Some aspects of analysis on almost complex manifolds with boundary

  • Bernard Coupet
  • Hervé Gaussier
  • Alexandre Sukhov


We present some results dealing with the local geometry of almost complex manifolds. We establish mainly the complete hyperbolicity of strictly pseudoconvex domains, the extension of plurisubharmonic functions through generic submanifolds, and the elliptic regularity of some diffeomorphisms of almost complex manifolds with boundary.


Complex Manifold Cotangent Bundle Real Hypersurface Pseudoconvex Domain Plurisubharmonic Function 
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  1. 1.
    M. Audin and J. Lafontaine, Holomorphic Curves in Symplectic Geometry, Birkhäuser, Progress in Math., 117 (1994).Google Scholar
  2. 2.
    Z. Balogh and Ch. Leuenberger, “Higher dimensional Riemann maps,” Internat. J. Math., 9, 421–442 (1998).MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    J. F. Barraud and E. Mazzilli, “Regular type of real hyper-surfaces in (almost) complex manifolds, ” Math. Z., 248, 757–772 (2004).MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    S. Bell and L. Lempert, “A C Schwarz reflection principle in one and several complex variables,” J. Differential Geom., 32, No. 2, 899–915 (1990).MATHMathSciNetGoogle Scholar
  5. 5.
    D. Bennequin, “Topologie symplectique, convexité holomorphe holomorphe et structures de contact” (d'après Y. Eliashberg, D. Mc Duff et al.), Astérisque, 189-190, 285–323 (1990).MathSciNetGoogle Scholar
  6. 6.
    F. Berteloot, “Attraction des disques analytiques et continuityé höldérienne d'applications holomorphes propres,” in: Topics in Complex Analysis (Warsaw, 1992), 91–98, Banach Center Publ., 31, Polish Acad. Sci., Warsaw (1995).Google Scholar
  7. 7.
    F. Berteloot, “Principe de Bloch et estimations de la métrique de Kobayashi dans les domains de ℂ2,” J. Geom. Anal., 13, No. 1, 29–37 (2003).MATHMathSciNetGoogle Scholar
  8. 8.
    F. Berteloot and G. Coeuré, “Domaines de ℂ2, pseudoconvexes et de type fini ayant un groupe non compact d'automorphismes,” Ann. Inst. Fourier, 41, 77–86 (1991).MATHGoogle Scholar
  9. 9.
    J. Bland, “Contact geometry and CR structures on S2,” Acta Math. 172, 1–49 (1994).MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    J. Bland and T. Duchamp, “Moduli for pointed convex domains,” Invent. Math., 104, 61–112 (1991).MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    J. Bland, T. Duchamp, and M. Kalka, “A characterization of ℂℙn by its automorphism group,” In: Lecture Notes in Math., 1268 (1987), pp. 60–65.CrossRefMathSciNetGoogle Scholar
  12. 12.
    S. Bu and W. Schachermayer, “Approximation of Jensen measures by image measures under holomorphic functions and applications,” Trans. Amer. Math. Soc., 331, 585–608 (1992).MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    É. Cartan, “Sur la géométrie pseudo-conforme des hypersurfaces de l'espace de deux variables complexes, I,” Annali di Mat., 11, 17–90 (1932).MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    U. Cegrell, “Sur les ensembles singuliers impropres des fonctions plurisousharmoniques,” C. R. Acad. Sci. Paris, 281, 905–908 (1975).MATHMathSciNetGoogle Scholar
  15. 15.
    M. Cerne, “Stationary disks of fibrations over the circle,” Internat. J. Math., 6, 805–823 (1995).MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    S. S. Chern and J. K. Moser, “Real hypersurfaces in complex manifolds,” Acta Math., 133, 219–271 (1974).CrossRefMathSciNetGoogle Scholar
  17. 17.
    E. Chirka, “Regularity of boundaries of analytic sets,” Mat. Sb., 45, 291–336 (1983).CrossRefMathSciNetGoogle Scholar
  18. 18.
    E. Chirka, “Introduction to the almost complex analysis,” in: Lecture Notes in Math. (2003).Google Scholar
  19. 19.
    E. Chirka, personal communication. Google Scholar
  20. 20.
    E. Chirka, B. Coupet, and A. Sukhov, “On boundary regularity of analytic disks,” Mich. Math. J., 46, 271–279 (1999).MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    K. Clancey and I. Gohberg, Factorization of Matrix Functions and Singular Integral Operators, Birkhauser, Basel-Boston-Stuttgart (1981).MATHGoogle Scholar
  22. 22.
    B. Coupet, “Precise regularity up to the boundary of proper holomorphic mappings,” Ann. Scuola Norm. Sup. Pisa, 20, 461–482 (1993).MATHMathSciNetGoogle Scholar
  23. 23.
    B. Coupet, H. Gaussier, and A. Sukhov, “Riemann maps in almost complex manifolds,” Ann. Sc. Norm. Super. Pisa Cl. Sci., 5, 761–785 (2003).MathSciNetGoogle Scholar
  24. 24.
    B. Coupet, H. Gaussier, and A. Sukhov, “Fefferman's mapping theorem on almost complex manifolds in complex dimension two,” Math. Z., 250, 59–90 (2005).MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    R. Debalme, Kobayashi Hyperbolicity of Almost Complex Manifolds, Preprint of the University of Lille, IRMA 50 (1999), math.CV/9805130.Google Scholar
  26. 26.
    R. Debalme and S. Ivashkovich, “Complete hyperbolicit neiborhoods in almost complex surfaces,” Int. J. Math., 12, 211–221 (2001).MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    K. Diederich and J. E. Fornaess, “Proper holomorphic maps onto pseudoconvex domains with real analytic boundary,” Ann. Math., 110, 575–592 (1979).CrossRefMathSciNetGoogle Scholar
  28. 28.
    K. Diederich and A. Sukhov, “Diffeomorphisms of Stein structures,” math. CV/0603416, to appear in J. Geom. Anal. Google Scholar
  29. 29.
    K. Diederich and A. Sukhov, “Plurisubharmonic exhaustion functions and almost complex Stein structures,” mat. CV/0603417.Google Scholar
  30. 30.
    G. A. Edgar, “Complex martingale convergence,” in: Lecture Notes in Math., 1116 (1985), pp. 38–59.CrossRefMathSciNetGoogle Scholar
  31. 31.
    Ch. Fefferman, “The Bergman kernel and biholomorphic mappings of pseudoconvex domains,” Invent. math., 26, 1–65 (1974).MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    F. Forstneric, “An elementary proof of Fefferman's theorem,” Exposition. Math., 10, 135–149 (1992).MATHMathSciNetGoogle Scholar
  33. 33.
    H. Gaussier and A. Sukhov, “Estimates of the Kobayashi metric in almost complex manifolds,” ArXiv, Bull. Soc. Math. France, 133, 259–273 (2005).MATHMathSciNetGoogle Scholar
  34. 34.
    H. Gaussier and A. Sukhov, “On the geometry of model almost complex manifolds with boundary,” math. CV/0412095, to appear in Math. Z. Google Scholar
  35. 35.
    J. Globevnik, “Perturbation by analytic discs along maximal real submanifolds of ℂN,” Math. Z., 217, 287–316 (1994).MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    J. Globevnik, “Perturbing analytic discs attached to a maximal totally real submanifolds of ℂ n,” Indag. Math., 7, 37–46 (1996).MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    I. Graham, “Boundary behavior of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains in C n with smooth boundary,” Trans. Amer. Math. Soc., 207, 219–240 (1975).MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    H. Grauert and I. Lieb, “Das Ramirezsche Integral und die Lösung der Gleichung {ie984-1} im Bereich der beschränkten Formen,” Rice Univ. Studies, 56, 29–50 (1970).MATHMathSciNetGoogle Scholar
  39. 39.
    M. Gromov, “Pseudoholomorphic curves in symplectic manifolds,” Invent. Math., 82, No. 2, 307–347 (1985).MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    R. Hamilton, “Deformation of complex structures on manifolds with boundary. I. The stable case,” J. Differential Geom., 12, 1–45 (1977).MATHMathSciNetGoogle Scholar
  41. 41.
    G. Henkin, “Integral representation of functions in strongly pseudoconvex regions, and applications to the {ie984-2},” Mat. Sb., 82(124), 300–308 (1970).MathSciNetGoogle Scholar
  42. 42.
    H. Hofer, “Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three,” Invent. Math., 114, 515–563 (1993).MATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    K. Yano Ishihara, “Tangent and cotangent bundles: Differential geometry,” in: Pure ans Appl. Math., No. 16. Marcel Dekker Inc., New York (1973).Google Scholar
  44. 44.
    S. Ivashkovish and J. P. Rosay, “Schwarz-type lemmas for solutions of {ie984-3} and complete hyperbolicity of almost complex manifolds,” Preprint.Google Scholar
  45. 45.
    S. Ivashkovish and V. Shevchishin, “Reflection principle and J-complex curves with boundary on totally real immersions,” Comm. in Contemp. Math., 4, No. 1, 65–106 (2002).CrossRefGoogle Scholar
  46. 46.
    N. C. Karpova, “On the removal of singularoties of plurisubharmonic functions,” Mat. Sb., 63, 252–256 (1989).Google Scholar
  47. 47.
    S. Kobayashi, Hyperbolic Complex Spaces, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 318. Springer-Verlag, Berlin (1998).MATHGoogle Scholar
  48. 48.
    S. Kobayashi, “Almost complex manifolds and hyperbolicity. Dedicated to Shiing-Shen Chern on his 90th birthday,” Results Math., 40, No. 1–4, 246–256 (2001).MATHMathSciNetGoogle Scholar
  49. 49.
    N. Kerzman and J. P. Rosay, “Fonctions plurisousharmoniques d'exhaustion bornées et domains taut,” Math. Ann., 257, No. 2, 171–184 (1981).MATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    B. S. Kruglikov, “Existence of close pseudoholomorphic disks for almost complex manifolds and their application to the Kobayashi-Royden pseudonorm,” Funkts. Anal. Prilozhen., 33, No. 1, 46–58 (1999).MathSciNetGoogle Scholar
  51. 51.
    B. S. Kruglikov, “Deformation of big pseudoholomorphic discs and application to the Hanh pseudonorm, ” arXiv: math. CV/0304166 v1 14.04.2003.Google Scholar
  52. 52.
    L. Lempert, “La métrique de Kobayashi et la representation des domaine sur la boule,” Bull. Math. Soc. France, 109, 427–474 (1981).MATHMathSciNetGoogle Scholar
  53. 53.
    L. Lempert, “Solving the degenerate complex Monge-Ampère equation with one concentrated singularity,” Math. Ann., 263, 515–532 (1983).MATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    L. Lempert, “A precise result on the boundary regularity of biholomorphic mappings,” Math. Z., 193, 559–579 (1986).MATHCrossRefMathSciNetGoogle Scholar
  55. 55.
    L. Lempert, “Holomorphic invariants, normal forms and moduli space of convex domains,” Ann. of Math., 128, 47–78 (1988).CrossRefMathSciNetGoogle Scholar
  56. 56.
    L. Lempert, “Erratum: A precise result on the boundary regularity of biholomorphic mappings,” Math. Z., 206, 501–504 (1991).MATHCrossRefMathSciNetGoogle Scholar
  57. 57.
    L. Lempert and R. Szöke, “The tangent bundle of an almost complex manifold,” Canad. Math. Bull., 44, 70–79 (2001).MATHMathSciNetGoogle Scholar
  58. 58.
    P. Libermann, “Problèmes d'équivalence relatifs à une structure presque complexe sur une variété à quatre dimensions,” Acad. Roy. Belgique Bull. Cl. Sci. (5), 36, 742–755 (1950).MathSciNetGoogle Scholar
  59. 59.
    D. McDuff and D. Salamon, “J-holomorphic curves and symplectic topology,” in: Amer. Math. Soc. Colloq. Publ., 52, Providence, RI (2004), xii+669 p.Google Scholar
  60. 60.
    A. Newlander and L. Nirenberg, “Complex analytic coordinates in almost complex manifolds,” Ann. of Math. (2), 65, 391–404 (1957).CrossRefMathSciNetGoogle Scholar
  61. 61.
    A. Nijenhuis and W. Woolf, “Some integration problems in almost-complex and complex manifolds, ” Ann. Math., 77, 429–484 (1963).MathSciNetCrossRefGoogle Scholar
  62. 62.
    L. Nirenberg, S. Webster, and P. Yang, “Local boundary regularity of holomorphic mappings,” Comm. Pure Appl. Math. 33, 305–338 (1980).MATHCrossRefMathSciNetGoogle Scholar
  63. 63.
    M. Y. Pang, “Smoothness of the Kobayashi metric of non-convex domains,” Int. J. Math., 4, 953–987 (1993).MATHCrossRefGoogle Scholar
  64. 64.
    P. Pflug, “Ein Fortsetzungsatz fur plurisubharmonische Funktionen über reell-2-kodimensionale Flachen,” Arch. Math. (Basel), 33, 559–663 (1979/80).MathSciNetGoogle Scholar
  65. 65.
    S. Pinchuk, “A boundary uniqueness theorem for holomorphic functions of several complex variables, ” Math. Notes, 15, 116–120 (1974).MathSciNetGoogle Scholar
  66. 66.
    Th. Ransford, Potential Theory in the Complex Plane, Cambridge Univ. Press (1995).Google Scholar
  67. 67.
    S. Pinchuk, “The scaling method and holomorphic mappings,” in: Several Complex Variables and Complex Geometry, Part 1 (Santa Cruz, CA, 1989), pp. 151–161, Proc. Sympos. Pure Math., 52 Part 1, Amer. Math. Soc., Providence, RI (1991).Google Scholar
  68. 68.
    S. Pinchuk and S. Khasanov, “Asymptotically holomorphic functions and their applications,” Mat. Sb., 62, 541–550 (1989).MATHCrossRefMathSciNetGoogle Scholar
  69. 69.
    H. L. Royden, “Remarks on the Kobayashi metric,” in: Lecture Notes in Math., 185, Springer-Verlag (1970), pp. 125–137.CrossRefMathSciNetGoogle Scholar
  70. 70.
    S. Semmes, “A generalization of Riemann mappings and geometric structures on a space of domains in ℂn,” Mem. Amer. Math. Soc., 98 (1992), vi+98 p.Google Scholar
  71. 71.
    B. Shiffman, “Extension of positive line bundles and meromorphic maps,” Invent. math., 15, 332–347 (1972).MATHCrossRefMathSciNetGoogle Scholar
  72. 72.
    N. Sibony, “A class of hyperbolic manifolds,” Ann. of Math. Stud., 100, 91–97 (1981).MathSciNetGoogle Scholar
  73. 73.
    J.-C. Sikorav, “Some properties of holomorphic curves in almost complex manifolds,” in: Holomorphic Curves and Symplectic Geometry (M. Audin and J. Lafontaine, Eds.), Birkhäuser (1994), pp. 165–189.Google Scholar
  74. 74.
    A. Spiro, “Total reality of the conormal bundle of a real hypersurface in an almost complex manifold, ” Preprint (2003).Google Scholar
  75. 75.
    A. Spiro and A. Sukhov, “An existence theorem for stationary discs in almost complex manifolds, ” math. CV/0502121.Google Scholar
  76. 76.
    A. Spiro and S. Trapani, “Eversive maps of bounded convex domains in ℂn+1,” J. Geom. Anal., 12, 695–715 (2002).MATHMathSciNetGoogle Scholar
  77. 77.
    A. Tumanov, “Analytic discs and the regularity of CR mappings in higher codimension,” Duke Math. J., 76, 793–807 (1994).MATHCrossRefMathSciNetGoogle Scholar
  78. 78.
    A. Tumanov, “Extremal discs and the regularity of CR mappings in higher codimension,” Amer. J. Math., 123, 445–473 (2001).MATHCrossRefMathSciNetGoogle Scholar
  79. 79.
    N. P. Vekua, Systems of Singular Integral Equations, Nordholf, Groningen (1967).Google Scholar
  80. 80.
    S. Webster, “On the reflection principle in several complex variables,” Proc. Amer. Math. Soc., 71, 26–28 (1978).MATHCrossRefMathSciNetGoogle Scholar
  81. 81.
    S. Webster, “A new proof of the Newlander-Nirenberg theorem,” Math. Z., 201, 303–316 (1989).MATHCrossRefMathSciNetGoogle Scholar
  82. 82.
    K. Yano and Sh. Ishihara, Tangent and Cotangent Bundles, Marcel Dekker, New York (1973).MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  • Bernard Coupet
    • 1
  • Hervé Gaussier
    • 2
  • Alexandre Sukhov
    • 3
  1. 1.C.M.I.Marseille Cedex 13France
  2. 2.I.M.J.Paris CedexFrance
  3. 3.U.S.T.L., Cité Scientifique, 59655Villeneuve d´Ascq CedexFrance

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