniqueness of Ricci flows on complete manifolds.

Introduces a new continuous dependence estimate independent of injectivity radius.

Ensures continuous dependence on initial data for Ricci flows.

Abstract

Based on the framework of Koch-Lamm and tensor heat kernel estimates, we obtain a uniform proof of the short-time existence, uniqueness, and continuous dependence for Ricci flows starting from a complete Riemannian metric with bounded curvature. A new ingredient is an effective continuous dependence estimate without the assumption of injectivity radius lower bound.