The traces of holomorphic functions on real submanifolds
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 , when does there exist a holomorphic mapping F such that ?
- [E-H] Paul M. Eakin and Gary A. Harris, When 𝐹(𝑓) convergent implies 𝑓 is convergent, Math. Ann. 229 (1977), no. 3, 201–210. MR 0444651, https://doi.org/10.1007/BF01391465
- [G-R] H. Grauert and R. Remmert, Analytische Stellenalgebren, Springer-Verlag, Berlin-New York, 1971 (German). Unter Mitarbeit von O. Riemenschneider; Die Grundlehren der mathematischen Wissenschaften, Band 176. MR 0316742
- [T] Giuseppe Tomassini, Tracce delle funzioni olomorfe sulle sottovarietà analitiche reali d’una varietà complessa, Ann. Scuola Norm. Sup. Pisa (3) 20 (1966), 31–43 (Italian). MR 0206992
- Paul Eakin and Gary Harris, When convergent implies f is convergent, Math. Ann. 229 (1977), 211-221. MR 0444651 (56:3001)
- Hans Grauert and Reinhold Remmert, Analytische Stellenalgebren, Springer-Verlag, New York, 1970. MR 0316742 (47:5290)
- Giuseppe Tomassini, Trace delle funzioni olomorfe sulle sottovarieta analitiche reali d'una varieta complessa, Ann. Scuola Norm. Sup. Pisa 20 (1966), 31-43. MR 0206992 (34:6808)
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Keywords: Holomorphic trace, real-analytic submanifold of
Article copyright: © Copyright 1978 American Mathematical Society