Preservation of product structures under the Ricci flow with instantaneous curvature bounds

TL;DR

This paper proves that under certain curvature bounds, the product structure of a Ricci flow solution at initial time persists throughout its evolution, highlighting stability of geometric splitting.

Contribution

The authors establish a new stability result for Ricci flow solutions with curvature bounds, showing product structures are preserved over time.

Findings

Product structure persists under Ricci flow with curvature bounds

Existence of a dimension-dependent constant epsilon

Stability of geometric splitting over time

Abstract

In this note, we prove that there exists a constant $\epsilon >0$, depending only on the dimension, such that if a complete solution to the Ricci flow splits as a product at time $t=0$ and has curvature bounded by $\frac{\epsilon}{t}$, then the solution splits for all time.