On the Gromov non-hyperbolicity of certain domains in $ {\mathbb{C}}^{n}$

Fathi Haggui; Abdelwahed Chrih

doi:10.1016/j.crma.2017.03.013

Complex analysis

On the Gromov non-hyperbolicity of certain domains in

C^{n}

Presented by: Jean-Pierre Demailly

Fathi Haggui ¹; Abdelwahed Chrih ¹

¹ Université de Monastir, Institut préparatoire aux études d'ingénieur de Monastir, 5019 Monastir, Tunisia

Comptes Rendus. Mathématique, Volume 355 (2017) no. 5, pp. 493-498.

Abstract
Résumé

In this paper, we prove that if Ω is a bounded convex domain in $C^{n}$ , $n \geq 2$ , and S is an affine complex hyperplane such that $Ω \cap S$ is not empty, then $Ω ∖ S$ is not Gromov hyperbolic with respect to the Kobayashi distance. Next, we show that if X is a bounded convex domain in $C^{n}$ , then $Ω = {(z, w) \in X \times C^{⁎}, | w | < e^{- φ (z)}}$ is not Gromov hyperbolic, where φ is a strictly plurisubaharmonic function on X continuous up to $\overline{X}$ .

Nous étudions dans cette Note l'hyperbolicité au sens de Gromov de certains domaines de $C^{n}$ . On démontre que, si Ω est un domaine convexe borné de $C^{n}$ , $n \geq 2$ , et si S est un hyperplan affine complexe tel que $Ω \cap S \neq \emptyset$ , alors $Ω ∖ S$ n'est pas hyperbolique au sens de Gromov. Si X est un domaine convexe de $C^{n}$ , alors $Ω = {(z, w) \in X \times C^{⁎}, | w | < e^{- φ (z)}}$ n'est pas Gromov hyperbolique, où φ est une fonction strictement plurisousharmonique sur X et continue sur $\overline{X}$ .

Received: 2016-08-28
Accepted: 2017-03-23
Published online: 2017-04-06

DOI: 10.1016/j.crma.2017.03.013

Author's affiliations:

Fathi Haggui ¹; Abdelwahed Chrih ¹

¹ Université de Monastir, Institut préparatoire aux études d'ingénieur de Monastir, 5019 Monastir, Tunisia

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     author = {Fathi Haggui and Abdelwahed Chrih},
     title = {On the {Gromov} non-hyperbolicity of certain domains in $ {\mathbb{C}}^{n}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {493--498},
     publisher = {Elsevier},
     volume = {355},
     number = {5},
     year = {2017},
     doi = {10.1016/j.crma.2017.03.013},
     language = {en},
}

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Fathi Haggui; Abdelwahed Chrih. On the Gromov non-hyperbolicity of certain domains in $ {\mathbb{C}}^{n}$. Comptes Rendus. Mathématique, Volume 355 (2017) no. 5, pp. 493-498. doi : 10.1016/j.crma.2017.03.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.03.013/

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