On some biholomorphic invariants in the analysis on manifolds

  • Jerzy Kalina
  • Julian Ławrynowicz
  • Ewa Ligocka
  • Maciej Skwarczyński
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 798)


The paper gives an exposition of recent results connected with applications of the Bergman function and complex capacities (intrinsic semi-norms), in particular in the case of a strictly pseudoconvex domain, smooth extensions of biholomorphic mappings (with the use of ∂ Neumann problem), nonlinear Dirichlet problems, and foliations related to the generalized complex Monge-Ampère equations. The paper includes recent, unpublished results of the first named author which extend some achievements of E. Bedford and M. Kalka to the case of foliations related to the generalized complex Monge-Ampère equations.


Dirichlet Problem Complex Manifold Neumann Problem Pseudoconvex Domain Bergman Kernel 
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  1. [1]
    ALEKSANDROV, A. D.: The Dirichlet problem for the equation Det||zi, j|| = ψ(z1, ..., zn, x1, ..., xn), I [in Russian], Vestnik Leningrad. Univ. 13 (1958), no. 1, 5–24.Google Scholar
  2. [2]
    ANDREOTTI, A. and J. ŁAWRYNOWICZ: On the generalized complex Monge-Ampère equation on complex manifolds and related questions, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), 943–948.zbMATHGoogle Scholar
  3. [3]
    — and —: The generalized complex Monge-Ampère equation and a variational capacity problem, ibid. 25 (1977), 949–955.zbMATHGoogle Scholar
  4. [4]
    BARKER, W.: Kernel functions on domains with hyperelliptic double, Trans. Amer. Math. Soc. 231 (1977), 339–347.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    BEDFORD, E. and J. E. FORNAESS: Counterexamples to regularity for the complex Monge-Ampère equation, Invent. Math. 50 (1979), 129–134.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    — and M. KALKA: Foliations and complex Monge-Ampère equations, Comm. Pure Appl. Math. 30 (1977), 543–571.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    — and B. A. TAYLOR: The Dirichlet problem for a complex Monge-Ampère equation, Inventiones Math. 37 (1976), 1–44.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    — and —: Variational properties of the complex Monge-Ampère equation. I. Dirichlet principle, Duke Math. J. 45 (1978), 375–403.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    — and —: Variational properties of the complex Monge-Ampère equation. II. Intrinsic norms, Amer. J. Math., to appear.Google Scholar
  10. [10]
    BELL, S.: Non-vanishing of the Bergman kernel function at boundary points of certain domains in ¢n, Ms.Google Scholar
  11. [11]
    BERGMAN, S.: The kernel function and conformal mapping, 2. ed. (Math. Surveys 5), Amer. Math. Soc., Providence, R.I., 1970.Google Scholar
  12. [12]
    BOUTET DE MONVEL, L. et S. SJÖSTRAND: Sur la singularité des noyaux de Bergman et de Szegö, Asterisque 34/35 (1976), 123–164.zbMATHGoogle Scholar
  13. [13]
    BREMERMANN, H.: Holomorphic continuation of the kernel function and the Bergman metric in several complex variables, Lectures on functions of a complex variable, Univ. of Michigan Press, Ann Arbor, Mich., 1955, pp. 349–383.Google Scholar
  14. [14]
    CARAMAN, P.: p-capacity in infinite dimensional spaces, this volume, pp. 69–109.Google Scholar
  15. [15]
    CHENG, S.-Y. and S.-T. YAU: On the regularity of the Monge-Ampère equation det(∂2u/∂xi∂xj) = F(x, u), Comm. Pure Appl. Math. 30 (1977), 41–68.MathSciNetCrossRefGoogle Scholar
  16. [16]
    CHERN, S. S., H. I. LEVINE and L. NIRENBERG: Intrinsic norms on a complex manifold, Global analysis, Papers in honor of K. Kodaira, ed. by D. C. Spencer and S. Iynaga, Univ. of Tokyo Press and Princeton Univ. Press, Tokyo 1969, pp. 119–139; reprinted in: S.-s. CHERN, Selected papers, Springer-Verlag, New York-Heidelberg-Berlin 1978, pp. 371–391.Google Scholar
  17. [17]
    CHRISTOFFERS, H.: Princeton University Thesis, Princeton, N. J., 1976.Google Scholar
  18. [18]
    DEBIARD, A. et B. GAVEAU: Méthodes de contrôle optimal en analyse complexe. IV. Applications aux algèbres de fonctions analytiques, this volume, pp. 111–142.Google Scholar
  19. [19]
    DIEDERICH, K.: Über die 1. und 2. Ableitungen der Bergmanschen Kernfunktion und ihr Randverhalten, Math. Ann. 203 (1963), 129–170.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    FEFFERMAN, C.: The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1–65.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    —: Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains, Ann. of Math. 103 (1976), 395–416, and 104 (1976), 393–394.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    FOLLAND, G. B. and J. J. KOHN: The Neumann problem for the Cauchy-Riemann complex, Ann. of Math. Studies no. 75, Princeton, N. J., 1972.Google Scholar
  23. [23]
    GAFFNEY, M.: Hilbert space methods in the theory of harmonic integrals, Trans. Amer. Math. Soc. 78 (1955), 426–444.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    GHIŞA, D.: The modulus and the hyperbolic measure, this volume, pp. 169–172.Google Scholar
  25. [25]
    HOLMANN, H.: On the stability of holomorphic foliations, this volume, pp. 201–211.Google Scholar
  26. [26]
    HÖRMANDER, L.: L2-estimates and existence theorems for the ∂-operator, Acta Math. 113 (1965), 89–152.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    HUA Lo-keng: Harmonic analysis of several complex variables in the classical domains [in Chinese], Science Press, Peking 1958; English transl.: Transl. Math. Monographs 6, Amer. Math. Soc., Providence, R. I., 1963.Google Scholar
  28. [28]
    KALINA, J.: A variational characterization of condenser capacities in IRn within a class of plurisubharmonic functions, Ann. Polon. Math., to appear.Google Scholar
  29. [29]
    —: Biholomorphic invariance of the capacity and the capacity of an annulus, Ann. Polon. Math., to appear.Google Scholar
  30. [30]
    KERZMAN, N.: The Bergman kernel function. Differentiability at the boundary. Math. Ann. 195 (1972), 149–158.MathSciNetCrossRefGoogle Scholar
  31. [31]
    ŁAWRYNOWICZ, J.: Condenser capacities and an extension of Schwarz's lemma for hermitian manifolds, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 23 (1975), 839–844.zbMATHGoogle Scholar
  32. [32]
    —: Electromagnetic field and the theory of conformal and biholomorphic invariants, Complex analysis and its applications III, International Atomic Energy Agency, Wien 1976, pp. 1–23.Google Scholar
  33. [33](a)
    — On a class of capacities on complex manifolds endowed with an hermitian structure and their relation to elliptic and hyperbolic quasiconformal mappings, Ann. Polon. Math. 33 (1976), 178 (abstract), (b) Dissertationes Math. 166 (1978), 44 pp. (in extenso).Google Scholar
  34. [34]
    —: On biholomorphic continuability of regular quasiconformal mappings, this volume, pp. 341–364.Google Scholar
  35. [35]
    LIGOCKA, E.: Some remarks on extension of biholomorphic mappings, this volume, pp. 365–378.Google Scholar
  36. [36]
    —: How to prove Fefferman's theorem without use of differential geometry, Ann. Polon. Math., to appear.Google Scholar
  37. [37]
    POGORELOV, A.V.: The Dirichlet problem for the multidimensional analogue of the Monge-Ampère equation [in Russian], Dokl. Akad. Nauk SSSR 201 (1971), 790–793; English transl.: Soviet Math. Dokl. 12 (1971), 1727–1731.MathSciNetGoogle Scholar
  38. [38]
    RAMADANOV, I.: Sur une propriété de la fonction de Bergman, C. R. Acad. Bulg. Sci. 20 (1967), 759–762.MathSciNetzbMATHGoogle Scholar
  39. [39]
    SKWARCZYŃSKI, M.: The Bergman function and semiconformal mappings, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 22 (1974), 667–673.MathSciNetzbMATHGoogle Scholar
  40. [40]
    —: A remark on holomorphic isometries with respect to the induced Bergman metrics, this volume, pp. 425–427.Google Scholar
  41. [41]
    —: The ideal boundary of a domain in ¢n, Ann. Polon. Math., to appear.Google Scholar
  42. [42]
    WEBSTER, S.: Biholomorphic mappings and the Bergman kernel off the diagonal, Ms.Google Scholar
  43. [43]
    ZINOVÉV, B.S.: Reproducing kernels for multicircular domains of holomorphy [in Russian], Sibirsk. Mat. Ž. 15 (1974), 35–48 and 236.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Jerzy Kalina
    • 1
  • Julian Ławrynowicz
    • 1
  • Ewa Ligocka
    • 2
  • Maciej Skwarczyński
    • 3
  1. 1.Institute of Mathematics of thePolish Academy of Sciences, Łódź BranchŁódźPoland
  2. 2.Osiedle Przyjaźń 15 m. 4Warszawa JelonkiPoland
  3. 3.Institute of MathematicsUniversity of Warsaw Pałac Kultury i Nauki, IX p.WarszawaPoland

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