On Wilking's criterion for the Ricci flow

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 $A{d}_{SO(n,\text{C})}$ invariant subset $S\subset \mathbf{s}\mathbf{o}(n,\text{C})$. For curvature operators of a K"ahler manifold of complex dimension $n$, one considers $A{d}_{GL(n,\text{C})}$ invariant subsets $S\subset \mathbf{g}\mathbf{l}(n,\text{C})$. In this article we show: (i) If $S$ is an $A{d}_{SO(n,\text{C})}$ subset, then $C(S)$ is contained in the cone of curvature operators with nonnegative isotropic curvature and if $S$ is an $A{d}_{GL(n,\text{C})}$ subset, then $C(S)$ is contained in the cone of K"ahler curvature operators with nonnegative orthogonal bisectional curvature. (ii) If $S\subset \mathbf{s}\mathbf{o}(n,\text{C})$ is a closed $A{d}_{SO(n,\text{C})}$ invariant subset and $C_{+}(S)\subset C(S)$ denotes the cone of curvature operators which are {\it 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.