The traces of holomorphic functions on real submanifolds
HTML articles powered by AMS MathViewer
- by Gary Alvin Harris
- Trans. Amer. Math. Soc. 242 (1978), 205-223
- DOI: https://doi.org/10.1090/S0002-9947-1978-0477120-1
- PDF | Request permission
Abstract:
Suppose M is a real-analytic submanifold of complex Euclidean n = space and consider the following question: Given a real-analytic function f defined on M, is f the restriction to M of an ambient holomorphic function? If M is a C.R. submanifold the question has been answered completely. Namely, f is the trace of a holomorphic function if and only if f is a C.R. function. The more general situation in which M need not be a C.R. submanifold is discussed in this paper. A complete answer is obtained in case the dimension of M is larger than or equal to n and M is generic in some neighborhood of each point off its C.R. singularities. The solution is of infinite order and follows from a consideration of the following problem: Given a holomorphic function f and a holomorphic mapping $\Phi$, when does there exist a holomorphic mapping F such that $f = F \circ \Phi$?References
- Paul M. Eakin and Gary A. Harris, When $F(f)$ convergent implies $f$ is convergent, Math. Ann. 229 (1977), no. 3, 201–210. MR 444651, DOI 10.1007/BF01391465
- H. Grauert and R. Remmert, Analytische Stellenalgebren, Die Grundlehren der mathematischen Wissenschaften, Band 176, Springer-Verlag, Berlin-New York, 1971 (German). Unter Mitarbeit von O. Riemenschneider. MR 0316742, DOI 10.1007/978-3-642-65033-8
- Giuseppe Tomassini, Tracce delle funzioni olomorfe sulle sottovarietà analitiche reali d’una varietà complessa, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 20 (1966), 31–43 (Italian). MR 206992
Bibliographic Information
- © Copyright 1978 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 242 (1978), 205-223
- MSC: Primary 32C05
- DOI: https://doi.org/10.1090/S0002-9947-1978-0477120-1
- MathSciNet review: 0477120