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Complete densely embedded complex lines in $ \mathbb{C}^2$


Authors: Antonio Alarcón and Franc Forstnerič
Journal: Proc. Amer. Math. Soc. 146 (2018), 1059-1067
MSC (2010): Primary 32H02; Secondary 32E10, 32M17, 53A10
DOI: https://doi.org/10.1090/proc/13873
Published electronically: November 10, 2017
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Abstract: In this paper we construct a complete injective holomorphic immersion $ \mathbb{C}\to \mathbb{C}^2$ whose image is dense in $ \mathbb{C}^2$. The analogous result is obtained for any closed complex submanifold $ X\subset \mathbb{C}^n$ for $ n>1$ in place of $ \mathbb{C}\subset \mathbb{C}^2$. We also show that if $ X$ intersects the unit ball $ \mathbb{B}^n$ of $ \mathbb{C}^n$ and $ K$ is a connected compact subset of $ X\cap \mathbb{B}^n$, then there is a Runge domain $ \Omega \subset X$ containing $ K$ which admits a complete injective holomorphic immersion $ \Omega \to \mathbb{B}^n$ whose image is dense in $ \mathbb{B}^n$.


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  • [1] Antonio Alarcón and Ildefonso Castro-Infantes, Complete minimal surfaces densely lying in arbitrary domains of $ \mathbb{R}^n$, Geom. Topol., in press.
  • [2] Antonio Alarcón, Isabel Fernández, and Francisco J. López, Harmonic mappings and conformal minimal immersions of Riemann surfaces into $ \mathbb{R}^{\rm N}$, Calc. Var. Partial Differential Equations 47 (2013), no. 1-2, 227-242. MR 3044138, https://doi.org/https://doi.org/10.1007/s00526-012-0517-0
  • [3] Antonio Alarcón and Josip Globevnik, Complete embedded complex curves in the ball of $ \Bbb{C}$2 can have any topology, Anal. PDE 10 (2017), no. 8, 1987-1999. MR 3694012, https://doi.org/https://doi.org/10.2140/apde.2017.10.1987
  • [4] Antonio Alarcón, Josip Globevnik, and Francisco J. López, A construction of complete complex hypersurfaces in the ball with control on the topology, J. Reine Angew. Math., in press. Online first version available at https://doi.org/10.1515/crelle-2016-0061.
  • [5] Antonio Alarcón and Francisco J. López, Proper holomorphic embeddings of Riemann surfaces with arbitrary topology into $ \mathbb{C}^2$, J. Geom. Anal. 23 (2013), no. 4, 1794-1805. MR 3107678, https://doi.org/https://doi.org/10.1007/s12220-012-9306-4
  • [6] Antonio Alarcón and Francisco J. López, Complete bounded embedded complex curves in $ \Bbb {C}^2$, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 8, 1675-1705. MR 3519537
  • [7] Rafael Andrist, Franc Forstnerič, Tyson Ritter, and Erlend Fornæss Wold, Proper holomorphic embeddings into Stein manifolds with the density property, J. Anal. Math. 130 (2016), 135-150. MR 3574650, https://doi.org/https://doi.org/10.1007/s11854-016-0031-y
  • [8] Miran Černe and Franc Forstnerič, Embedding some bordered Riemann surfaces in the affine plane, Math. Res. Lett. 9 (2002), no. 5-6, 683-696. MR 1906070, https://doi.org/https://doi.org/10.4310/MRL.2002.v9.n5.a10
  • [9] Tobias H. Colding and William P. Minicozzi II, The Calabi-Yau conjectures for embedded surfaces, Ann. of Math. (2) 167 (2008), no. 1, 211-243. MR 2373154, https://doi.org/https://doi.org/10.4007/annals.2008.167.211
  • [10] H. M. Farkas and I. Kra, Riemann surfaces, 2nd ed., Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York, 1992. MR 1139765
  • [11] Francisco Fontenele and Frederico Xavier, A Riemannian Bieberbach estimate, J. Differential Geom. 85 (2010), no. 1, 1-14. MR 2719407
  • [12] Franc Forstnerič, Stein manifolds and holomorphic mappings: The homotopy principle in complex analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 56, Springer, Heidelberg, 2011. MR 2975791
  • [13] Franc Forstneric, Josip Globevnik, and Berit Stensønes, Embedding holomorphic discs through discrete sets, Math. Ann. 305 (1996), no. 3, 559-569. MR 1397436, https://doi.org/https://doi.org/10.1007/BF01444237
  • [14] Franc Forstnerič and Jean-Pierre Rosay, Approximation of biholomorphic mappings by automorphisms of $ {\bf C}^n$, Invent. Math. 112 (1993), no. 2, 323-349. MR 1213106, https://doi.org/https://doi.org/10.1007/BF01232438
  • [15] Franc Forstnerič and Erlend Fornæss Wold, Embeddings of infinitely connected planar domains into $ {\mathbb{C}}^2$, Anal. PDE 6 (2013), no. 2, 499-514. MR 3071396, https://doi.org/https://doi.org/10.2140/apde.2013.6.499
  • [16] Josip Globevnik, A complete complex hypersurface in the ball of $ \mathbb{C}^N$, Ann. of Math. (2) 182 (2015), no. 3, 1067-1091. MR 3418534, https://doi.org/https://doi.org/10.4007/annals.2015.182.3.4
  • [17] Josip Globevnik, Embedding complete holomorphic discs through discrete sets, J. Math. Anal. Appl. 444 (2016), no. 2, 827-838. MR 3535737, https://doi.org/https://doi.org/10.1016/j.jmaa.2016.06.053
  • [18] William H. Meeks III, Joaquín Pérez, and Antonio Ros, The embedded Calabi-Yau conjectures for finite genus, preprint, http://www.ugr.es/local/jperez/papers/papers.htm.
  • [19] Dror Varolin, The density property for complex manifolds and geometric structures, J. Geom. Anal. 11 (2001), no. 1, 135-160. MR 1829353, https://doi.org/https://doi.org/10.1007/BF02921959
  • [20] Paul Yang, Curvature of complex submanifolds of $ C^{n}$, Several complex variables (Proc. Sympos. Pure Math., Vol. XXX, Part 2, Williams Coll., Williamstown, Mass., 1975) Amer. Math. Soc., Providence, R.I., 1977, pp. 135-137. MR 0450606
  • [21] Kentaro Yano, Anti-invariant submanifolds of a Sasakian manifold with vanishing contact Bochner curvature tensor, J. Differential Geometry 12 (1977), no. 2, 153-170. MR 0487884

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Additional Information

Antonio Alarcón
Affiliation: Departamento de Geometría y Topología e Instituto de Matemáticas (IEMath-GR), Universidad de Granada, Campus de Fuentenueva s/n, E–18071 Granada, Spain
Email: alarcon@ugr.es

Franc Forstnerič
Affiliation: Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI–1000 Ljubljana, Slovenia—and—Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI–1000 Ljubljana, Slovenia.
Email: franc.forstneric@fmf.uni-lj.si

DOI: https://doi.org/10.1090/proc/13873
Received by editor(s): February 25, 2017
Published electronically: November 10, 2017
Additional Notes: The first author was supported by the Ramón y Cajal program of the Spanish Ministry of Economy and Competitiveness and by the MINECO/FEDER grant No. MTM2014-52368-P, Spain.
The second author was partially supported by the research grants P1-0291 and J1-7256 from ARRS, Republic of Slovenia.
Communicated by: Filippo Bracci
Article copyright: © Copyright 2017 American Mathematical Society

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