A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities

Abstract

We consider a subset $S$ of the complex Lie algebra $\so(n,\C)$ and the cone $C(S)$ of curvature operators which are nonnegative on $S$. We show that $C(S)$ defines a Ricci flow invariant curvature condition if $S$ is invariant under $\Ad_{\SO(n,\C)}$. The analogue for K\"ahler curvature operators holds as well. Although the proof is very simple and short it recovers all previously known invariant nonnegativity conditions. As an application we reprove that a compact K\"ahler manifold with positive orthogonal bisectional curvature evolves to a manifold with positive bisectional curvature and is thus biholomorphic to $\CP^n$. Moreover, the methods can also be applied to prove Harnack inequalities. In addition to an earlier version the paper contains some remarks on negative results for Harnack inequalities.