On the distance control problem in Ricci flows

TL;DR

This paper proves that the distance function in Ricci flows remains uniformly continuous over time when the scalar curvature is bounded, providing insights into the stability of geometric structures during the flow.

Contribution

It establishes a new uniform continuity result for the distance function in Ricci flows under scalar curvature bounds, a condition weaker than previous assumptions.

Findings

Distance function is uniformly continuous in time under Ricci flow.

Scalar curvature boundedness suffices for distance control.

Results improve understanding of geometric stability in Ricci flows.

Abstract

We show that the distance function under the Ricci flow is uniformly continuous in the time direction, assuming only the scalar curvature is bounded.