Thomas Richard

# 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+εI𝒞 at the initial time, then it satisfies R+KεI𝒞 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+I𝒞 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.

The author is grateful to Gilles Carron and Harish Seshadri for helpful discussions during the elaboration of this paper. The author also thanks his supervisor Gérard Besson for his interest and support.