Kobayashi-Royden vs. Hahn pseudometric in ℂ²

Witold Jarnicki

Annales Polonici Mathematici (2000)

  • Volume: 75, Issue: 3, page 289-294
  • ISSN: 0066-2216

Abstract

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For a domain D ⊂ ℂ the Kobayashi-Royden ϰ and Hahn h pseudometrics are equal iff D is simply connected. Overholt showed that for D ⊂ ℂ n , n ≥ 3, we have h D ≡ ϰ D . Let D₁, D₂ ⊂ ℂ. The aim of this paper is to show that h D ₁ × D ₂ iff at least one of D₁, D₂ is simply connected or biholomorphic to ℂ 0. In particular, there are domains D ⊂ ℂ² for which h D ≢ ϰ D .

How to cite

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Witold Jarnicki. "Kobayashi-Royden vs. Hahn pseudometric in ℂ²." Annales Polonici Mathematici 75.3 (2000): 289-294. <http://eudml.org/doc/208402>.

@article{WitoldJarnicki2000,
abstract = {For a domain D ⊂ ℂ the Kobayashi-Royden ϰ and Hahn h pseudometrics are equal iff D is simply connected. Overholt showed that for $D ⊂ ℂ^n$, n ≥ 3, we have $h_D ≡ ϰ_D$. Let D₁, D₂ ⊂ ℂ. The aim of this paper is to show that $h_\{D₁ × D₂\}$ iff at least one of D₁, D₂ is simply connected or biholomorphic to ℂ 0. In particular, there are domains D ⊂ ℂ² for which $h_D ≢ ϰ_D$.},
author = {Witold Jarnicki},
journal = {Annales Polonici Mathematici},
keywords = {Hahn pseudometric; Kobayashi pseudometric},
language = {eng},
number = {3},
pages = {289-294},
title = {Kobayashi-Royden vs. Hahn pseudometric in ℂ²},
url = {http://eudml.org/doc/208402},
volume = {75},
year = {2000},
}

TY - JOUR
AU - Witold Jarnicki
TI - Kobayashi-Royden vs. Hahn pseudometric in ℂ²
JO - Annales Polonici Mathematici
PY - 2000
VL - 75
IS - 3
SP - 289
EP - 294
AB - For a domain D ⊂ ℂ the Kobayashi-Royden ϰ and Hahn h pseudometrics are equal iff D is simply connected. Overholt showed that for $D ⊂ ℂ^n$, n ≥ 3, we have $h_D ≡ ϰ_D$. Let D₁, D₂ ⊂ ℂ. The aim of this paper is to show that $h_{D₁ × D₂}$ iff at least one of D₁, D₂ is simply connected or biholomorphic to ℂ 0. In particular, there are domains D ⊂ ℂ² for which $h_D ≢ ϰ_D$.
LA - eng
KW - Hahn pseudometric; Kobayashi pseudometric
UR - http://eudml.org/doc/208402
ER -

References

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  1. [Cho] K. S. Choi, Injective hyperbolicity of product domain, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 5 (1998), 73-78. 
  2. [Hah] K. T. Hahn, Some remark on a new pseudo-differential metric, Ann. Polon. Math. 39 (1981), 71-81. 
  3. [Jar-Pfl] M. Jarnicki and P. Pflug, Invariant Distances and Metrics in Complex Analysis, de Gruyter Exp. Math. 9, de Gruyter, Berlin, 1993. 
  4. [Ove] M. Overholt, Injective hyperbolicity of domains, Ann. Polon. Math. 62 (1995), 79-82. Zbl0847.32027
  5. [Roy] H. L. Royden, Remarks on the Kobayashi metric, in: Several Complex Variables, II, Lecture Notes in Math. 189, Springer, 1971, 125-137. 
  6. [Two-Win] P. Tworzewski and T. Winiarski, Continuity of intersection of analytic sets, Ann. Polon. Math. 42 (1983), 387-393. Zbl0576.32013
  7. [Ves] E. Vesentini, Injective hyperbolicity, Ricerche Mat. 36 (1987), 99-109. 
  8. [Vig] J.-P. Vigué, Une remarque sur l'hyperbolicité injective, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 83 (1989), 57-61. Zbl0741.32019

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