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The conical complex Monge–Ampère equations on Kähler manifolds

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Abstract

In this paper, by providing the uniform gradient estimates for approximating equations, we prove the existence, uniqueness and regularity of conical parabolic complex Monge–Ampère equation with weak initial data. As an application, we obtain a regularity estimate, that is, any \(L^{\infty }\)-solution of the conical complex Monge–Ampère equation admits the \(C^{2,\alpha ,\beta }\)-regularity.

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Acknowledgements

The authors would like to thank their advisors Professor Jiayu Li and Professor Xi Zhang for providing many suggestions and encouragements. The J. Liu also would like to thank Professor Miles Simon and Professor Xiaohua Zhu for their constant help. The J. Liu is partially supported by SPP2026 from the German Research Foundation (DFG), and the C. Zhang is partially supported by NSF in China No. 11625106.

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Authors and Affiliations

  1. Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106, Magdeburg, Germany

    Jiawei Liu

  2. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, People’s Republic of China

    Chuanjing Zhang

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  1. Jiawei Liu
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  2. Chuanjing Zhang
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Correspondence to Chuanjing Zhang.

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Communicated by A. Chang.

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Liu, J., Zhang, C. The conical complex Monge–Ampère equations on Kähler manifolds. Calc. Var. 57, 44 (2018). https://doi.org/10.1007/s00526-018-1318-x

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