Quantitative hyperconvexity and the Bergman distance
Abstract
Let Ω⊂ℂn be a bounded hyperconvex domain admitting a negative plurisubharmonic exhaustion function u such that −u(z)≤ Cexp[−η(log(1/δΩ(z)))] near the boundary, where η is nondecreasing with limt→+∞η(t)=+∞. We present a lower bound for the Bergman distance dB,Ω(z0,z) by counting the iterates of a function determined by the generalized inverse of η. The bound tends to infinity as z→∂Ω for every such η; moreover, it recovers several known estimates. The proof combines a one-sided localization of Green sublevel sets with boundary estimates for the Bergman kernel and a capacity--volume inequality. We also obtain an upper bound for the Bergman distance along the axis of a family of complete Hartogs domains whose fiber radii decay faster than every positive power of the boundary distance in the base disc.
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BibTeX
@misc{chen2026quantitative,
title = {{Quantitative hyperconvexity and the Bergman distance}},
author = {Bo-Yong Chen and Yuanpu Xiong},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34339},
eprint = {2610.34339},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34339},
}
RIS
TY - UNPB
TI - Quantitative hyperconvexity and the Bergman distance
AU - Bo-Yong Chen
AU - Yuanpu Xiong
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34339
UR - https://arxiv.org/abs/2610.34339
AB - Let Ω⊂ℂ^n be a bounded hyperconvex domain admitting a negative plurisubharmonic exhaustion function u such that −u(z)≤ Cexp[−η(log(1/δ_Ω(z)))] near the boundary, where η is nondecreasing with lim_t→+∞η(t)=+∞. We present a lower bound for the Bergman distance d_B,Ω(z_0,z) by counting the iterates of a function determined by the generalized inverse of η. The bound tends to infinity as z→∂Ω for every such η; moreover, it recovers several known estimates. The proof combines a one-sided localization of Green sublevel sets with boundary estimates for the Bergman kernel and a capacity--volume inequality. We also obtain an upper bound for the Bergman distance along the axis of a family of complete Hartogs domains whose fiber radii decay faster than every positive power of the boundary distance in the base disc.
ER -