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Transactions of the American Mathematical Society

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Analytic continuation, envelopes of holomorphy, and projective and direct limit spaces


Author: Robert Carmignani
Journal: Trans. Amer. Math. Soc. 209 (1975), 237-258
MSC: Primary 32D10
DOI: https://doi.org/10.1090/S0002-9947-1975-0385165-2
MathSciNet review: 0385165
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Abstract: For a Riemann domain $ \Omega $, a connected complex manifold where $ n(n = dimension)$ globally defined functions form a local system of coordinates at every point, and an arbitrary holomorphic function $ f$ in $ \Omega $, the ``Riemann surface'' $ {\Omega _f}$, a maximal holomorphic extension Riemann domain for $ f$, is formed from the direct limit of a sequence of Riemann domains. Projective limits are used to construct an envelope of holomorphy for $ \Omega $, a maximal holomorphic extension Riemann domain for all holomorphic functions in $ \Omega $, which is shown to be the projective limit space of the ``Riemann surfaces'' $ {\Omega _f}$. Then it is shown that the generalized notion of envelope of holomorphy of an arbitrary subset of a Riemann domain can also be characterized in a natural way as the projective limit space of a family of ``Riemann surfaces".


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  • [1] H. Behnke and P. Thullen, Theorie der Funktionen mehrerer komplexer Veränderlichen, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 51, Zwiete, erweiterte Auflage, Springer-Verlag, Berlin and New York, 1970. MR 42 #6274. MR 0271391 (42:6274)
  • [2] E. Bishop, Holomorphic completions, analytic continuation, and the interpolation of semi-norms, Ann. of Math. (2) 78 (1963), 468-500. MR 27 #4958. MR 0155016 (27:4958)
  • [3] H.-J. Bremermann, Construction of the envelopes of holomorphy of arbitrary domains, Rev. Mat. Hisp.-Amer. (4) 17 (1957), 175-200. MR 19, 880. MR 0090844 (19:880c)
  • [4] -, On the conjecture of the equivalence of the plurisubharmonic functions and the Hartogs functions, Math. Ann. 131 (1956), 76-86. MR 17, 1070. MR 0077644 (17:1070h)
  • [5] R. Carmignani, Envelopes of holomorphy and holomorphic convexity, Trans. Amer. Math. Soc. 179 (1973), 415-431. MR 47 #5296. MR 0316748 (47:5296)
  • [6] H. Cartan und P. Thullen, Zur theorie der Singularitäten der Funktionen mehrer Veränderlichen. Regularitäts-und Konvergenzbereiche, Math. Ann. 106 (1932), 617-647. MR 1512777
  • [7] Séminaires H. Cartan, École Normale Supérieure, 1951/52, Secrétariat mathématique, Paris, 1953. MR 16, 233.
  • [8] J. Dugundji, Topology, Allyn and Bacon, Boston, Mass., 1966. MR 33 #1824. MR 0193606 (33:1824)
  • [9] R. Gunning and H. Rossi, Analytic functions of several complex variables, Prentice-Hall Ser. in Modern Analysis, Prentice-Hall, Englewood Cliffs, N. J., 1965. MR 31 #4927. MR 0180696 (31:4927)
  • [10] R. Hartogs, Über die aus den singulären Stellen einer analytischen Funktion mehrerer Veränderlichen bestchenden Gebilde, Acta. Math. 32 (1909), 57-79. MR 1555046
  • [11] R. Harvey and R. O. Wells, Jr., Compact holomorphically convex subsets of a Stein manifold, Trans. Amer. Math. Soc. 136 (1969), 509-516. MR 38 #3470. MR 0235158 (38:3470)
  • [12] L. Hörmander, An introduction to complex analysis in several variables, Van Nostrand, Princeton, N. J., 1966. MR 34 #2933. MR 0203075 (34:2933)
  • [13] B. Malgrange, Lectures on the theory of functions of complex variables, Tata Institute of Fundamental Research, Bombay, 1958.
  • [14] K. Oka, Sur les fonctions analytiques de plusieurs variables. IX. Domaines finis sans point critique intérieur, Japan J. Math. 23 (1953), 97-155 (1954). MR 17, 82. MR 0071089 (17:82b)
  • [15] H. Rossi, On envelopes of holomorphy, Comm. Pure Appl. Math. 16 (1963), 9-17. MR 26 #6436. MR 0148940 (26:6436)
  • [16] P. Thullen, Zur Theorie der Singuläritäten der Funktionen zweier komplexer Veränderlichen. Die Regularitätshüllen, Math. Ann. 106 (1932), 64-76. MR 1512749
  • [17] V. S. Vladimirov, Methods of the theory of functions of several complex variables, ``Nauka", Moscow, 1964; English transl., M. I. T. Press, Cambridge, Mass., 1966. MR 30 #2163; 34 #1551.
  • [18] R. O. Wells, Jr., Function theory on differentiable submanifolds, Contribution to Analysis--A Collection of Papers Dedicated To Lipman Bers, Academic Press, New York and London, 1974, pp. 407-441. MR 0357856 (50:10322)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0385165-2
Keywords: Riemann domain, direct limit space, domain of holomorphy, projective limit, projective limit spaces, envelope of holomorphy, Stein manifold, holomorphically convex sets, convex hull
Article copyright: © Copyright 1975 American Mathematical Society

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