Non-comparable invariant metrics and failure of Gromov hyperbolicity on a real-analytic pseudoconvex domain
Abstract
We study the explicit bounded pseudoconvex domain Ω={(z1,z2,w)∈ℂ3: |w|2+|z1z2(z1−z2)|2+|z1|8+|z2|8<1}. Its boundary is real analytic and has finite D'Angelo type everywhere. We prove two assertions about the invariant metrics of this domain. First, (Ω,kΩ) is not Gromov hyperbolic; in fact arbitrarily fat Kobayashi geodesic triangles occur in every neighborhood of the type-$8$ point (0,0,1). Second, the Carathéodory--Reiffen and Kobayashi--Royden metrics are not uniformly comparable. Along an explicit family of tangential directions the Carathéodory metric tends to zero while the Kobayashi metric has a positive limit. For the corresponding distances there are points xj,yj→(0,0,1) such that cΩ(xj,yj)→0 while kΩ(xj,yj)→∞. This gives, among bounded real-analytic pseudoconvex domains, an affirmative answer to Problem~7.3.2 of the Several Complex Variables Problem List.
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BibTeX
@misc{edigarian2026noncomparable,
title = {{Non-comparable invariant metrics and failure of Gromov hyperbolicity on a real-analytic pseudoconvex domain}},
author = {Armen Edigarian},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36124},
eprint = {2610.36124},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36124},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Non-comparable invariant metrics and failure of Gromov hyperbolicity on a real-analytic pseudoconvex domain
AU - Armen Edigarian
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.36124
UR - https://arxiv.org/abs/2610.36124
AB - We study the explicit bounded pseudoconvex domain Ω={(z_1,z_2,w)∈ℂ^3: |w|^2+|z_1z_2(z_1−z_2)|^2+|z_1|^8+|z_2|^8<1}. Its boundary is real analytic and has finite D'Angelo type everywhere. We prove two assertions about the invariant metrics of this domain. First, (Ω,k_Ω) is not Gromov hyperbolic; in fact arbitrarily fat Kobayashi geodesic triangles occur in every neighborhood of the type-$8$ point (0,0,1). Second, the Carathéodory--Reiffen and Kobayashi--Royden metrics are not uniformly comparable. Along an explicit family of tangential directions the Carathéodory metric tends to zero while the Kobayashi metric has a positive limit. For the corresponding distances there are points x_j,y_j→(0,0,1) such that c_Ω(x_j,y_j)→0 while k_Ω(x_j,y_j)→∞. This gives, among bounded real-analytic pseudoconvex domains, an affirmative answer to Problem~7.3.2 of the Several Complex Variables Problem List.
ER -