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Noncoercive Ricci flow invariant curvature cones

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 "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^2xS^2 or CP^2. We also investigate conditions which are satisfied by all conformally flat manifolds with nonnegative scalar curvature.