Bergman Kernel in Complex Analysis

  • Łukasz Kosiński
  • Włodzimierz ZwonekEmail author
Reference work entry


In this survey a brief review of results on the Bergman kernel and Bergman distance concentrating on those fields of complex analysis which remain in the focus of the research interest of the authors is presented. The topics discussed contain general discussion of \(\mathcal{L}_{\mathrm{h}}^{2}\) spaces, behavior of the Bergman distance, regularity of extension of proper holomorphic mappings, and recent development in the theory of Bergman distance stemming from the pluripotential theory and very short discussion of the Lu Qi Keng problem.


Pseudoconvex Domain Bergman Kernel Plurisubharmonic Function Biholomorphic Mapping Bergman Projection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Bell, S.: Biholomorphic mappings and the \(\bar{\partial }\)-problem. Ann. Math. (2) 114(1), 103–113 (1981)Google Scholar
  2. 2.
    Bell, S.: Proper holomorphic mappings between circular domains. Comment. Math. Helv. 57(4), 532–538 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bell, S.: The Bergman kernel function and proper holomorphic mappings. Trans. Am. Math. Soc. 270(2), 685–691 (1982)zbMATHGoogle Scholar
  4. 4.
    Bell, S., Catlin, D.: Boundary regularity of proper holomorphic mappings. Duke Math. J. 49(2), 385–396 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Bell, S., Ligocka, E.: A simplification and extension of Fefferman’s theorem on biholomorphic mappings. Invent. Math. 57(3), 283–289 (1980)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Bergman, S.: Zur Theorie von pseudokonformen Abbildungen. Mat. Sb. 43, 79–96 (1936)Google Scholar
  7. 7.
    Błocki, Z.: The Bergman metric and the pluricomplex Green function. Trans. Am. Math. Soc. 357(7), 2613–2625 (2005)zbMATHCrossRefGoogle Scholar
  8. 8.
    Błocki, Z.: The Bergman kernel and pluripotential theory. In: Potential theory in Matsue, 1–9. Advanced Studies in Pure Mathematics, vol. 44, Mathematics Society of Japan, Tokyo (2006)Google Scholar
  9. 9.
    Błocki, Z.: Suita conjecture and the Ohsawa-Takegoshi extension theorem. Invent. Math. 193(1), 149–158 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Błocki, Z.: Bergman kernel and pluripotential theory. In: Proceedings of the Conference in Honor of Duong Phong. Contemporary Mathematics. American Mathematical Society (2014, to appear)Google Scholar
  11. 11.
    Błocki, Z.: Cauchy-Riemann meet Monge-Ampére. Bull. Math. Sci. 4, 433–480 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Błocki, Z., Pflug, P.: Hyperconvexity and Bergman completeness. Nagoya Math. J. 151, 221–225 (1998)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Błocki, Z., Zwonek, W.: Estimates for the Bergman kernel and the multidimensional Suita conjecture (2014, preprint)Google Scholar
  14. 14.
    Boas, H.P.: Counterexample to the Lu Qi-Keng conjecture. Proc. Am. Math. Soc. 97, 374–375 (1986)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Boas, H.P.: Lu Qi-Keng’s problem. J. Korean Math. Soc. 37(2), 253–267 (2000). Several complex variables (Seoul, 1998)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Boas, H.P., Fu, S., Straube, E.J.: The Bergman kernel function: explicit formulas and zeroes. Proc. Am. Math. Soc. 127(3), 805–811 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Chakrabarti, D., Verma, K.: Condition R and proper holomorphic maps between equidimensional product domains. Adv. Math. 248, 820–842 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Chen, B.Y.: Completeness of the Bergman metric on non-smooth pseudoconvex domains. Ann. Pol. Math. 71(3), 241–251 (1999)zbMATHGoogle Scholar
  19. 19.
    Chen, B.Y.: A note on Bergman completeness. Int. J. Math. 12(4), 383–392 (2001)zbMATHCrossRefGoogle Scholar
  20. 20.
    Christ, M.: Global \({\mathcal{C}}^{\infty }\)-irregularity of the \(\bar{\partial }\)-Neumann problem for worm domains. J. Am. Math. Soc. 9(4), 1171–1185 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Conway, J.B.: Functions of One Complex Variable. II. Springer, New York (1995)zbMATHCrossRefGoogle Scholar
  22. 22.
    Diederich, K., Fornaess, J.E.: Boundary regularity of proper holomorphic mappings. Invent. Math. 67(3), 363–384 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Diederich, K., Ohsawa, T.: An estimate for the Bergman distance on pseudoconvex domains. Ann. Math. (2) 141(1), 181–190 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Fefferman, C.: The Bergman kernel and biholomorphic mappings of pseudoconvex domains. Invent. Math. 26, 1–65 (1974)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Herbig, A.-K., McNeal, J.D., Straube, E.J.: Duality of holomorphic function spaces and smoothing properties of the Bergman projection. Trans. Am. Math. Soc. 366(2), 647–665 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Herbort, G.: The Bergman metric on hyperconvex domains. Math. Z. 232, 183–196 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Jarnicki, M., Pflug, P.: Extension of Holomorphic Functions. Walter de Gruyter, Berlin/New York (2000)zbMATHCrossRefGoogle Scholar
  28. 28.
    Jarnicki, M., Pflug, P.: Invariant Distances and Metrics in Complex Analysis, 2nd extended edn. Walter de Gruyter, Berlin (2013)CrossRefGoogle Scholar
  29. 29.
    Josefson, B.: On the equivalence between locally polar and globally polar sets for plurisubharmonic functions on \(\mathbb{C}^{n}\). Ark. Mat. 16(1), 109–115 (1978)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Jucha, P.: Bergman completeness of Zalcman type domains. Studia Math. 163(1), 71–83 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Jucha, P.: A remark on the dimension of the Bergman space of some Hartogs domains. J. Geom. Anal. 22(1), 23–37 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Klimek, M.: Extremal plurisubharmonic functions and invariant pseudodistances. Bull. Soc. Math. Fr. 113(2), 231–240 (1985)zbMATHMathSciNetGoogle Scholar
  33. 33.
    Klimek, M.: Pluripotential Theory, Clarendon Press, Oxford/New York (1991)zbMATHGoogle Scholar
  34. 34.
    Kobayashi, S.: Geometry of bounded domains. Trans. Am. Math. Soc. 92, 267–290 (1959)zbMATHCrossRefGoogle Scholar
  35. 35.
    Kosiński, Ł.: Geometry of quasi-circular domains and applications to tetrablock. Proc. Am. Math. Soc. 139(2), 559–569 (2011)zbMATHCrossRefGoogle Scholar
  36. 36.
    Lempert, L.: La métrique de Kobayashi et la représentation des domaines sur la boule. Bull. Soc. Math. Fr. 109, 427–474 (1981)zbMATHMathSciNetGoogle Scholar
  37. 37.
    Qi-Keng, L.: On Kaehler manifolds with constant curvature. Chin. Math. Acta 8, 283–298 (1966)Google Scholar
  38. 38.
    Ohsawa, T.: Addendum to “On the Bergman kernel of hyperconvex domains”. Nagoya Math. J. 137, 145–148 (1995)zbMATHMathSciNetGoogle Scholar
  39. 39.
    Ohsawa, T., Takegoshi, K.: On the extension of \(\mathcal{L}^{2}\) holomorphic functions. Math. Z. 195(2), 197–204 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Pflug, P., Zwonek, W.: L h 2-domains of holomorphy and the Bergman kernel. Studia Math. 151(2), 99–108 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    Pflug, P., Zwonek, W.: Logarithmic capacity and Bergman functions. Arch. Math. 80, 536–552 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    Ransford, T.: Potential Theory in the Complex Plane. London Mathematical Society Student Texts, vol. 28. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  43. 43.
    Rosenthal, P.: On the zeros of the Bergman function in double-connected domains. Proc. Am. Math. Soc. 21, 33–35 (1969)zbMATHCrossRefGoogle Scholar
  44. 44.
    Skwarczyński, M.: Biholomorphic invariants related to the Bergman function. Diss. Math. 173, 1–59 (1980)Google Scholar
  45. 45.
    Suita, N.: Capacities and kernels on Riemann surfaces. Arch. Ration. Mech. Anal. 46, 212–217 (1972)zbMATHMathSciNetCrossRefGoogle Scholar
  46. 46.
    Suita, N., Yamada, A.: On the Lu Qi-Keng conjecture. Proc. Am. Math. Soc. 59, 222–224 (1976)zbMATHMathSciNetCrossRefGoogle Scholar
  47. 47.
    Szafraniec, F.H.: The reproducing kernel property and its space: the basics. In: Alpay, D. (ed.) Operator Theory, chapter 1, pp. 3-30, Springer, Basel (2015). doi: 10.1007/978-3-0348-0692-3_65CrossRefGoogle Scholar
  48. 48.
    Wiegerinck, J.: Domains with finite dimensional Bergman space. Math. Z. 187, 559–562 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    Zwonek, W.: Regularity properties of the Azukawa metric. J. Math. Soc. Jpn. 52(4), 899–914 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  50. 50.
    Zwonek, W.: An example concerning Bergman completeness. Nagoya Math. J. 164, 89–101 (2001)zbMATHMathSciNetGoogle Scholar
  51. 51.
    Zwonek, W.: Wiener’s type criterion for Bergman exhaustiveness. Bull. Pol. Acad. Sci. Math. 50(3), 297–311 (2002)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer Science, Department of Mathematics, Jagiellonian UniversityKrakówPoland

Personalised recommendations