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Lower Bounds on the Bergman Metric Near Points of Infinite Type

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Abstract

Let Ω be a pseudoconvex domain in \(\mathbb C^{n}\) satisfying an f -property for some function f. We show that the Bergman metric associated with Ω has the lower bound \(\tilde {g}(\delta _{\Omega }(z)^{-1})\) where δΩ(z) is the distance from z to the boundary Ω and \(\tilde g\) is a specific function defined by f. This refines Khanh–Zampieri’s work in Khanh and Zampieri (Invent. Math. 188, 729–750, 2012) with weaker smoothness assumption of the boundary.

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Acknowledgements

This work was completed when the second author was visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank the VIASM for their hospitality. It is a pleasure to thank Tran Vu Khanh for stimulating discussions. Especially, we would like to express our gratitude to the referees. Their valuable comments on the first version of this paper led to significant improvements.

Funding

The second author received financial support from the VIASM. The second author was supported by NAFOSTED under grant number 101.02-2017.311.

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Correspondence to Dau The Phiet.

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Phiet, D.T., Van Thu, N. Lower Bounds on the Bergman Metric Near Points of Infinite Type. Vietnam J. Math. 48, 1–10 (2020). https://doi.org/10.1007/s10013-019-00337-7

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