Lower bounds on Ricci flow invariant curvatures and geometric applications

Abstract

Abstract We consider Ricci flow invariant cones 𝒞 in the space of curvature operators lying between the cones “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 the Ricci flow has its curvature operator which satisfies R + ε I ∈ 𝒞 $\textup {R}+\varepsilon \textup {I}\in \mathcal {C}$ at the initial time, then it satisfies R + K ε I ∈ 𝒞 $\textup {R}+K\varepsilon \textup {I}\in \mathcal {C}$ 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 ∈ 𝒞 $\textup {R}+\textup {I}\in \mathcal {C}$ 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 allows 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.