The ball embedding property of the open unit disc

Borell, Stefan

doi:10.1090/S0002-9939-2011-10798-6

The ball embedding property of the open unit disc
HTML articles powered by AMS MathViewer

by Stefan Borell PDF

Proc. Amer. Math. Soc. 139 (2011), 3573-3581 Request permission

Abstract:

We prove that the open unit disc $\triangle$ in $\mathbb {C}$ satisfies the ball embedding property in $\mathbb {C}^2$; i.e., given any discrete set of discs in $\mathbb {C}^2$ there exists a proper holomorphic embedding $\triangle \hookrightarrow \mathbb {C}^2$ which passes arbitrarily close to the discs. It is already known that $\mathbb {C}$ does not satisfy the ball embedding property in $\mathbb {C}^2$ and that $\triangle$ satisfies the ball embedding property in $\mathbb {C}^n$ for $n>2$.

References

Shreeram S. Abhyankar and Tzuong Tsieng Moh, Embeddings of the line in the plane, J. Reine Angew. Math. 276 (1975), 148–166. MR 379502
Stefan Borell and Frank Kutzschebauch, Non-equivalent embeddings into complex Euclidean spaces, Internat. J. Math. 17 (2006), no. 9, 1033–1046. MR 2274009, DOI 10.1142/S0129167X06003795
Stefan Borell and Frank Kutzschebauch, Embeddings through discrete sets of balls, Ark. Mat. 46 (2008), no. 2, 251–269. MR 2430726, DOI 10.1007/s11512-008-0079-8
Gregery T. Buzzard and John Erik Fornæss, An embedding of $\mathbf C$ in $\mathbf C^2$ with hyperbolic complement, Math. Ann. 306 (1996), no. 3, 539–546. MR 1415077, DOI 10.1007/BF01445264
Donald A. Eisenman, Intrinsic measures on complex manifolds and holomorphic mappings, Memoirs of the American Mathematical Society, No. 96, American Mathematical Society, Providence, R.I., 1970. MR 0259165
Franc Forstneric, Interpolation by holomorphic automorphisms and embeddings in $\textbf {C}^n$, J. Geom. Anal. 9 (1999), no. 1, 93–117. MR 1760722, DOI 10.1007/BF02923090
Franc Forstneric, Josip Globevnik, and Jean-Pierre Rosay, Nonstraightenable complex lines in $\mathbf C^2$, Ark. Mat. 34 (1996), no. 1, 97–101. MR 1396625, DOI 10.1007/BF02559509
Franc Forstnerič and Jean-Pierre Rosay, Approximation of biholomorphic mappings by automorphisms of $\textbf {C}^n$, Invent. Math. 112 (1993), no. 2, 323–349. MR 1213106, DOI 10.1007/BF01232438
Reinhold Remmert, Sur les espaces analytiques holomorphiquement séparables et holomorphiquement convexes, C. R. Acad. Sci. Paris 243 (1956), 118–121 (French). MR 79808
Jean-Pierre Rosay and Walter Rudin, Holomorphic maps from $\textbf {C}^n$ to $\textbf {C}^n$, Trans. Amer. Math. Soc. 310 (1988), no. 1, 47–86. MR 929658, DOI 10.1090/S0002-9947-1988-0929658-4
Jean-Pierre Rosay and Walter Rudin, Holomorphic embeddings of $\mathbf C$ in $\mathbf C^n$, Several complex variables (Stockholm, 1987/1988) Math. Notes, vol. 38, Princeton Univ. Press, Princeton, NJ, 1993, pp. 563–569. MR 1207881
Masakazu Suzuki, Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l’espace $\textbf {C}^{2}$, J. Math. Soc. Japan 26 (1974), 241–257 (French). MR 338423, DOI 10.2969/jmsj/02620241
Erlend Fornæss Wold, Fatou-Bieberbach domains, Internat. J. Math. 16 (2005), no. 10, 1119–1130. MR 2182211, DOI 10.1142/S0129167X05003235