Invariant pseudodistances and pseudometrics - completeness and product property

Marek Jarnicki; Peter Pflug

Annales Polonici Mathematici (1991)

  • Volume: 55, Issue: 1, page 169-189
  • ISSN: 0066-2216

Abstract

top
A survey of properties of invariant pseudodistances and pseudometrics is given with special stress put on completeness and product property.

How to cite

top

Marek Jarnicki, and Peter Pflug. "Invariant pseudodistances and pseudometrics - completeness and product property." Annales Polonici Mathematici 55.1 (1991): 169-189. <http://eudml.org/doc/262259>.

@article{MarekJarnicki1991,
abstract = {A survey of properties of invariant pseudodistances and pseudometrics is given with special stress put on completeness and product property.},
author = {Marek Jarnicki, Peter Pflug},
journal = {Annales Polonici Mathematici},
keywords = {survey; intrinsic pseudodistances; pseudometrics; Kobayashi; Carathéodory; product domains},
language = {eng},
number = {1},
pages = {169-189},
title = {Invariant pseudodistances and pseudometrics - completeness and product property},
url = {http://eudml.org/doc/262259},
volume = {55},
year = {1991},
}

TY - JOUR
AU - Marek Jarnicki
AU - Peter Pflug
TI - Invariant pseudodistances and pseudometrics - completeness and product property
JO - Annales Polonici Mathematici
PY - 1991
VL - 55
IS - 1
SP - 169
EP - 189
AB - A survey of properties of invariant pseudodistances and pseudometrics is given with special stress put on completeness and product property.
LA - eng
KW - survey; intrinsic pseudodistances; pseudometrics; Kobayashi; Carathéodory; product domains
UR - http://eudml.org/doc/262259
ER -

References

top
  1. [1] K. Azukawa, Two intrinsic pseudo-metrics with pseudoconvex indicatrices and starlike circular domains, J. Math. Soc. Japan 38 (1986), 627-647. Zbl0607.32015
  2. [2] K. Azukawa, The invariant pseudometric related to negative pluri-subharmonic functions, Kodai Math. J. 10 (1987), 83-92. Zbl0618.32020
  3. [3] K. Azukawa, A note on Carathéodory and Kobayashi pseudodistances, preprint, 1990. 
  4. [4] T. J. Barth, The Kobayashi distance induces the standard topology, Proc. Amer. Math. Soc. 35 (1972), 439-441. Zbl0259.32007
  5. [5] T. J. Barth, Some counterexamples concerning intrinsic distances, ibid. 66 (1977), 49-53. Zbl0331.32019
  6. [6] T. J. Barth, The Kobayashi indicatrix at the center of a circular domain, ibid. 88 (1983), 527-530. Zbl0494.32008
  7. [7] E. Bedford and J. E. Fornæss, A construction of peak functions on weakly pseudoconvex domains, Ann. of Math. 107 (1978), 555-568. Zbl0392.32004
  8. [8] J. Burbea, The Carathéodory metric in plane domains, Kodai Math. Sem. Rep. 29 (1977), 157-166. Zbl0419.30010
  9. [9] J. Burbea, Inequalities between intrinsic metrics, Proc. Amer. Math. Soc. 67 (1977), 50-54. Zbl0346.32030
  10. [10] H. Busemann, Recent Synthetic Differential Geometry, Springer, Berlin 1970. Zbl0194.53701
  11. [11] D. Catlin, Boundary behavior of holomorphic functions on pseudoconvex domains, J. Differential Geom. 15 (1980), 605-625. Zbl0484.32005
  12. [12] D. Catlin, Estimates of invariant metrics on pseudoconvex domains of dimension two, Math. Z. 200 (1989), 429-466. Zbl0661.32030
  13. [13] R. Courant und D. Hilbert, Methoden der mathematischen Physik I, Springer, Berlin 1968. Zbl0156.23201
  14. [14] J.-P. Demailly, Mesures de Monge-Ampère et mesures pluriharmoniques, Math. Z. 194 (1987), 519-564. Zbl0595.32006
  15. [15] S. Dineen, The Schwarz Lemma, Clarendon Press, Oxford 1989. Zbl0708.46046
  16. [16] A. Eastwood, A propos des variétés hyperboliques complètes, C. R. Acad. Sci. Paris 280 (1975), 1071-1075. Zbl0301.32021
  17. [17] A. A. Fadlalla, Quelques propriétés de la distance de Carathéodory, in: 7th. Arab. Sc. Congr., Cairo II (1973), 1-16. 
  18. [18] J. E. Fornæss and N. Sibony, Construction of p.s.h. functions on weakly pseudoconvex domains, Duke Math. J. 58 (1989), 633-655. Zbl0679.32017
  19. [19] T. Franzoni and E. Vesentini, Holomorphic Maps and Invariant Distances, North-Holland Math. Stud. 40, North-Holland, Amsterdam 1980. Zbl0447.46040
  20. [20] K. T. Hahn, On the completeness of the Bergman metric and its subordinate metrics, II, Pacific J. Math. 68 (1977), 437-446. Zbl0356.32017
  21. [21] M. Hakim et N. Sibony, Spectre de A(Ω̅) pour des domaines bornés faiblement pseudoconvexes réguliers, J. Funct. Anal. 37 (1980), 127-135. Zbl0441.46044
  22. [22] L. A. Harris, Schwarz-Pick systems of pseudometrics for domains in normed linear spaces, in: Advances in Holomorphy, J. A. Barroso (ed.), North-Holland Math. Stud. 34, North-Holland, Amsterdam 1979, 345-406. 
  23. [23] M. Jarnicki and P. Pflug, Effective formulas for the Carathéodory distance, Manusripta Math. 62 (1988), 1-20. Zbl0656.32016
  24. [24] M. Jarnicki and P. Pflug, The Carathéodory pseudodistance has the product property, Math. Ann. 285 (1989), 161-164. Zbl0662.32023
  25. [25] M. Jarnicki and P. Pflug, Bergman completeness of complete circular domains, Ann. Polon. Math. 50 (1989), 219-222. Zbl0701.32002
  26. [26] M. Jarnicki and P. Pflug, A counterexample for Kobayashi completeness of balanced domains, Proc. Amer. Math. Soc., to appear. 
  27. [27] M. Jarnicki and P. Pflug, The simplest example for the non-innerness of the Carathéodory distance, Results in Math. 18 (1990), 57-59. Zbl0709.32015
  28. [28] M. Jarnicki and P. Pflug, Some remarks on the product property for invariant pseudometrics, in: Proc. Sympos. Pure Math., to appear. Zbl0747.32018
  29. [29] M. Klimek, Extremal plurisubharmonic functions and invariant pseudodistances, Bull. Soc. Math. France 113 (1985), 123-142. 
  30. [30] M. Klimek, Infinitesimal pseudo-metrics and the Schwarz Lemma, Proc. Amer. Math. Soc. 105 (1989), 134-140. 
  31. [31] S. Kobayashi, Intrinsic distances, measures and geometric function theory, Bull. Amer. Math. Soc. 82 (3) (1976), 357-416. Zbl0346.32031
  32. [32] A. Kodama, On boundedness of circular domains, Proc. Japan. Acad. 58 (1982), 227-230. Zbl0515.32011
  33. [33] S. G. Krantz, Function Theory of Several Complex Variables, Wiley-Interscience, New York 1982. Zbl0471.32008
  34. [34] S. Lang, Introduction to Complex Hyberbolic Spaces, Springer, Berlin 1987. 
  35. [35] L. Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France 109 (1981), 427-474. 
  36. [36] L. Lempert, Holomorphic retracts and intrinsic metrics in convex domains, Anal. Math. 8 (1982), 257-261. Zbl0509.32015
  37. [37] L. Lempert, Intrinsic distances and holomorphic retracts, in: Complex Analysis and Applications '81, Sofia 1984, 341-364. 
  38. [38] T. Mazur, P. Pflug and M. Skwarczyński, Invariant distances related to the Bergman function, Proc. Amer. Math. Soc. 94 (1985), 72-76. Zbl0534.32010
  39. [39] T. Ohsawa, A remark on the completeness of the Bergman metric, Proc. Japan Acad. 57 (1981), 238-240. Zbl0508.32008
  40. [40] P. Pflug, About the Carathéodory completeness of all Reinhardt domains, in: Functional Analysis, Holomorphy and Approximation Theory II, North-Holland, 1984, 331-337. 
  41. [41] E. A. Poletskiĭ and B. V. Shabat, Invariant metrics, in: Encyclopaedia of Mathematical Sciences, Vol. 9, Springer, 1989, 63-111. 
  42. [42] H. J. Reiffen, Die Carathéodorysche Distanz und ihr zugehörige Differentialmetrik, Math. Ann. 161 (1965), 315-324. Zbl0141.08803
  43. [43] W. Rinow, Die innere Geometrie der metrischen Räume, Grundlehren Math. Wiss. 105, Springer, Berlin 1961. 
  44. [44] R. M. Robinson, Analytic functions on circular rings, Duke Math. J. 10 (1943), 341-354. Zbl0060.21804
  45. [45] H. L. Royden, Remarks on the Kobayashi metric, in: Lecture Notes in Math. 185 Springer, 1971, 125-137. 
  46. [46] N. Sibony, Prolongement des fonctions holomorphes bornées et métrique de Carathéodory, Invent. Math. 29 (1975), 205-230. Zbl0333.32011
  47. [47] N. Sibony, A class of hyperbolic manifolds, in: Ann. of Math. Stud. 100, Princeton Univ. Press, Princeton, N.J., 1981, 357-372. 
  48. [48] J. Siciak, Balanced domains of holomorphy of type H , Mat. Vesnik 37 (1985), 134-144. Zbl0575.32009
  49. [49] R. R. Simha, The Carathéodory metric on the annulus, Proc. Amer. Math. Soc. 50 (1975), 162-166. Zbl0281.30010
  50. [50] M. Suzuki, The generalized Schwarz Lemma for the Bergman metric, Pacific J. Math. 117 (1985), 429-442. Zbl0573.32025
  51. [51] S. Venturini, Comparison between the Kobayashi and Carathéodory distances on strongly pseudoconvex bounded domains in n , Proc. Amer. Math. Soc. 107 (1989), 725-730. Zbl0692.32013
  52. [52] E. Vesentini, Complex geodesics and holomorphic maps, Sympos. Math. 26 (1982), 211-230. Zbl0506.32008
  53. [53] J.-P. Vigué, La distance de Carathéodory n'est pas intérieure, Resultate Math. 6 (1983), 100-104. Zbl0552.32022
  54. [54] J.-P. Vigué, The Carathéodory distance does not define the topology, Proc. Amer. Math. Soc. 91 (1984), 223-224. Zbl0555.32016
  55. [55] M. Jarnicki, P. Pflug and J.-P. Vigué, The Carathéodory distance does not define the topology - the case of domains, C. R. Acad. Sci. Paris 312 (1991), 77-79. Zbl0721.32010
  56. [56] M. Jarnicki and P. Pflug, The inner Carathéodory distance for the annulus, Math. Ann. 289 (1991), 335-339. Zbl0719.30036

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.