Uniqueness of the Ricci Flow on Complete Noncompact Manifolds

TL;DR

This paper proves that the Ricci flow with bounded curvature on complete noncompact manifolds has a unique solution, addressing a longstanding open problem in geometric analysis.

Contribution

It establishes the first rigorous proof of uniqueness for Ricci flow solutions on complete noncompact manifolds with bounded curvature.

Findings

Uniqueness of Ricci flow solutions on complete noncompact manifolds proven.

Addresses an open problem in the theory of Ricci flow.

Supports the development of Ricci flow with surgery techniques.

Abstract

The Ricci flow is an evolution system on metrics. For a given metric as initial data, its local existence and uniqueness on compact manifolds was first established by Hamilton \cite{Ha1}. Later on, De Turck \cite{De} gave a simplified proof. In the later of 80's, Shi \cite{Sh1} generalized the local existence result to complete noncompact manifolds. However, the uniqueness of the solutions to the Ricci flow on complete noncompact manifolds is still an open question. Recently it was found that the uniqueness of the Ricci flow on complete noncompact manifolds is important in the theory of the Ricci flow with surgery. In this paper, we give an affirmative answer for the uniqueness question. More precisely, we prove that the solution of the Ricci flow with bounded curvature on a complete noncompact manifold is unique.