A smoothly bounded pseudoconvex domain with incomplete Kobayashi distance

Armen Edigarian

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arXiv · 2610.28833

A smoothly bounded pseudoconvex domain with incomplete Kobayashi distance

Armen Edigarian

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Abstract

We construct a bounded pseudoconvex domain in $\C^3$ with $C^\infty$ boundary which is not Kobayashi complete. The domain has a smooth plurisubharmonic defining function, and its Carathéodory distance is also incomplete. We then prove that every bounded pseudoconvex domain with $C^\infty$ boundary of finite D'Angelo type is Kobayashi complete. Consequently, every bounded pseudoconvex domain with real-analytic boundary is Kobayashi complete. We also obtain quantitative lower estimates for the Kobayashi--Royden metric and the Kobayashi distance in the finite-type case.

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Armen Edigarian. 2026-10-20. A smoothly bounded pseudoconvex domain with incomplete Kobayashi distance. https://arxiv.org/abs/2610.28833

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