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Complex geodesics and complex Monge–Ampère equations with boundary singularity

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Abstract

We study complex geodesics and complex Monge–Ampère equations on bounded strongly linearly convex domains in \(\mathbb C^n\). More specifically, we prove the uniqueness of complex geodesics with prescribed boundary value and direction in such a domain, when its boundary is of minimal regularity. The existence of such complex geodesics was proved by the first author in the early 1990s, but the uniqueness was left open. Based on the existence and the uniqueness proved here, as well as other previously obtained results, we solve a homogeneous complex Monge–Ampère equation with prescribed boundary singularity, which was first considered by Bracci et al. on smoothly bounded strongly convex domains in \({\mathbb {C}}^n\).

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Notes

  1. That \(g_{\Omega }(\,\cdot \,,w)\in C^1({\overline{\Omega }}{\setminus }\{w\})\) is enough for our purpose here.

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Acknowledgements

Part of this work was done while both authors were visiting Huzhou University in part of the summers of 2017 and 2018. Both authors would like to thank this institute for its hospitality during their visit. Part of this work was also carried out while the second author was a postdoctor at the Institute of Mathematics, AMSS, Chinese Academy of Sciences. He would like to express his deep gratitude to his mentor, Professor Xiangyu Zhou, for constant supports and encouragements. He would also like to thank Professors F. Bracci, L. Lempert, and E. A. Poletsky for patiently answering his questions during his reading of their related work. Special thanks also go to Professor L. Lempert for providing the second author with a copy of his very limitedly accessible work [37]. Last but not least, both authors thank the anonymous referee for his/her reading of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Rutgers University, New Brunswick, NJ, 08903, USA

    Xiaojun Huang

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

    Xieping Wang

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  1. Xiaojun Huang
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Correspondence to Xieping Wang.

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Communicated by Ngaiming Mok.

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Xiaojun Huang was partially supported by NSF grants DMS-1665412 and DMS-2000050. Xieping Wang was partially supported by NSFC grants 11771412, 12001513, NSF of Anhui Province grant 2008085QA18, and CPSF grant 2017M620072.

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Huang, X., Wang, X. Complex geodesics and complex Monge–Ampère equations with boundary singularity. Math. Ann. 382, 1825–1864 (2022). https://doi.org/10.1007/s00208-020-02111-4

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