On Wilking's criterion for the Ricci flow

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 $S \subset {\bf so}(n,\C)$. For curvature operators of a K\"ahler manifold of complex dimension $n$, one considers $Ad_{GL(n,\C)}$ invariant subsets $S \subset {\bf gl}(n,\C)$. In this article we show: (i) If $S$ is an $Ad_{SO(n,\C)}$ subset, then $C(S)$ 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 K\"ahler curvature operators with nonnegative orthogonal bisectional curvature. (ii) If $S \subset {\bf so}(n,\C)$ is a closed $Ad_{SO(n,\C)}$ 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.