Abstract
Let n > 1 and let C n denote the complex n-dimensional Euclidean space. We prove several jet-interpolation results for nowhere degenerate entire mappings F:C n →C n and for holomorphic automorphisms of C n on discrete subsets of C n.We also prove an interpolation theorem for proper holomorphic embeddings of Stein manifolds into C n.For each closed complex submanifold (or subvariety) M ⊂ C n of complex dimension m < n we construct a domain Ω ⊂C n containing M and a biholomorphic map F: Ω → C n onto C n with J F ≡ 1such that F(M) intersects the image of any nondegenerate entire map G:C n−m →C n at infinitely many points. If m = n − 1, we construct F as above such that C n ∖F(M) is hyperbolic. In particular, for each m ≥ 1we construct proper holomorphic embeddings F:C m →C m−1 such that the complement C m+1 ∖F(C m)is hyperbolic.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andersén, E. Volume-preserving automorphisms of Cn,Complex Variables,14, 223–235, (1990).
Andersén, E. and Lempert, L. On the group of holomorphic automorphisms ofC n,Invent. Math.,110, 371–388, (1992).
Buzzard, G. Kupka-Smale Theorem for Automorphisms ofC n,Duke Math. J.,93, 487–503, (1998).
Buzzard, G. and Fornæss, J.E. An embedding ofC intoC 2 with hyperbolic complement,Math. Ann.,306, 539–546, (1996).
Buzzard, G. and Forstneric, F. An interpolation theorem for holomorphic automorphisms ofC n. Preprint, 1996.
Chirka, E.Complex Analytic Sets, Kluwer, Dordrecht, 1989.
Eliashberg, Y. and Gromov, M. Embeddings of Stein manifolds of dimensionn into the affine space of dimension3n/2 + 1,Ann. Math.,136(2), 123–135, (1992).
Forstneric, F. Approximation by automorphisms on smooth submanifolds ofC n,Math. Ann.,300, 719–738, (1994).
Forstneric, F. Equivalence of real submanifolds under volume preserving holomorphic automorphisms ofC n,Duke Math. J.,77, 431–445, (1995).
Forstneric, F. Holomorphic automorphism groups ofC n: A survey.The Proceedings Complex Analysis and Geometry, Ancona, V., Ballico, E., and Silva, A., Eds., 173–200,Lecture Notes in Pure and Applied Mathematics,173, Marcel Dekker, New York, 1996.
Forstneric, F., Globevnik, J., and Rosay, J.-P. Non straightenable complex lines inC 2,Arkiv Math,34, 97–101, (1996).
Forstneric, F., Globevnik, J., and Stensønes, B. Embedding holomorphic discs through discrete sets,Math. Ann.,304, 559–596, (1995).
Forstneric, F. and Rosay, J.-P. Approximation of biholomorphic mappings by automorphisms ofC n,Invent. Math.,112, 323–349, (1993). Correction,Invent. Math.,118, 573–574, (1994).
Globevnik, J. and Stensønes, B. Holomorphic embeddings of planar domains intoC 2,Math. Ann.,303, 579–597, (1995).
Hörmander, L.An Introduction to Complex Analysis in Several Variables, 3rd ed., North Holland, Amsterdam, 1990.
Jacobson, N.Basic Algebra I, Freeman, San Francisco, 1974.
Rosay, J.-P. and Rudin, W. Holomorphic maps fromC n toC n,Trans. Am. Math. Soc.,310, 47–86, (1988).
Rossi, H. The local maximum modulus principle,Ann. Math.,72(2), 1–11, (1960).
Stolzenberg, G. Polynomially and rationally convex sets,Acta Math.,109, 259–289, (1963).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Forstneric, F. Interpolation by holomorphic automorphisms and embeddings in Cn . J Geom Anal 9, 93–117 (1999). https://doi.org/10.1007/BF02923090
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02923090