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

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.

Rank andk-nullity of contact manifolds

We prove that the dimension of the1-nullity distributionN(1)on a closed Sasakian manifoldMof ranklis at least equal to2l−1provided thatMhas an isolated closed characteristic. The result is then used to provide some examples ofk-contact manifolds which are not Sasakian. On a closed,2n+1-dimensional Sasakian manifold of positive bisectional curvature, we show that either the dimension ofN(1)is less than or equal ton+1orN(1)is the entire tangent bundleTM. In the latter case, the Sasakian manifoldMis isometric to a quotient of the Euclidean sphere under a finite group of isometries. We also point out some interactions betweenk-nullity, Weinstein conjecture, and minimal unit vector fields.