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Part of the book series: Aspects of Mathematics ((ASMA,volume E 26))

Abstract

This paper is devoted to the theory of removable singularities in the boundary of a domain in ℂn, n ≥ 2, or in a complex manifold of dimension at least two.

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Bibliography

  1. Alexander, H.: Polynomial approximation and hulls in sets of finite linear measure. Amer. J. Math. 93 (1971), 65–74

    Article  MathSciNet  MATH  Google Scholar 

  2. Alexander, H.:Removable sets for CR functions. To appear in the proceedings of the Mittag-Leffler special year in several complex variables

    Google Scholar 

  3. Anderson, J. T. and Cima, J. A.: Removable singularities for LP CR functions. Michigan Math. J (To appear)

    Google Scholar 

  4. Andreotti, A. and Grauert, H.: Théorèmes de finitude pour la cohomologie des espaces complexes. Bull. Soc. Math. France 90 (1962), 193–259

    MathSciNet  MATH  Google Scholar 

  5. Andreotti, A. and Kas, A.: Duality on complex spaces. Ann. Scuola Norm. Sup. Pisa XXVII (1973), 187–263

    Google Scholar 

  6. Bânicà, C. and Stânâsila, O.: “Algebraic Methods in the Global Theory of Complex Spaces”, editor John Wiley, London 1976

    Google Scholar 

  7. Battelli, F.: Singolarità eliminabili per le tracce delle funzioni pluriarmoniche. Boll. U.M.I. (6) 1 (1982), no. 1, 133–146

    MathSciNet  MATH  Google Scholar 

  8. Bedford, E. and Fornwss, J.-E.: A construction of peak functions on weakly pseudoconvex domains. Ann. Math. (2) 107 (1978), 555–568

    Article  MATH  Google Scholar 

  9. Bedford, E. and Gaveau, B.: Envelopes of holomorphy of certain 2-spheres in C2. Amer. J. Math. 105 (1983), 975–1009

    Article  MathSciNet  MATH  Google Scholar 

  10. Björk, J. E.: Holomorphic convexity and analytic structures in Banach algebras. Ark. Mat. 9 (1971), 39–54

    Article  MathSciNet  MATH  Google Scholar 

  11. Bourbaki, N.: “Topological Vector Spaces”, Chapters 1–5, Springer-Verlag, Berlin 1987 [12] Bredon, G. E.: “Sheaf Theory”, McGraw-Hill, New York 1967

    Google Scholar 

  12. Cartan, H.: Variétés analytiques réeles et variétés analytiques complexes. Bull. Soc. math. France 85 (1957), 77–99

    MathSciNet  MATH  Google Scholar 

  13. Cassa, A.: Coomologia separata sulle varietà analitiche complesse. Ann. Scuola Norm. Sup. Pisa (3) 25 (1971), 290–323

    Google Scholar 

  14. Chirka, E. M.: Analytic representation of CR-functions. Mat. Sb. 98 (4) (1975), 591–623

    MathSciNet  Google Scholar 

  15. Chirka, E. M.: Regularization and 0-homotopy on a complex manifold. Dokl. Akad. Nauk SSSR 244 no. 2 (1979), 300–303

    MathSciNet  Google Scholar 

  16. Chirka, E. M.: “Complex Analytic Sets”, Nauka, Moscow 1985 (Russian). English translation, Kluwer, Dortrecht and Boston, 1989

    Google Scholar 

  17. Chirka. E. M.: Introduction to the geometry of CR-manifolds. Uspekhi Mat. Nauk 46 (277) no. 1 (1991), 80–164 (Russian). English translation, Russian Math. Surveys, 46 no. 1 (1991), 81–197

    Google Scholar 

  18. deRham, G.: “Variétés Différentiables”, Hermann, Paris 1973

    Google Scholar 

  19. Dieudonné, J. “Treatise on Analysis, vol. III”, Academic Press, New York 1972

    Google Scholar 

  20. Dieudonné, J. and Schwartz, L.: La dualité dans les espaces (F) et (LT). Ann. Inst. Fourier (Grenoble) 1 (1949) 61–101

    Google Scholar 

  21. Dini, G. and Parrini, C.: Singolarità rimovibili per CR-distribuzioni su domini piatti. Boll. Un. Mat. Ital. B(5) 17 (1980), 286–297

    MathSciNet  MATH  Google Scholar 

  22. Dini, G. and Parrini, C.: Extending CR-distributions. Bull. Sc. Math. (2)(106) (1982), 3–18

    Google Scholar 

  23. Dolbeault, P.: Formes differentielles et cohomolgie sur une variétè analytique complexe, I. Ann. Math. (2) 64 (1956), 83–130

    Google Scholar 

  24. Duval, J.:Surfaces convexes dans un bord pseudoconvex. (to appear)

    Google Scholar 

  25. Ehrenpreis, L.: A new proof and extension of Hartog’s theorem. Bull. Amer. Math. Soc. 67 (1967), 507–509

    Article  MathSciNet  Google Scholar 

  26. Federer, H: “Geometric Measure Theory”, Springer-Verlag, Berlin, Heidelberg, New York 1969

    Google Scholar 

  27. Forstnerié, E. and Stout, E. L.: A new class of polynomially convex sets. Ark. Mat. 29 (1) (1991), 51–62

    Article  MathSciNet  Google Scholar 

  28. Guenot, J. and Narasimhan, R.: Introduction à la théorie des surfaces de Riemann. L’Enseignment Mathématique (II) XXI (1975), 123–328

    Google Scholar 

  29. Gunning, R. C.: “Introduction to Holomorphic Functions of Several Complex Variables”, Wadsworth and Brooks/Cole, Belmont 1990

    Google Scholar 

  30. Gunning, R. and Rossi, H.: “Analytic Functions of Several Complex Variables”, Prentice-Hall, Englewood Cliffs 1965

    Google Scholar 

  31. Harvey, F. R. and Lawson, B.: On boundaries of complex analytic varieties. I. Ann. Math. (II) 102 (1975), 233–290

    Google Scholar 

  32. Harvey, E R. and Wells, R. O.: Compact holomorphically convex subsets of a Stein manifold. Trans. Amer. Math. Soc. 136 (1969), 509–516

    Article  MathSciNet  MATH  Google Scholar 

  33. Harvey, E. R. and Wells, R. O.: Holomorphic approximation and hyperfunction theory on a C totally real submanifold of a complex manifold. Math. Ann. 197 (1972), 287–318

    Article  MathSciNet  MATH  Google Scholar 

  34. Hatziafratis, T.: On certain integrals associated to CR-functions. Trans. Amer. Math. Soc. 314 (1989), 781–802

    Article  MathSciNet  MATH  Google Scholar 

  35. Henkin, G. and Leiterer, J.: “Theory of Functions on Complex Manifolds”, Birkhäuser Verlag, Basel, Boston, Stuttgart 1984

    Google Scholar 

  36. Henkin, G. and Leiterer, J.: “Andreotti-Grauert Theory by Integral Formulas”, Birkhäuser Verlag, Basal, Boston, Stuttgart 1988

    Google Scholar 

  37. Hörmander, L.: “An Introduction to Complex Analysis in Several Variables”, Van Nostrand, Princeton 1966

    Google Scholar 

  38. Horvath, J.: “Topological Vector Spaces and Distributions”, Addison-Wesley, Reading 1966

    Google Scholar 

  39. Hurewicz, W. and Waliman, H.: “Dimension Theory”, Princeton University Press, Princeton 1948

    Google Scholar 

  40. Jöricke, B.: Removable singularities for CR-functions. Ark. Mat. 26 (1988), 117–143

    Article  MathSciNet  MATH  Google Scholar 

  41. Jöricke, B.: Envelopes of holomorphy and CR-invariant subsets of CR-manifolds. (to appear)

    Google Scholar 

  42. Kytmanov, A. M.: Holomorphic continuation of integrable CR-functions from part of the boundary of a domain. Mat. Zametki 48 (2) (1990), 64–70

    MathSciNet  Google Scholar 

  43. Kytmanov,A. M.: Holomorphic extension of CR-function with singularities on a hypersurface. Izvestiya Akad. Nauk CCCP 54 No. 6 (1990), 1320–1330 (Russian), English translation Math. USSR Izvestiya 37 (1991), 681–691

    Article  MathSciNet  Google Scholar 

  44. Kytmanov, A. M. and Nikitina, T. N.: On the removable singularities of CR-functions given on a generic manifold. (to appear)

    Google Scholar 

  45. Laurent-Thiebaut, C.: Sur l’extension des fonctions CR dans une variété da Stein. Ann. Mat. Pura Appl.(IV) 150 (1988), 1–21

    Article  MathSciNet  Google Scholar 

  46. Laurent-Thiebaut, C. and Leiterer, J.: On the Hartogs-Bochner extension phenomenon for differential forms. Math. Ann. 284 (1989), 103–119

    Article  MathSciNet  MATH  Google Scholar 

  47. Lawrence, M.: Hulls of tame Cantor sets. (to appear)

    Google Scholar 

  48. Lupacciolu, G.: A theorem on holomorphic extension of CR-functions. Pacific J. Math. 124 (1986), 177–191

    MathSciNet  MATH  Google Scholar 

  49. Lupacciolu, G.: Holomorphic continuation in several complex variables. Pacific J. Math. 128 (1987), 117–125

    MathSciNet  MATH  Google Scholar 

  50. Lupacciolu, G.: On the removal of singular sets for the tangential Cauchy-Riemann operator. Arkiv for Mat. 28 (1990), 119–130

    Article  MathSciNet  MATH  Google Scholar 

  51. Lupacciolu, G.: Some global results on extensions of CR-objects in complex manifolds. Trans. Amer. Math. Soc. 321 (1990), 761–774

    Article  MathSciNet  MATH  Google Scholar 

  52. Lupacciolu, G.: Characterization of removable sets in strongly pseudoconvex boundaries. (to appear)

    Google Scholar 

  53. Lupacciolu, G.: On the envelopes of holomorphy of strictly Levi-convex hypersurfaces. (to appear)

    Google Scholar 

  54. Lupacciolu, G.: Topological properties of q-convex sets. Trans. Amer. Math. Soc. (to appear)

    Google Scholar 

  55. Lupacciolu, G.: Holomorphic and meromorphic q-hulls. (to appear)

    Google Scholar 

  56. Lupacciolu, G.: Approximation and cohomology vanishing properties of low-dimensional compact sets in a Stein manifold. Math. Z. 211 (1992), 523–532

    Article  MathSciNet  MATH  Google Scholar 

  57. L upacciolu, G. and Stout, E. L.: Removable singularities for ób. (to appear in the proceedings of the Mittag-Leffler special year on several complex variables)

    Google Scholar 

  58. Lupacciolu, G. and Tomassini, G.: Un teorema di estensione per le CR-functioni. Ann. Mat. Pura App. (IV) 137 (1984), 257–263

    Article  MathSciNet  MATH  Google Scholar 

  59. Malgrange, B.: Existence et approximation des solutions des équations aux dérivées partielle et des équations de convolution. Ann. Inst. Fourier (Grenoble) 6 (1955), 271–354

    Article  MathSciNet  Google Scholar 

  60. Narasimhan, R.: A note on Stein spaces and their normalizations. Ann. Scuola Norm. Sup. Pisa (3) 16 (1962), 327–333

    MathSciNet  MATH  Google Scholar 

  61. Narasimhan, R.: “Introduction to Analytic Spaces” in Springer Lecture Notes, vol. 25, Springer-Verlag, Berlin, Heidelberg, New York 1966

    Google Scholar 

  62. Rosay, J.-P. and Stout, E. L.: Radó’s theorem for CR-functions. Proc. Amer. Math. Soc. 106 (1989), 1017–1026

    MathSciNet  MATH  Google Scholar 

  63. Rossi, H.: The local maximum modulus principle. Ann. Math. (2) 72 (1960), 1–11

    Google Scholar 

  64. Rossi, H.: On envelopes of holomorphy. Comm. Pure Appl. Math. 16 (1963), 9–19

    MATH  Google Scholar 

  65. Rushing, T. B.: “Topological Embeddings”, Academic Press, New York 1973

    Google Scholar 

  66. Sadullaev, A. and Chirka, E. M.: On the continuation of functions with polar singularities. Mat. Sb. 132 (174) no. 3 (1987), 383–390 (Russian), English translation, Math. USSR Sbornik 60 (1988), 377–384

    Google Scholar 

  67. Schaefer, H. H.: “Topological Vector Spaces”, Springer-Verlag, New York, Heidelberg, Berlin 1971

    Google Scholar 

  68. Schneider, M.: Tubenumgebungen Steinscher Räume. Manuscripta Math. 18 (1976), 391–397

    Article  MathSciNet  MATH  Google Scholar 

  69. Schwartz, L.: “Théorie des Distributions”, Hermann, Paris 1966

    Google Scholar 

  70. Serre, J. P.: “Quelques problèmes globaux relatifs aux variétés de Stein” in Colloque sur les Fonctions de Plusieurs Variables Complex, Bruxelles, Mars, 1953; Georges Throne, Liege; Masson, Paris, 1953

    Google Scholar 

  71. Serre, J. P.: Un théorème de dualité. Comm. Math. Helv. 29 (1955), 9–26

    Article  MATH  Google Scholar 

  72. Shcherbina, N. V.: On fibering into analytic curves of the common boundary of two domains of holomorphy. Izv. Akad. Nauk SSSR 46 no. 5 (1982), 1106–1123 (Russian), English translation, Math. USSR Izvestiya 21 (1983) 399–413

    MATH  Google Scholar 

  73. Shcherbina, N. V.: The polynomial hull of a sphere embedded in C2. Mat. Zametki 49 (1991), 127–134 (Russian), English translation, Math. Notes, 49 (1991), 89–93

    Article  MathSciNet  MATH  Google Scholar 

  74. Shcherbina, N. V.: On the polynomial hull of a two-dimensional sphere in C2. Dokl. Akal. Nauk SSSR 306 (no. 6) (1991), 1315–1319

    MathSciNet  Google Scholar 

  75. Siu, Y.-T.: Every Stein subvariety admits a Stein neighborhood. Invent. Math. 38 (1976/1977), 89–100

    Google Scholar 

  76. Slodkowski, Z.: Analytic set-valued functions and spectra. Math. Ann. 256 (1981), 363–386

    Article  MathSciNet  MATH  Google Scholar 

  77. Slodkowski, Z.: Local maximum property and q-plurisubharmonic functions in uniform algebras. J. Math. Anal. Appl. 115 (1986), 105–130

    Article  MathSciNet  MATH  Google Scholar 

  78. Stolzenberg, G.: Uniform approximation on smooth curves. Acta. Math. 115 (1966), 185–198

    Article  MathSciNet  MATH  Google Scholar 

  79. Stout, E. L.: Analytic continuation and boundary continuity of functions of several complex variables. Proc. Royal Soc. Edinburgh 89 A (1981), 63–74

    Google Scholar 

  80. Stout, E. L.:Removable singularities for the boundary values of holomorphic functions. (to appear in the proceedings of the Mittag-Leffler special year in several complex variables)

    Google Scholar 

  81. Trépreau, J.-M.: Sur le prolongement holomorphe des fonctions C-R défines sur une hypersurface réele de class CE dans CN. Invent. Math. 83 (1986), 583–592

    Article  MathSciNet  MATH  Google Scholar 

  82. ), 951–964 (Russian), English translation, Math. USSR Sbornik 70 (1991), 385–398

    Google Scholar 

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Chirka, E.M., Stout, E.L. (1994). Removable singularities in the boundary. In: Skoda, H., Trépreau, JM. (eds) Contributions to Complex Analysis and Analytic Geometry / Analyse Complexe et Géométrie Analytique. Aspects of Mathematics, vol E 26. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-663-14196-9_3

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  • DOI: https://doi.org/10.1007/978-3-663-14196-9_3

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