Lower bounds on Ricci flow invariant curvatures and geometric applications

TL;DR

This paper establishes lower bounds on Ricci flow invariant curvatures, linking convergence properties and stability of manifolds under curvature conditions, and extends Ricci flow techniques to certain singular spaces.

Contribution

It introduces new bounds on Ricci flow invariant cones, connects Gromov-Hausdorff and Ricci flow convergence, and constructs Ricci flows for specific singular Alexandrov spaces.

Findings

Bounded curvature operators remain in invariant cones under Ricci flow.

Gromov-Hausdorff convergence aligns with Ricci flow convergence under curvature conditions.

Constructed Ricci flows for certain singular Alexandrov spaces.

Abstract

We consider Ricci flow invariant cones C in the space of curvature operators lying between nonnegative Ricci curvature and nonnegative curvature operator. Assuming some mild control on the scalar curvature of the Ricci flow, we show that if a solution to Ricci flow has its curvature operator which satsisfies R+\epsilon I \in C at the initial time, then it satisfies R +K\epsilon I \in C on some time interval depending only on the scalar curvature control. This allows us to link Gromov-Hausdorff convergence and Ricci flow convergence when the limit is smooth and R + I \in C along the sequence of initial conditions. Another application is a stability result for manifolds whose curvature operator is almost in C. Finally, we study the case where C is contained in the cone of operators whose sectional curvature is nonnegative. This allow us to weaken the assumptions of the previously mentioned applications. In particular, we construct a Ricci flow for a class of (not too) singular Alexandrov spaces.