scholarly journals The Chern-Ricci flow and holomorphic bisectional curvature

2016 ◽  
Vol 59 (11) ◽  
pp. 2199-2204 ◽  
Author(s):  
XiaoKui Yang
Keyword(s):  
Ricci Flow ◽  
Bisectional Curvature ◽  
Holomorphic Bisectional Curvature

Bounded properties of curvatures of perturbed Ricci metric on curve moduli

2014 ◽  
Vol 25 (12) ◽  
pp. 1450113
Author(s):  
Xiaorui Zhu
Keyword(s):  
Finite Volume ◽  
Ricci Curvature ◽  
Moduli Space ◽  
Point Of View ◽  
Bisectional Curvature ◽  
Bounded Geometry ◽  
Holomorphic Bisectional Curvature ◽  
Chern Form ◽  
Thick Part ◽  
Kahler Metric

As is well-known, the Weil–Petersson metric ωWP on the moduli space ℳg has negative Ricci curvature. Hence, its negative first Chern form defines the so-called Ricci metric ωτ. Their combination [Formula: see text], C > 0, introduced by Liu–Sun–Yau, is called the perturbed Ricci metric. It is a complete Kähler metric with finite volume. Furthermore, it has bounded geometry. In this paper, we investigate the finiteness of this new metric from another point of view. More precisely, we will prove in the thick part of ℳg, the holomorphic bisectional curvature of [Formula: see text] is bounded by a constant depending only on the thick constant and C0 when C ≥ (3g - 3)C0, but not on the genus g.

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.

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
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