Recent progress and future directions in several complex variables

  • Steven G. Krantz
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1268)


Finite Type Real Hypersurface Pseudoconvex Domain Bergman Kernel Biholomorphic Mapping 
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  1. [BJT]
    S. Baouendi, H. Jacobowitz, and F. Treves, On the analyticity of CR mappings, preprint.Google Scholar
  2. [BAR1]
    D. Barrett, Irregularity of the Bertgman projection on a smooth, bounded domain in ℂ2, Ann. Math. 119(1984), 431–436.CrossRefGoogle Scholar
  3. [BAR2]
    D. Barrett, Regularity of the Bergman projeciton and local geometry of domains, Duke Jour. Math. 53(1986), 333–343.CrossRefzbMATHGoogle Scholar
  4. [BED1]
    E. Bedford, Invariant forms on complex manifolds with applications to holomorphic mappings, Math. Annalen 265(1983), 377–396.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [BED2]
    E. Bedford, Proper holomorphic mappings, Bull. Am. Math. Soc. 10(1984), 157–175.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [BB]
    E. Bedford and S. Bell, Boundary continuity of proper holomorphic correspondences, Seminaire Dolbeault — Lelong — Skoda, 1983.Google Scholar
  7. [BDA]
    E. Bedford and J. Dadok, Bounded domains with prescribed automorphism groups, preprint.Google Scholar
  8. [BEF]
    E. Bedford and J. E. Fornaess, A construction of peak functions on weakly pseudoconvex domains, Ann. Math. 107(1978), 555–568.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [BTU]
    H. Behnke and P. Thullen, Theorie der Funktionen Mehrerer Komplexer Vera \(\ddot n\) derlichen, Second Edition, Springer, Berlin, 1970.CrossRefGoogle Scholar
  10. [BEL]
    S. Bell, Biholomorphic mappings and the \(\bar \partial \)problem, Ann. Math. 114(1981), 103–112.CrossRefGoogle Scholar
  11. [BEC]
    S. Bell and D. Catlin, Proper holomorphic mappings extend smoothly to the boundary, Bull. Am. Math. Soc. 7(1982), 269–272.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [BELL]
    S. Bell and E. Ligocka, A simplification and extension of Fefferman's theorem on biholomorphic mappings, Invent. Math. 57(1980), 283–289.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [BERG]
    S. Bergman, The kernel function and conformal mapping, Am. Math. Soc., Providence, 1970.zbMATHGoogle Scholar
  14. [BER]
    L. Bers, Introduction to Several Complex Variables, New York University Press, New York, 1964.Google Scholar
  15. [BL1]
    T. Bloom, C peak functionns for pseudoconvex domains of strict type, Duke Math. J. 45(1978), 133–147.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [BL2]
    T. Bloom, On the contact between complex manifolds and real hypersurfaces in ℂ3, Trans. Am. Math. Soc. 263(1981), 515–529.MathSciNetzbMATHGoogle Scholar
  17. [BG]
    T. Bloom and I. Graham, A geometric characterization of points of finite type m on real submanifolds of ℂn, J. Diff. Geom. 12(1977), 171–182.MathSciNetzbMATHGoogle Scholar
  18. [BOM]
    S. Bochner and W. Martin, Several Complex Variables, Princeton University Press, Princeton, 1948.zbMATHGoogle Scholar
  19. [BMS]
    L. Boutet de Monvel and J. Sjöstrand, Sur la singularité des noyaux de Bergman et Szegö, Soc. Mat. de France Asterisque 34–35(1976), 123–164.zbMATHGoogle Scholar
  20. [BS]
    D. Burns and S. Shnider, Geometry of hypersurfaces and mapping theorems in ℂn, Comment. Math. Helv. 54(1979), 199–217.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [BSW]
    D. Burns, S. Shnider, and R. Wells, On deformations of strongly pseudoconvex domains, Invent. Math. 46(1978), 237–253.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [CAR]
    H. Cartan, Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes, Hermann, Paris, 1964.zbMATHGoogle Scholar
  23. [CAT1]
    D. Catlin, Invariant metrics on pseudoconvex domains, in Several Complex Variables; Proceedings of the 1981 Hangzhou conference, ed. J. J. Kohn, Q-K Lu, R. Remmert, Y. T. Siu, Birkhäuser, Boston, 1984.Google Scholar
  24. [CAT2]
    D. Catlin, Subelliptic estimates for the \(\bar \partial \)— Neumann problem on pseudoconvex domains, preprint.Google Scholar
  25. [CM]
    S. S. Chern and J. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133(1974), 219–271.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [CI]
    E. Cirka, The theorems of Lindelöf and Fatou in ℂn, Mat. Sb. 92(134)(1973), 622–644; Math. USSR Sb. 21(1973), 619–639.MathSciNetzbMATHGoogle Scholar
  27. [DAN]
    J. D'Angelo, Real hypersurfaces, orders of contact and applications, Ann. Math. 115(1982), 615–638.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [DF1]
    K. Diederich and J. E. Fornaess, Pseudoconvex domains: an example with nontrivial Nebenhülle, Math. Annalen 225(1977), 272–292.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [DF2]
    K. Diederich and J.E. Fornaess, Pseudoconvex domains: bounded plurisubharmonic exhaustion functions, Invent. Math. 39(1977), 129–141.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [DF3]
    K. Diederich and J. E. Fornaess, Pseudoconvex domains with real analytic boundaries, Ann. Math. 107(1978), 371–384.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [DF4]
    K. Diederich and J. E. Fornaess, Biholomorphic mappings between certain real analytic domains in ℂn, Math. Annalen 245(1979), 255–272.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [DF5]
    K. Diederich and J. E. Fornaess, Smooth extendability of proper holomorphic mappings, Bull. Am. Math. Soc. 7(1982), 264–268.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [FE1]
    C. Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26(1974), 1–65.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [FE2]
    C. Fefferman, Monge Ampère equations, the Bergman kernel, and the geometry of pseudoconvex domains, Ann. Math. 103(1976), 395–416.MathSciNetCrossRefzbMATHGoogle Scholar
  35. [FE3]
    C. Fefferman, Parabolic invariant theory in complex analysis, Adv. Math 31(1979), 131–262.MathSciNetCrossRefzbMATHGoogle Scholar
  36. [FES]
    C. Fefferman and E. M. Stein, Hp spaces of several variables, Acta Math. 129(1972), 137–193.MathSciNetCrossRefzbMATHGoogle Scholar
  37. [FOK]
    G. B. Folland and J. J. Kohn, The Neumann Problem for the Cauchy-Riemann Complex, Princeton University Press, Princeton, 1972.zbMATHGoogle Scholar
  38. [FOS1]
    G. B. Folland and E. M. Stein, Estimates for the \(\bar \partial _b \) complex and analysis of the Heisenberg group, Comm. Pure and Appl. Math. 27(1974), 429–522.MathSciNetCrossRefGoogle Scholar
  39. [FOS2]
    G. B. Folland and E. M. Stein, Hardy spaces on Homogeneous Groups, Math. Notes 28, Princeton University Press, Princeton, 1982.zbMATHGoogle Scholar
  40. [FO1]
    J. E. Fornaess, Peak points on weakly pseudoconvex domains, Math. Annalen 227(1977), 173–175.MathSciNetCrossRefzbMATHGoogle Scholar
  41. [FO2]
    J. E. Fornaess, Sup norm estimates for \(\bar \partial \) in ℂ2, preprint.Google Scholar
  42. [FOR]
    A. R. Forsyth, Lectures Introductory to the Theory of Two Complex Variables, Cambridge University Press, Cambridge, 1914.zbMATHGoogle Scholar
  43. [FR]
    B. Fridman, One example of the boundary behavior of biholomorphic transformations, preprint.Google Scholar
  44. [FUK1]
    B. A. Fuks, Introduction to the Theory of Analytic Functions of Several Complex Variables, Translations of Mathematical Monographs, American Math. Society, Providence, 1963.zbMATHGoogle Scholar
  45. [FUK2]
    B. A. Fuks, Special Chapters in the Theory of Analytic Functions of Several Complex Variables, Translations of Mathematical Monographs, American Math. Society, Providence, 1965.zbMATHGoogle Scholar
  46. [GK1]
    R. E. Greene and S. G. Krantz, Stability of the Bergman kernel and curvature properties of bounded domains, in Recent Developments in Several Complex Variables. Annals of Math. Studies 100, Princeton University Press, Princeton, 1981.Google Scholar
  47. [GK2]
    R. E. Greene and S. G. Krantz, Deformations of complex structures, estimates for the \(\bar \partial \) equation, and stability of the Bergman kernel, Adv. in Math. 43(1982), 1–86.MathSciNetCrossRefGoogle Scholar
  48. [GK3]
    R. E. Greene and S. G. Krantz, The automorphism groups of strongly pseudoconvex domains, Math. Annalen 261(1982), 425–446.MathSciNetCrossRefzbMATHGoogle Scholar
  49. [GK4]
    R. E. Greene and S. G. Krantz, Biholomorphic self-maps of domains, Proceedings of a Special Year in Complex Analysis at the University of Maryland, Springer Lecture Notes, Springer, 1987.Google Scholar
  50. [GK5]
    R. E. Greene and S. G. Krantz, A new invariant metric in complex analysis and some applications, to appear.Google Scholar
  51. [GR]
    P. Greiner, Subelliptic estimates for the \(\bar \partial \)-Neumann problem in C2, J. Diff. Geom. 9(1974), 239–250.MathSciNetGoogle Scholar
  52. [GRS]
    P. Greiner and E. M. Stein, On the solvability of some differential operators of type □b, Proceedings of International Conferences, Cortona, Italy, Scuola Normale Superiore Pisa, 1978.Google Scholar
  53. [HS1]
    M. Hakim and N. Sibony, Quelques conditions pour l'existence de fonctions pics dans les domains pseudoconvexes, Duke Math. Jour. 44(1977), 399–406.MathSciNetCrossRefzbMATHGoogle Scholar
  54. [HS2]
    M. Hakim and N. Sibony, Frontière de Šilov et spectre de A\((\bar D)\) pour des domaines faiblement pseudoconvex, C. R. Acad. Sci. Paris, Ser. A–B 281(1975), A959–A962.MathSciNetGoogle Scholar
  55. [HAR]
    F. Hartogs, Zur Theorie der analytischen Functionen mehrener unabhänghiger Veränderlichen insbesondere über die Darstellung derselben durch Reihen, welche nach Potenzen einer Veränderlichen fortschreiten, Math. Annalen 62(1906), 1–88.MathSciNetCrossRefzbMATHGoogle Scholar
  56. [HEL]
    S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, New York, 1962.zbMATHGoogle Scholar
  57. [HEN1]
    G.M. Henkin, Integral representations of functions holomorphic in strictly pseudoconvex domains and some applications, Mat. Sb. 78(120)(1969), 611–632; Math. USSR Sb. 7(1969), 597–616.MathSciNetGoogle Scholar
  58. [HEN2]
    G. M. Henkin, An analytic polyhedron is not holomorphically equivalent to a strictly pseudoconvex domain (Russian), Dokl. Acad. Nauk. SSSR 210(1973), 1026–1029.MathSciNetzbMATHGoogle Scholar
  59. [HOR]
    L. Hörmander, Introduction to Complex Analysis in Several Variables, North Holland, Amsterdam, 1973.zbMATHGoogle Scholar
  60. [HUA]
    L. K. Hua, Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, American Mathematical Society, Providence, 1963.Google Scholar
  61. [KER]
    N. Kerzman, The Bergman Kernel function. Differentiability at the boundary, Math. Annalen 195(1972), 149–158.MathSciNetCrossRefGoogle Scholar
  62. [KL]
    P. Klembeck, Kähler metrics of negative curvature, the Bergman metric near the boundary and the Kobayashi metric on smooth bounded strictly pseudoconvex domains sets, Indiana Univ. Math. J. 27(1978), 275–282.MathSciNetCrossRefzbMATHGoogle Scholar
  63. [KO1]
    J. J. Kohn, Boundary behavior of \(\bar \partial \) on weakly pseudoconvex manifolds of dimension two, J. Diff. Geom. 6(1972), 523–542.MathSciNetGoogle Scholar
  64. [KO2]
    J. J. Kohn, Sufficient conditions for subellipticity of weakly pseudoconvex domains, Proc. Nat. Acad. Sci. (USA) 74(1977), 2214–2216.MathSciNetCrossRefzbMATHGoogle Scholar
  65. [KO3]
    J. J. Kohn, Global regularity for \(\bar \partial \) on weakly pseudoconvex manifolds, Trans. Am. Math. Soc. 181(1973), 273–292.MathSciNetGoogle Scholar
  66. [KON]
    J. J. Kohn and L. Nireberg, A pseudo-convex domain not admitting a holomorphic support function, Math. Annalen 201(1973), 265–268.MathSciNetCrossRefzbMATHGoogle Scholar
  67. [KR1]
    S. Krantz, Function Theory of Several Complex Variables, John Wiley and Sons, New York, 1982.zbMATHGoogle Scholar
  68. [KR2]
    S. Krantz, Characterizations of various domains of holomorphy via \(\bar \partial \) estimates and applications to a problem of Kohn, III. J. Math. 23(1979), 267–286.MathSciNetGoogle Scholar
  69. [KR3]
    S. Krantz, Fatou theorems on domains in Cn, Bull. Am. Math. Soc., to appear.Google Scholar
  70. [KR4]
    S. Krantz, Invariant metrics and harmonic analysis on domains in Cn, to appear.Google Scholar
  71. [KR5]
    S. Krantz, A compactness principle complex analysis, Division de Matematicas U. A. M. Seminarios, to appear.Google Scholar
  72. [MAL]
    B. Malgrange, Lectures on the Theory of Functions of Several Complex Variables, Tata Institute of Fundamental Research, Bombay, 1958.zbMATHGoogle Scholar
  73. [MOW]
    J. Moser and S. Webster, Normal forms for real surfaces in C2 near complex tangents and hyperbolic surface transformations, Acta Math. 150(1983), 255–298.MathSciNetCrossRefzbMATHGoogle Scholar
  74. [NSW1]
    A. Nagel, E.M. Stein, and S. Wainger, Boundary behavior of functions holomorphic in domains of finite type, Proc. Nat. Acad. Sci. (USA) 78(1981), 6596–6599.MathSciNetCrossRefzbMATHGoogle Scholar
  75. [NSW2]
    A. Nagel, E.M. Stein, and S. Wainger, Balls and metrics defined by vector fields I: Basic properties, Acta Math. 155(1985), 103–148.MathSciNetCrossRefzbMATHGoogle Scholar
  76. [NSW3]
    A. Nagel, oral communication.Google Scholar
  77. [NWY]
    L. Nirenberg, S. Webster, and P. Yang, Local boundary behavior of holomorphic mappings, Comm. Pure Appl. Math. 33(1980), 305–338.MathSciNetCrossRefzbMATHGoogle Scholar
  78. [OKA]
    K. Oka, Collected Papers, Springer, Berlin, 1984.CrossRefzbMATHGoogle Scholar
  79. [RAM]
    E. Ramirez, Divisions problem in der komplexen analysis mit einer Anwendung auf Rand integral darstellung, Math. Ann. 184(1970), 172–187.MathSciNetCrossRefGoogle Scholar
  80. [RAN1]
    R.M. Range, The Caratheodory metric and holomorphic maps on a class of weakly pseudoconvex domains, Pac. Jour. Math. 78(1978), 173–189.MathSciNetCrossRefzbMATHGoogle Scholar
  81. [RAN2]
    R. M. Range, Holomorphic Functions and Integral Representations in Several Complex Variables, Springer Verlag, New York, 1986.CrossRefzbMATHGoogle Scholar
  82. [SI1]
    N. Sibony, Sur le plongement des domaines faiblement pseudoconvexes dans des domaines convex, Math. Ann. 273(1986), 209–214.MathSciNetCrossRefzbMATHGoogle Scholar
  83. [SI2]
    N. Sibony, Un exemple de domaine pseudonconvexe regulier ou l'equation \(\bar \partial u = f\) n'admet pas de solution bornee pour f bornee, Invent. Math. 62(1980), 235–242.MathSciNetCrossRefGoogle Scholar
  84. [ST1]
    E. M. Stein, Boundary Behavior of Holomorphic Functions of Several Complex Variables, Princeton University Press, Princeton, 1972.zbMATHGoogle Scholar
  85. [ST2]
    E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.zbMATHGoogle Scholar
  86. [TA]
    N. Tanaka, On generalized graded Lie algebras and geometric structures I, J. Math. Soc. Japan 19(1967), 215–254.MathSciNetCrossRefzbMATHGoogle Scholar
  87. [VLA]
    V. Vladimirov, Methods of the Theory of Functions of Many Complex Variables, MIT Press, Cambridge, 1964.Google Scholar
  88. [WE]
    S. Webster, Biholomorphic mappings and the Bergman kernel off the diagonal, Invent. Math. 51 (1979), 155–169.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • Steven G. Krantz
    • 1
  1. 1.Department of MathematicsWashington UniversitySt. Louis

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