Approximation and accumulation results of holomorphic mappings with dense image
DOI:
https://doi.org/10.7146/math.scand.a-136450Abstract
We present four approximation theorems for manifold–valued mappings. The first one approximates holomorphic embeddings on pseudoconvex domains in $\mathbb{C}^n$ with holomorphic embeddings with dense images. The second theorem approximates holomorphic mappings on complex manifolds with bounded images with holomorphic mappings with dense images. The last two theorems work the other way around, constructing (in different settings) sequences of holomorphic mappings (embeddings in the first one) converging to a mapping with dense image defined on a given compact minus certain points (thus in general not holomorphic).
References
Alarcón, A., and Forstnerič, F., Complete densely embedded complex lines in $C^2$, Proc. Amer. Math. Soc. 146 (2018), no. 3, 1059–1067. https://doi.org/10.1090/proc/13873
Deng, F., Fornæss, J. E., and Wold, E. F., Exposing boundary points of strongly pseudoconvex subvarieties in complex spaces, Proc. Amer. Math. Soc. 146 (2018), no. 6, 2473–2487. https://doi.org/10.1090/proc/13693
Fornæss, J. E., and Stout, E. L., Polydiscs in complex manifolds., Math. Ann. 227 (1977), no. 2, 145–153. https://doi.org/10.1007/BF01350191
Forstnerič, F., Stein manifolds and holomorphic mappings. The homotopy principle in complex analysis, Second edition, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 56. Springer, Cham, 2017. https://doi.org/10.1007/978-3-319-61058-0
Forstnerič, F., and Winkelmann, J., Holomorphic discs with dense images. Math. Res. Lett. 12 (2005), no. 2–3, 265–268. https://doi.org/10.4310/MRL.2005.v12.n2.a11
Hörmander, L., An introduction to complex analysis in several variables, Third edition, North-Holland Publishing Co., Amsterdam, 1990.
Range, R. M., Holomorphic functions and integral representations in several complex variables, Graduate Text in Mathematics 108, Springer-Verlag, New York, 1986. https://doi.org/10.1007/978-1-4757-1918-5
Serre, J. P., Un théorème de dualité, Comment. Math. Helv. 29 (1955), 9–26. https://doi.org/10.1007/BF02564268