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

Wilking, Burkhard

Zusammenfassung

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 AdSO(n,ℂ). The analogue for Kähler 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 Kähler manifold with positive orthogonal bisectional curvature evolves to a manifold with positive bisectional curvature and is thus biholomorphic to ℂℙn. Moreover, the methods can also be applied to prove Harnack inequalities. © Walter de Gruyter Berlin · Boston 2013.