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Generalized Kähler–Einstein Metrics andEnergy Functionals

Published online by Cambridge University Press:  20 November 2018

Xi Zhang  and
Xiangwen Zhang
Xi Zhang
Affiliation:
Department of Mathematics, University of Science and Technology of China, P. R. China. e-mail: mathzx@ustc.edu.cn
Xiangwen Zhang
Affiliation:
Department of Mathematics, Columbia University, New York, NY 10027, USA. e-mail: xzhang@math.columbia.edu
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Abstract.

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In this paper, we consider a generalized Kähler–Einstein equation on a 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

53C5532W20complex Monge–Ampère equationenergy functionalgeneralized Kähler–Einstein metricMoser–Trudinger type inequality
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 6 , 01 December 2014 , pp. 1413 - 1435
Copyright
Copyright © Canadian Mathematical Society 2014

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