Effective formulas for invariant functions - case of elementary Reinhardt domains

Peter Pflug; Włodzimierz Zwonek

Annales Polonici Mathematici (1998)

  • Volume: 69, Issue: 2, page 175-196
  • ISSN: 0066-2216

Abstract

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We find effective formulas for the invariant functions, appearing in the theory of several complex variables, of the elementary Reinhardt domains. This gives us the first example of a large family of domains for which the functions are calculated explicitly.

How to cite

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Peter Pflug, and Włodzimierz Zwonek. "Effective formulas for invariant functions - case of elementary Reinhardt domains." Annales Polonici Mathematici 69.2 (1998): 175-196. <http://eudml.org/doc/270362>.

@article{PeterPflug1998,
abstract = {We find effective formulas for the invariant functions, appearing in the theory of several complex variables, of the elementary Reinhardt domains. This gives us the first example of a large family of domains for which the functions are calculated explicitly.},
author = {Peter Pflug, Włodzimierz Zwonek},
journal = {Annales Polonici Mathematici},
keywords = {invariant functions; invariant metrics and pseudodistances; pseudometrics; convex domains; Reinhardt domains},
language = {eng},
number = {2},
pages = {175-196},
title = {Effective formulas for invariant functions - case of elementary Reinhardt domains},
url = {http://eudml.org/doc/270362},
volume = {69},
year = {1998},
}

TY - JOUR
AU - Peter Pflug
AU - Włodzimierz Zwonek
TI - Effective formulas for invariant functions - case of elementary Reinhardt domains
JO - Annales Polonici Mathematici
PY - 1998
VL - 69
IS - 2
SP - 175
EP - 196
AB - We find effective formulas for the invariant functions, appearing in the theory of several complex variables, of the elementary Reinhardt domains. This gives us the first example of a large family of domains for which the functions are calculated explicitly.
LA - eng
KW - invariant functions; invariant metrics and pseudodistances; pseudometrics; convex domains; Reinhardt domains
UR - http://eudml.org/doc/270362
ER -

References

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  1. [A] K. Azukawa, Two intrinsic pseudo-metrics with pseudoconvex indicatrices and starlike circular domains, J. Math. Soc. Japan 38 (1986), 627-647. Zbl0607.32015
  2. [BFKKMP] B. E. Blank, D. Fan, D. Klein, S. G. Krantz, D. Ma and M.-Y. Pang, The Kobayashi metric of a complex ellipsoid in ℂ², Experiment. Math. 1 (1992), 47-55. Zbl0783.32012
  3. [C] C. Carathéodory, Über eine spezielle Metrik die in der Theorie der analytischen Funktionen auftritt, Atti Pontifica Acad. Sci. Nuovi Lincei 80 (1927), 135-141. Zbl53.0314.01
  4. [E] A. Edigarian, On extremal mappings in complex ellipsoids, Ann. Polon. Math. 62 (1995), 83-96. Zbl0851.32025
  5. [Ge] G. Gentili, Regular complex geodesics in the domain Dₙ = {(z₁,...,zₙ) ∈ ℂⁿ: |z₁|+...+|zₙ| < 1}, in: Lecture Notes in Math. 1277, Springer, 1987, 35-45. 
  6. [Gi] B. Gilligan, On the Kobayashi pseudometric reduction of homogeneous spaces, Canad. Math. Bull. 31 (1988), 45-51. Zbl0617.32037
  7. [HW] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford Sci. Publ., 1978. 
  8. [HD] V. Z. Hristov and T. Davidov, Examples of typical Carathéodory and Kobayashi pseudodistances, C. R. Acad. Bulgare Sci. 39 (1986), 23-25. Zbl0595.32035
  9. [JP1] M. Jarnicki and P. Pflug, Some remarks on the product property, in: Proc. Sympos. Pure Math. 52 (Part 2), Amer. Math. Soc., 1991, 263-272. Zbl0747.32018
  10. [JP2] M. Jarnicki and P. Pflug, Invariant Distances and Metrics in Complex Analysis, Walter de Gruyter, 1993. Zbl0789.32001
  11. [JPZ] M. Jarnicki, P. Pflug and R. Zeinstra, Geodesics for convex complex ellipsoids, Ann. Scuola Norm. Sup. Pisa 20 (1993), 535-543. Zbl0812.32010
  12. [Kl1] M. Klimek, Extremal plurisubharmonic functions and invariant pseudodistances, Bull. Soc. Math. France 113 (1985), 231-240. Zbl0584.32037
  13. [Kl2] M. Klimek, Pluripotential Theory, Oxford Univ. Press, 1991. 
  14. [Ko1] S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings, Pure and Appl. Math. 2, M. Dekker, 1970. 
  15. [Ko2] S. Kobayashi, Intrinsic distances, measures and geometric function theory, Bull. Amer. Math. Soc. 82 (1976), 357-416. Zbl0346.32031
  16. [L1] L. Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France 109 (1981), 327-479. 
  17. [L2] L. Lempert, Intrinsic distances and holomorphic retracts, Complex Analysis and Applications (Varna, 1981), Publ. House Bulgar. Acad. Sci., Sophia, 1984, 341-364. 
  18. [N] S. Nivoche, Pluricomplex Green function, capacitative notions and approximation problems in ℂⁿ, Indiana Univ. Math. J. 44 (1995), 489-510. Zbl0846.31008
  19. [Pa] M.-Y. Pang, Smoothness of the Kobayashi metric of non-convex domains, Internat. J. Math. 4 (1993), 953-987. Zbl0795.32008
  20. [PZ] P. Pflug and W. Zwonek, The Kobayashi metric for non-convex complex ellipsoids, Complex Variables 29 (1996), 59-71. Zbl0843.32015
  21. [R] H.-J. Reiffen, Die Carathéodory Distanz und ihre zugehörige Differentialmetrik, Math. Ann. 161 (1965), 315-324. Zbl0141.08803

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