PreprintCC BY 4.0arXiv
Kobayashi completeness of pseudoconvex domains: finite type and a smooth counterexample
Abstract
We prove that every bounded pseudoconvex domain in ℂN with C∞ boundary of finite D'Angelo one-type is Kobayashi complete. As a separate consequence of our criterion of completeness and an estimate of Diederich--Fornæss, we prove that every bounded pseudoconvex domain with real-analytic boundary is Kobayashi complete. We then show that smoothness alone does not imply completeness: there exists a bounded pseudoconvex domain in ℂ3 with C∞ boundary and a global smooth plurisubharmonic defining function whose Kobayashi and Carathéodory distances are incomplete.
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BibTeX
@misc{edigarian2026kobayashi,
title = {{Kobayashi completeness of pseudoconvex domains: finite type and a smooth counterexample}},
author = {Armen Edigarian},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28833},
eprint = {2610.28833},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28833},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Kobayashi completeness of pseudoconvex domains: finite type and a smooth counterexample
AU - Armen Edigarian
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.28833
UR - https://arxiv.org/abs/2610.28833
AB - We prove that every bounded pseudoconvex domain in ℂ^N with C^∞ boundary of finite D'Angelo one-type is Kobayashi complete. As a separate consequence of our criterion of completeness and an estimate of Diederich--Fornæss, we prove that every bounded pseudoconvex domain with real-analytic boundary is Kobayashi complete. We then show that smoothness alone does not imply completeness: there exists a bounded pseudoconvex domain in ℂ^3 with C^∞ boundary and a global smooth plurisubharmonic defining function whose Kobayashi and Carathéodory distances are incomplete.
ER -