On Wilking’s criterion for the Ricci flow

Abstract

Abstract. B Wilking has recently shown that one can associate a Ricci flow invariant cone of curvature operators C(S), which are nonnegative in a suitable sense, to every Ad SO(n,C) invariant subset S ⊂ so(n, C). For curvature operators of a Kähler manifold of complex dimension n, one considers Ad GL(n,C) invariant subsets S ⊂ gl(n, C). In this article we show:is contained in the cone of curvature operators with nonnegative isotropic curvature and if S is an Ad GL(n,C) subset, then C(S) is contained in the cone of Kähler curvature operators with nonnegative orthogonal bisectional curvature.(ii) If S ⊂ so(n, C) is a closed Ad SO(n,C) invariant subset and C + (S) ⊂ C(S) denotes the cone of curvature operators which are positive in the appropriate sense then one of the two possibilities holds: (a) The connected sum of any two Riemannian manifolds with curvature operators in C + (S) also admits a metric with curvature operator in C + (S) (b) The normalized Ricci flow on any compact Riemannian manifold M with curvature 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.

Surgery stable curvature conditions

“…Since this latter condition is seen to be equivalent to the requirement R S n−1 ×R ∈ C(S), this case also is covered by Theorem A. Furthermore, it was shown in [GMS11] that C(S) ⊂ C iso>0 and that C(S 0 ) = C iso>0 can be achieved by taking S 0 = X ∈ so(n, C) | rk(X) = 2 and X 2 = 0 , thus it is not surprising that the proof in [GMS11] consists in a generalization of the proof given in [MW93].…”

Positive isotropic curvature and self-duality in dimension 4

“…Remark 2.12. This result contradicts Theorem 1.1 in [13], which says that any Wilking cone is contained in the cone of curvature operators with nonnegative isotropic curvature. The proof is however valid in dimension n ≥ 5.…”

“…We note that the above result applies to all Ricci flow invariant cones. In [13] a similar result is proved in dimensions ≥ 5: All the Ricci flow invariant cones constructed by Wilking [24] are contained in the N N IC cone.…”

“…Proof. This follows from [13] since the curvature operator of R × S 3 is in the interior of C IC + . See also the recent work of Hoelzel [17].…”

“…The proof is however valid in dimension n ≥ 5. The point is that it uses (p. 4 of [13]) the simplicity of the Lie algebra so(n, C), which holds only if n = 4. However, on should notice that the cone C IC + is only SO(4, R)-invariant.…”