Positivity of Ricci Curvature under the Kähler–ricci Flow

An invariant cone in the space of curvature operators is one that is preserved by a flow. For Ricci flow, the condition R ≥ 0 is preserved in all dimensions, while the conditionR ≤ 0 is preserved only in real dimension two. Positive curvature operator is preserved in all dimensions [11], but positive sectional curvature is not preserved in dimensions four and above. The known counterexamples, constructed recently by Ni [18], are non-product metrics on the total space of the tangent bundles R →֒ TSn ։ Sn for n ≥ 2. However, positive sectional curvature and positive Ricci curvature are preserved in dimension three [10]. In higher dimensions, Huisken [14] and Nishikawa [19, 20] have demonstrated the invariance of certain sets defined by curvature pinching conditions. Margerin [16] improved Huisken’s constant, thereby establishing a sharp invariant set in dimension four. Also in dimension four, H. Chen proved that 2-positivity of the curvature operator is preserved [5], while Hamilton later proved that positive isotropic curvature constitutes an invariant cone [13]. An attractive cone in the space of curvature operators is one that is entered asymptotically at a finite-time singularity. For example, the estimate R ≥ Rmin(0) implies that the half-space R ≥ 0 constitutes an attractive cone in all dimensions. Attractive cones place valuable restrictions on which singularity models may appear. Important examples are the pinching estimates for 3-manifolds proved independently by Ivey [15] and Hamilton [12]. These estimates imply that any rescaled limit formed at a finite-time singularity in dimension three has nonnegative sectional curvature. In turn, this fact is of fundamental importance in Perelman’s use of κ-solutions [21, 22] in his recent spectacular progress toward the Geometrization Conjecture. For Kähler–Ricci flow, Bando [1] and Mok [17] proved that positive bisectional curvature defines an invariant cone. Positive Ricci curvature is a more problematic condition (except of course in complex dimension one). It is not expected to be an attractive cone. Indeed, certain Kähler–Ricci solitons constructed by Feldman, Ilmanen, and the author [9] have Ricci curvature of mixed sign, yet are strongly