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Search: MSC category 32W20 ( Complex Monge-Ampere operators )

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1. CJM Online first

Zheng, Tao
The Chern-Ricci flow on Oeljeklaus-Toma manifolds
We study the Chern-Ricci flow, an evolution equation of Hermitian metrics, on a family of Oeljeklaus-Toma (OT-) manifolds which are non-Kähler compact complex manifolds with negative Kodaira dimension. We prove that, after an initial conformal change, the flow converges, in the Gromov-Hausdorff sense, to a torus with a flat Riemannian metric determined by the OT-manifolds themselves.

Keywords:Chern-Ricci flow, Oeljeklaus-Toma manifold, Calabi-type estimate, Gromov-Hausdorff convergence
Categories:53C44, 53C55, 32W20, 32J18, 32M17

2. CJM 2013 (vol 66 pp. 1413)

Zhang, Xi; Zhang, Xiangwen
Generalized Kähler--Einstein Metrics and Energy Functionals
In this paper, we consider a generalized Kähler-Einstein equation on Kähler manifold $M$. Using the twisted $\mathcal K$-energy introduced by Song and Tian, we show that the existence of generalized Kähler-Einstein metrics with semi-positive twisting $(1, 1)$-form $\theta $ is also closely related to the properness of the twisted $\mathcal K$-energy functional. Under the condition that the twisting form $\theta $ is strictly positive at a point or $M$ admits no nontrivial Hamiltonian holomorphic vector field, we prove that the existence of generalized Kähler-Einstein metric implies a Moser-Trudinger type inequality.

Keywords:complex Monge--Ampère equation, energy functional, generalized Kähler--Einstein metric, Moser--Trudinger type inequality
Categories:53C55, 32W20

3. CJM 2009 (vol 62 pp. 218)

Xing, Yang
The General Definition of the Complex Monge--Ampère Operator on Compact Kähler Manifolds
We introduce a wide subclass ${\mathcal F}(X,\omega)$ of quasi-plurisubharmonic functions in a compact Kähler manifold, on which the complex Monge-Ampère operator is well defined and the convergence theorem is valid. We also prove that ${\mathcal F}(X,\omega)$ is a convex cone and includes all quasi-plurisubharmonic functions that are in the Cegrell class.

Keywords:complex Monge--Ampère operator, compact Kähler manifold
Categories:32W20, 32Q15

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