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...re operator in C+(S) converges to either to a metric of constant positive sectional curvature or constant positive holomorphic sectional curvature or M is a rank-1 symmetric space. 1. Introduction In =-=[15]-=-, B. Wilking gave a simple criterion for the positivity of certain curvatures to be preserved under the Ricci flow. Wilking’s criterion captures many of the known Ricci flow invariant positivity condi...

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...n the sense that for any X ∧ Y , Z ∧W 〈Rm(X ∧ Y ), Z ∧W 〉 + Rm(X ∧ Y , Z ∧W ) = RXYWZ . Hence for any Ω = (Ωij̄), it is easy to check that 〈Rm(Ω),Ω〉 = Rij̄kl̄Ωij̄Ωkl̄ ∈ R. Hence one can identify (cf. =-=[16]-=-) the condition (NB) as (7.2) {Rm | 〈Rm(Ω),Ω〉 ≥ 0, for any Ω, rank(Ω) = 1}. POINCARÉ-LELONG EQUATION 19 Similarly, condition (NOB) is equivalent to (7.3) {Rm | 〈Rm(Ω),Ω〉 ≥ 0, for any Ω, rank(Ω) = 1,Ω...

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...ly 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 w...

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...nce CP2 is half conformally flat (it satisfies RW − = 0), it is also PIC−. These examples show that the PIC+ condition is strictly weaker than the usual PIC condition. By applying Wilking’s criterion =-=[24]-=- it is easy to see (see Proposition 2.11) that the NNIC± conditions are preserved by Ricci flow. We were informed by S. Brendle that R. Hamilton knew this fact and the proof is similar to Hamilton’s p...

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...ler-ricci flow on the Kähler manifold with Ricci flat metric as its limit at time infinity. We remark that one may formulate more convergence results using the invariant set constructed by B.Wilking =-=[22]-=-, who finds almost all invariant curvature conditions of Ricci flows [11] [5][6][3] [15][16]. The plan of the paper is below. In section 2, we consider the non-local collapsing result obtained by usin...

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...king’s construction We now describe a construction due to Wilking which recovers most of the Ricci flow invariant curvature conditions in one relatively easy proof. This construction was published in =-=[Wil13]-=-. Before stating the criterion, recall that Λ2Rn is naturally isomorphic to so(n,R). And not the action of SO(n,R) on Λ2Rn is actually just the adjoint action of SO(n,R) on its Lie algebra so(n,R), th...

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...e’s book [6] together with precise definitions and additional references. It should also be noted that Wilking has given a unified proof of the preservation of these conditions (along with others) in =-=[27]-=-. Some continuous families of such cones have also been constructed in [4] and [14]. It should be noted that in dimension greater or equal to 4, nonnegative Ricci curvature is not preserved, see [21]....

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...olomorphic to CPn or is isometrically biholomorphic to an irreducible compact Hermitian symmetric space of rank at least 2. While the condition B⊥ ≥ 0 seems weaker, by the works of Chen [8] (see also =-=[19]-=-) and Gu-Zhang [11] we know that a compact simply connected irreducible Kähler manifold with B⊥ ≥ 0 is also either biholomorphic to CPn or is isometrically biholomorphic to an irreducible Date: Febru...

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...riant subset S ⊂ so(n,C) the definition C(S) := { R ∈ CB(R n) | 〈 RC(X), X 〉 C > 0 for all X ∈ S } yields an open, convex O(n)-invariant cone (which moreover turns out to be Ricci flow invariant; see =-=[W11]-=-). Recently, it was proved in [GMS11] that C(S) is stable under connected sum constructions, if S does not contain any elements of the form v∧w, with v ∈ Rn, w ∈ Cn and (gRn)C (v, w) = 0. Since this l...