Every Gromov elliptic domain is meromorphically pseudoconvex
Abstract
We prove that every Gromov elliptic domain Ω in a complex manifold has the following property, which we call meromorphic pseudoconvexity: each boundary point has a neighborhood V such that Ω∩ V becomes Stein after removing a set which is, locally in Ω∩ V, the union of the polar sets of finitely many meromorphic functions on Ω∩ V. Since meromorphic functions on Hartogs figures extend to their hulls, it follows that, for a domain Ω=intΩ̅ in a Stein manifold, Gromov ellipticity is equivalent to being Oka and Stein, and also to being Oka and admitting a complete Kähler metric. This complements a theorem of Matsumura and the author that every weakly pseudoconvex Oka manifold admitting a positive line bundle is elliptic. As applications, we show that the Oka manifolds ℂn∖𝔹n̅ and ℂn∖ℝn, n≥2, are not elliptic, and that no proper domain with Lipschitz boundary in ℙn, n≥2, is elliptic.
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BibTeX
@misc{kusakabe2026every,
title = {{Every Gromov elliptic domain is meromorphically pseudoconvex}},
author = {Yuta Kusakabe},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20574},
eprint = {2610.20574},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20574},
}
RIS
TY - UNPB
TI - Every Gromov elliptic domain is meromorphically pseudoconvex
AU - Yuta Kusakabe
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20574
UR - https://arxiv.org/abs/2610.20574
AB - We prove that every Gromov elliptic domain Ω in a complex manifold has the following property, which we call meromorphic pseudoconvexity: each boundary point has a neighborhood V such that Ω∩ V becomes Stein after removing a set which is, locally in Ω∩ V, the union of the polar sets of finitely many meromorphic functions on Ω∩ V. Since meromorphic functions on Hartogs figures extend to their hulls, it follows that, for a domain Ω=intΩ̅ in a Stein manifold, Gromov ellipticity is equivalent to being Oka and Stein, and also to being Oka and admitting a complete Kähler metric. This complements a theorem of Matsumura and the author that every weakly pseudoconvex Oka manifold admitting a positive line bundle is elliptic. As applications, we show that the Oka manifolds ℂ^n∖𝔹^n̅ and ℂ^n∖ℝ^n, n≥2, are not elliptic, and that no proper domain with Lipschitz boundary in ℙ^n, n≥2, is elliptic.
ER -