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Les Annales de l'institut Fourier sont membres du Centre Mersenne pour l'édition scienti que ouverte www.centre-mersenne.org Ann. Inst. Fourier, Grenoble Article à paraître Mis en ligne le 15 avril 2021.

This note is a study of nonnegativity conditions on curvature which are preserved by the Ricci flow. We focus on specific kinds of curvature conditions which we call noncoercive, these are the conditions for which nonnegative curvature and vanishing scalar curvature doesn't imply flatness.We show that, in dimensions greater than 4, if a Ricci flow invariant condition is weaker than "Einstein with nonnegative scalar curvature", then this condition has to be (if not void) the condition "nonnegative scalar curvature". As a corollary, we obtain that a Ricci flow invariant curvature condition which is stronger than "nonnegative scalar curvature" cannot be (strictly) satisfied by compact Einstein symmetric spaces such as S 2 × S 2 or CP 2 . We also investigate conditions which are satisfied by all conformally flat manifolds with nonnegative scalar curvature.When studying Ricci flow, it is useful to know that some "nonnegative curvature"-type geometric condition is preserved along the flow. Although Ricci flow has been studied extensively since R. Hamilton's seminal paper, there is still no comprehensive theory of curvature conditions which are preserved by Ricci flow. A significant advance in this direction is the work of Wilking ([Wil10]) which gives a unified construction for almost all known Ricci flow invariant curvature conditions. The paper [GMS11] gives general results on curvature conditions coming from this construction.We want to gain a better understanding of general Ricci flow invariant curvature conditions. Curvature conditions are encoded by convex cones C (called curvature cones) in the space of algebraic curvature operators S 2 B Λ 2 R n which are invariant under the natural action of the orthogonal group. As a consequence of the maximum principle for systems, a sufficient condition for a curvature condition to be preserved under the Ricci flow is the preservation of the cone C by the flow of some explicit vector field. Readers not familiar with these notions will find a quick exposition and references in section 1.

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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 +ε I ∈ C at the initial time, then it satisfies R +Kε I ∈ 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 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.