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 Injectivity Theorem on a Non-Compact Kähler Manifold

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2222
Author(s):  
Jingcao Wu
Keyword(s):  
Sectional Curvature ◽  
Symmetric Spaces ◽  
Kähler Manifold ◽  
Bounded Domains ◽  
Bounded Symmetric Domains ◽  
Curvature Condition ◽  
Kahler Manifold ◽  
Weakly Pseudoconvex ◽  
Compact Kähler Manifold ◽  
Negative Sectional Curvature

In this paper, we establish an injectivity theorem on a weakly pseudoconvex Kähler manifold X with negative sectional curvature. For this purpose, we develop the harmonic theory in this circumstance. The negative sectional curvature condition is usually satisfied by the manifolds with hyperbolicity, such as symmetric spaces, bounded symmetric domains in Cn, hyperconvex bounded domains, and so on.

scholarly journals BOUNDING SECTIONAL CURVATURE ALONG THE KÄHLER–RICCI FLOW

2009 ◽  
Vol 11 (06) ◽  
pp. 1067-1077 ◽  
Author(s):  
WEI-DONG RUAN ◽  
YUGUANG ZHANG ◽  
ZHENLEI ZHANG
Keyword(s):  
Sectional Curvature ◽  
Chern Class ◽  
Ricci Flow ◽  
Kähler Manifold ◽  
Ricci Soliton ◽  
Curvature Operator ◽  
Kahler Manifold ◽  
First Chern Class ◽  
Compact Kähler Manifold ◽  
Uniformly Bounded

If a normalized Kähler–Ricci flow g(t), t ∈ [0,∞), on a compact Kähler manifold M, dim ℂ M = n ≥ 3, with positive first Chern class satisfies g(t) ∈ 2πc1(M) and has curvature operator uniformly bounded in Ln-norm, the curvature operator will also be uniformly bounded along the flow. Consequently, the flow will converge along a subsequence to a Kähler–Ricci soliton.

scholarly journals The collapsing rate of the Kähler–Ricci flow with regular infinite time singularity

2015 ◽  
Vol 2015 (703) ◽  
Author(s):  
Frederick Tsz-Ho Fong ◽  
Zhou Zhang
Keyword(s):  
Ricci Flow ◽  
Kähler Manifold ◽  
Infinite Time ◽  
Time Singularity ◽  
Kahler Manifold ◽  
Compact Kähler Manifold

AbstractWe study the collapsing behavior of the Kähler–Ricci flow on a compact Kähler manifold

scholarly journals Relative non-pluripolar product of currents

2021 ◽  
Author(s):  
Duc-Viet Vu
Keyword(s):  
Kähler Manifold ◽  
Monotonicity Property ◽  
Necessary Condition ◽  
Positive Current ◽  
Lelong Numbers ◽  
Kahler Manifold ◽  
Closed Positive Current ◽  
Positive Currents ◽  
Compact Kähler Manifold

AbstractLet X be a compact Kähler manifold. Let $$T_1, \ldots , T_m$$ T 1 , … , T m be closed positive currents of bi-degree (1, 1) on X and T an arbitrary closed positive current on X. We introduce the non-pluripolar product relative to T of $$T_1, \ldots , T_m$$ T 1 , … , T m . We recover the well-known non-pluripolar product of $$T_1, \ldots , T_m$$ T 1 , … , T m when T is the current of integration along X. Our main results are a monotonicity property of relative non-pluripolar products, a necessary condition for currents to be of relative full mass intersection in terms of Lelong numbers, and the convexity of weighted classes of currents of relative full mass intersection. The former two results are new even when T is the current of integration along X.

A POSITIVITY PROPERTY FOR FOLIATIONS ON COMPACT KÄHLER MANIFOLDS

2006 ◽  
Vol 17 (01) ◽  
pp. 35-43 ◽  
Author(s):  
MARCO BRUNELLA
Keyword(s):  
Kähler Manifold ◽  
Rational Curves ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Canonical Bundle ◽  
Positivity Property ◽  
Kahler Manifold ◽  
Compact Kähler Manifold

We prove that the canonical bundle of a foliation by curves on a compact Kähler manifold is pseudoeffective, unless the foliation is a (special) foliation by rational curves.

scholarly journals Zeros of Holomorphic One-Forms and Topology of Kähler Manifolds

2020 ◽  
Author(s):  
Stefan Schreieder
Keyword(s):  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Joint Paper ◽  
And Topology ◽  
Kahler Manifold ◽  
Compact Kähler Manifold

Abstract A conjecture of Kotschick predicts that a compact Kähler manifold $X$ fibres smoothly over the circle if and only if it admits a holomorphic one-form without zeros. In this paper we develop an approach to this conjecture and verify it in dimension two. In a joint paper with Hao [ 10], we use our approach to prove Kotschick’s conjecture for smooth projective three-folds.

scholarly journals TWISTING OF N=1 SUSY GAUGE THEORIES AND HETEROTIC TOPOLOGICAL THEORIES

1995 ◽  
Vol 10 (30) ◽  
pp. 4325-4357 ◽  
Author(s):  
A. JOHANSEN
Keyword(s):  
Complex Structure ◽  
Kähler Manifold ◽  
Dolbeault Cohomology ◽  
Kähler Metric ◽  
Yang Mills ◽  
Kahler Manifold ◽  
Brst Charge ◽  
Kahler Metric ◽  
Compact Kähler Manifold ◽  
Topological Theories

It is shown that D=4N=1 SUSY Yang-Mills theory with an appropriate supermultiplet of matter can be twisted on a compact Kähler manifold. The conditions for cancellation of anomalies of BRST charge are found. The twisted theory has an appropriate BRST charge. We find a nontrivial set of physical operators defined as classes of the cohomology of this BRST operator. We prove that the physical correlators are independent of the external Kähler metric up to a power of a ratio of two Ray-Singer torsions for the Dolbeault cohomology complex on a Kähler manifold. The correlators of local physical operators turn out to be independent of antiholomorphic coordinates defined with a complex structure on the Kähler manifold. However, a dependence of the correlators on holomorphic coordinates can still remain. For a hyper-Kähler metric the physical correlators turn out to be independent of all coordinates of insertions of local physical operators.

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