The conical complex Monge–Ampère equations on Kähler manifolds

  • Jiawei Liu
  • Chuanjing ZhangEmail author


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.

Mathematics Subject Classification

53C55 32W20 



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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für Analysis und NumerikOtto-von-Guericke-Universität MagdeburgMagdeburgGermany
  2. 2.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiPeople’s Republic of China

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