Short-time persistence of bounded curvature under the Ricci flow

TL;DR

This paper proves that under Ricci flow, bounded Ricci curvature ensures the sectional curvature remains bounded for a short time, highlighting a stability property of the flow.

Contribution

It establishes a new short-time curvature bound result for Ricci flow solutions with bounded Ricci curvature on noncompact manifolds.

Findings

Sectional curvature remains bounded for short time if initially bounded

Weyl curvature cannot become unbounded instantaneously under bounded Ricci curvature

Strengthened uniqueness statement for Ricci flow

Abstract

We use a first-order energy quantity to prove a strengthened statement of uniqueness for the Ricci flow. One consequence of this statement is that if a complete solution on a noncompact manifold has uniformly bounded Ricci curvature, then its sectional curvature will remain bounded for a short time if it is bounded initially. In other words, the Weyl curvature tensor of a complete solution to the Ricci flow cannot become unbounded instantaneously if the Ricci curvature remains bounded.