On Wilking's criterion for the Ricci flow

TL;DR

This paper explores Wilking's Ricci flow invariant cones of curvature operators, establishing their containment within known curvature positivity cones and analyzing their stability under Ricci flow, with implications for geometric classification.

Contribution

The paper demonstrates that certain invariant cones are contained within classical curvature positivity cones and characterizes the behavior of manifolds with curvature in these cones under Ricci flow.

Findings

Invariant cones are contained in nonnegative isotropic and orthogonal bisectional curvature cones.

Connected sums of manifolds with curvature in these cones also admit such metrics.

Ricci flow on manifolds with these curvature conditions converges to constant curvature or symmetric spaces.

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.