Pseudolocality and strong uniqueness for complete Ricci flows

Abstract

In this paper, we prove a uniform pseudolocality theorem for smooth complete Ricci flows whose initial metrics have Ricci curvature bounded below and uniformly positive injectivity radius. No a priori curvature bound is assumed on the flow. As a consequence, we obtain strong uniqueness for smooth complete Ricci flows from a complete metric with the above geometric bounds. We also prove that every smooth complete Ricci flow from a complete Ricci-flat metric with uniformly positive injectivity radius is stationary. The same conclusion holds for any complete flat initial metric. In particular, we give an affirmative answer to Chen's question on the strong uniqueness of Ricci flow from Euclidean space in every dimension.