Envelopes of holomorphy and holomorphic convexity
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- by Robert Carmignani
- Trans. Amer. Math. Soc. 179 (1973), 415-431
- DOI: https://doi.org/10.1090/S0002-9947-1973-0316748-1
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Abstract:
This paper is primarily a study of generalized notions of envelope of holomorphy and holomorphic convexity for special (algebraically restricted) subsets of ${{\mathbf {C}}^n}$ and in part for arbitrary subsets of ${{\mathbf {C}}^n}$. For any special set S in ${{\mathbf {C}}^n}$, we show that every function holomorphic in a neighborhood of S not only can be holomorphically continued but also holomorphically extended to a neighborhood in ${{\mathbf {C}}^n}$ of a maximal set $\tilde {S}$, the “envelope of holomorphy” of S, which is also a special set of the same type as S. Formulas are obtained for constructing $\tilde {S}$ for any special set S. “Holomorphic convexity” is characterized for these special sets. With one exception, the only topological restriction on these special sets is connectivity. Examples are given which illustrate applications of the theorems and help to clarify the concepts of “envelope of holomorphy” and “holomorphic convexity."References
- Salomon Bochner and William Ted Martin, Several Complex Variables, Princeton Mathematical Series, vol. 10, Princeton University Press, Princeton, N. J., 1948. MR 0027863
- H. J. Bremermann, Complex convexity, Trans. Amer. Math. Soc. 82 (1956), 17–51. MR 79100, DOI 10.1090/S0002-9947-1956-0079100-2
- H. J. Bremermann, Construction of the envelopes of holomorphy of arbitrary domains, Rev. Mat. Hisp.-Amer. (4) 17 (1957), 175–200. MR 90844 H. Cartan, Séminaires École Normale Supérieure, 1951/52, Secrétariat mathématique, Paris, 1953. MR 16, 233.
- Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR 0410387
- Reese Harvey and R. O. Wells Jr., Compact holomorphically convex subsets of a Stein manifold, Trans. Amer. Math. Soc. 136 (1969), 509–516. MR 235158, DOI 10.1090/S0002-9947-1969-0235158-8
- Lars Hörmander, An introduction to complex analysis in several variables, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0203075 B. Malgrange, Lectures on the theory of functions of complex variables, Tata Institute of Fundamental Research, Bombay, 1958.
- Kiyoshi Oka, Sur les fonctions analytiques de plusieurs variables. IX. Domaines finis sans point critique intérieur, Jpn. J. Math. 23 (1953), 97–155 (1954) (French). MR 71089, DOI 10.4099/jjm1924.23.0_{9}7
- Hugo Rossi, On envelopes of holomorphy, Comm. Pure Appl. Math. 16 (1963), 9–17. MR 148940, DOI 10.1002/cpa.3160160103
- Peter Thullen, Zur Theorie der Singularitäten der Funktionen zweier komplexen Veränderlichen, Math. Ann. 106 (1932), no. 1, 64–76 (German). MR 1512749, DOI 10.1007/BF01455877 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; MR 34 #1551.
Bibliographic Information
- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 179 (1973), 415-431
- MSC: Primary 32D10; Secondary 32E05, 32E30
- DOI: https://doi.org/10.1090/S0002-9947-1973-0316748-1
- MathSciNet review: 0316748