Families of holomorphic maps into Riemann surfaces
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- by Theodore J. Barth PDF
- Trans. Amer. Math. Soc. 207 (1975), 175-187 Request permission
Abstract:
In analogy with the Hartogs theorem that separate analyticity of a function implies analyticity, it is shown that a separately normal family of holomorphic maps from a polydisk into a Riemann surface is a normal family. This contrasts with examples of discontinuous separately analytic maps from a bidisk into the Riemann sphere. The proof uses a theorem on pseudoconvexity of normality domains, which is proved via the following convergence criterion: a sequence $\{ {f_j}\}$ of holomorphic maps from a complex manifold into a Riemann surface converges to a nonconstant holomorphic map if and only if the sequence $\{ f_j^{ - 1}\}$ of set-valued maps, defined on the Riemann surface, converges to a suitable set-valued map. Extending Osgood’s theorem, it is also shown that a separately analytic map (resp. a separately normal family of holomorphic maps) from a polydisk into a hyperbolic complex space is analytic (resp. normal).References
- H. Alexander, B. A. Taylor, and J. L. Ullman, Areas of projections of analytic sets, Invent. Math. 16 (1972), 335–341. MR 302935, DOI 10.1007/BF01425717
- Theodore J. Barth, Families of nonnegative divisors, Trans. Amer. Math. Soc. 131 (1968), 223–245. MR 219751, DOI 10.1090/S0002-9947-1968-0219751-3
- Theodore J. Barth, Normality domains for families of holomorphic maps, Math. Ann. 190 (1971), 293–297. MR 277753, DOI 10.1007/BF01431157
- Theodore J. Barth, The Kobayashi distance induces the standard topology, Proc. Amer. Math. Soc. 35 (1972), 439–441. MR 306545, DOI 10.1090/S0002-9939-1972-0306545-X
- James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
- Hans Grauert and Helmut Reckziegel, Hermitesche Metriken und normale Familien holomorpher Abbildungen, Math. Z. 89 (1965), 108–125 (German). MR 194617, DOI 10.1007/BF01111588
- Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0180696
- Fritz Hartogs, Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselben durch Reihen, welche nach Potenzen einer Veränderlichen fortschreiten, Math. Ann. 62 (1906), no. 1, 1–88 (German). MR 1511365, DOI 10.1007/BF01448415
- F. Hausdorff, Mengenlehre, Dover Publications, New York, N.Y., 1944 (German). MR 0015445
- Lars Hörmander, An introduction to complex analysis in several variables, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0203075
- Gaston Julia, Sur les familles de fonctions analytiques de plusieurs variables, Acta Math. 47 (1926), no. 1-2, 53–115 (French). MR 1555211, DOI 10.1007/BF02544108
- Shoshichi Kobayashi, Hyperbolic manifolds and holomorphic mappings, Pure and Applied Mathematics, vol. 2, Marcel Dekker, Inc., New York, 1970. MR 0277770
- Toshio Nishino, Sur une propriété des familles de fonctions analytiques de deux variables complexes, J. Math. Kyoto Univ. 4 (1965), 255–282 (French). MR 179384, DOI 10.1215/kjm/1250524660
- W. F. Osgood, Note über analytische Functionen mehrerer Veränderlichen, Math. Ann. 52 (1899), no. 2-3, 462–464 (German). MR 1511068, DOI 10.1007/BF01476172
- Wilhelm Stoll, Normal families of non-negative divisors, Math. Z. 84 (1964), 154–218. MR 165142, DOI 10.1007/BF01117123 B. L. van der Waerden, Modern Algebra. Vol. 1, Springer, Berlin, 1937; English transl., Ungar, New York, 1949. MR 10, 587.
- Hassler Whitney, Complex analytic varieties, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1972. MR 0387634
- H. Wu, Normal families of holomorphic mappings, Acta Math. 119 (1967), 193–233. MR 224869, DOI 10.1007/BF02392083
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 207 (1975), 175-187
- MSC: Primary 32A17; Secondary 32H20
- DOI: https://doi.org/10.1090/S0002-9947-1975-0374462-2
- MathSciNet review: 0374462