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# Generalized Kähler--Einstein Metrics and Energy Functionals

Published:2013-10-12
Printed: Dec 2014
• Xi Zhang,
Department of Mathematics, University of Science and Technology of China, P. R. China
• Xiangwen Zhang,
Department of Mathematics, Columbia University, New York, NY 10027, USA
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## Abstract

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
 MSC Classifications: 53C55 - Hermitian and Kahlerian manifolds [See also 32Cxx] 32W20 - Complex Monge-Ampere operators

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