scholarly journals The Kähler–Ricci flow with positive bisectional curvature

2008 ◽  
Vol 173 (3) ◽  
pp. 651-665 ◽  
Author(s):  
D.H. Phong ◽  
Jian Song ◽  
Jacob Sturm ◽  
Ben Weinkove
Keyword(s):  
Ricci Flow ◽  
Bisectional Curvature ◽  
Positive Bisectional Curvature

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

2013 ◽  
Vol 2013 (679) ◽  
pp. 223-247 ◽  
Author(s):  
Burkhard Wilking
Keyword(s):  
Lie Algebra ◽  
Ricci Flow ◽  
Algebraic Approach ◽  
Kähler Manifold ◽  
Curvature Condition ◽  
Bisectional Curvature ◽  
Harnack Inequalities ◽  
Kahler Manifold ◽  
Positive Bisectional Curvature ◽  
Compact Kähler Manifold

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 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.

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