Riemann surfaces in Stein manifolds with the Density property

Andrist, Rafael B.; Wold, Erlend Fornæss

doi:10.5802/aif.2862

Riemann surfaces in Stein manifolds with the Density property
[Surfaces de Riemann dans les variétés de Stein avec Density property]

Andrist, Rafael B.¹ ; Wold, Erlend Fornæss ²

¹ Bergische Universität Wuppertal, Fachbereich C - Mathematik und Naturwissenschaften, Gaußstraße 20, 42119 Wuppertal, Germany
² Matematisk Institutt, Universitetet i Oslo, Postboks 1053 Blindern, 0316 Oslo, Norway

Annales de l'Institut Fourier, Tome 64 (2014) no. 2, pp. 681-697.

Résumé
Abstract

Nous montrons que toute surface de Riemann ouverte peut être immergée proprement dans toute variété de Stein avec la propriété de densité (volumique) et de dimension au moins 2. Si la dimension est au moins 3, cette immersion est en fait un plongement propre. Les résultats obtenus sont appliqués pour montrer que toutes les variétés de Stein avec la propriété de densité (volumique) et de dimension au moins 3 sont caractérisées entre toutes les variétés complexes par leur demi-groupe d’endomorphismes holomorphes.

We show that any open Riemann surface can be properly immersed in any Stein manifold with the (Volume) Density property and of dimension at least 2. If the dimension is at least 3, we can actually choose this immersion to be an embedding. As an application, we show that Stein manifolds with the (Volume) Density property and of dimension at least 3, are characterized among all other complex manifolds by their semigroup of holomorphic endomorphisms.

MR Zbl

DOI : 10.5802/aif.2862

Classification : 32H02, 32E30, 20M20
Keywords: Riemann surface, Stein manifold, proper holomorphic map, Andersen-Lempert theory, Density property, Volume Density property
Mot clés : surface de Riemann, variété de Stein, application holomorphe propre, théorie de Andersen-Lempert, Density property, Volume Density property

Affiliations des auteurs :

Andrist, Rafael B. ¹ ; Wold, Erlend Fornæss ²

@article{AIF_2014__64_2_681_0,
     author = {Andrist, Rafael B. and Wold, Erlend Forn{\ae}ss},
     title = {Riemann surfaces in {Stein} manifolds with the {Density} property},
     journal = {Annales de l'Institut Fourier},
     pages = {681--697},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {64},
     number = {2},
     year = {2014},
     doi = {10.5802/aif.2862},
     zbl = {06387289},
     mrnumber = {3330919},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2862/}
}

TY  - JOUR
AU  - Andrist, Rafael B.
AU  - Wold, Erlend Fornæss
TI  - Riemann surfaces in Stein manifolds with the Density property
JO  - Annales de l'Institut Fourier
PY  - 2014
SP  - 681
EP  - 697
VL  - 64
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2862/
DO  - 10.5802/aif.2862
LA  - en
ID  - AIF_2014__64_2_681_0
ER  -

%0 Journal Article
%A Andrist, Rafael B.
%A Wold, Erlend Fornæss
%T Riemann surfaces in Stein manifolds with the Density property
%J Annales de l'Institut Fourier
%D 2014
%P 681-697
%V 64
%N 2
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2862/
%R 10.5802/aif.2862
%G en
%F AIF_2014__64_2_681_0

Andrist, Rafael B.; Wold, Erlend Fornæss. Riemann surfaces in Stein manifolds with the Density property. Annales de l'Institut Fourier, Tome 64 (2014) no. 2, pp. 681-697. doi : 10.5802/aif.2862. http://www.numdam.org/articles/10.5802/aif.2862/

Bibliographie
Cité par

[1] Alarcón, A.; Galvéz, J. A. Proper harmonic maps from hyperbolic Riemann surfaces into the Euclidean plane, Results Math., Volume 60 (2011) no. 1-4, pp. 487-505 | DOI | MR | Zbl

[2] Alarcon, A.; López, F. J. Minimal surfaces in $ℝ^{3}$ properly projecting into $ℝ^{2}$ , J. Differential Geom., Volume 90 (2012) no. 3, pp. 351-381 | MR | Zbl

[3] Alexander, H.; Wermer, J. Several complex variables and Banach algebras, Graduate Texts in Mathematics, 35, Springer-Verlag, New York, 1998 | MR | Zbl

[4] Andersén, E. Volume-preserving automorphisms of $ℂ^{n}$ , Complex Variables Theory Appl., Volume 14 (1990) no. 1-4, pp. 223-235 | DOI | MR | Zbl

[5] Andersén, E.; Lempert, L. On the group of holomorphic automorphisms of $ℂ^{n}$ , Invent. Math., Volume 10 (1992) no. 2, pp. 371-388 | DOI | MR | Zbl

[6] Andrist, R. B. Stein Spaces Characterized by their Endomorphisms, Tran. AMS, Volume 363 (2011), pp. 2341-2355 | DOI | MR | Zbl

[7] Bishop, E. Mappings of partially analytic spaces, Amer. J. Math., Volume 83 (1961), pp. 209-242 | DOI | MR | Zbl

[8] Buzzard, G. T.; Merenkov, A. Maps Conjugating Holomorphic Maps in $ℂ^{n}$ , Indiana Univ. Math. J., Volume 52 (2003) no. 5, pp. 1135-1146 | DOI | MR | Zbl

[9] Drinovec-Drnovšek, B.; Forstnerič, F. Holomorphic curves in complex spaces, Duke Math. J., Volume 139 (2007) no. 2, pp. 203-254 | DOI | MR | Zbl

[10] Drinovec-Drnovšek, B.; Forstnerič, F. Approximation of Holomorphic Mappings on Strongly Pseudoconvex Domains, Forum Mathematicum, Volume 20 (2008) no. 5, pp. 817-840 | DOI | MR | Zbl

[11] Drinovec-Drnovšek, B.; Forstnerič, F. Strongly Pseudoconvex Domains as Subvarieties of Complex Manifolds, American Journal of Mathematics, Volume 132 (2010) no. 2, pp. 331-360 | DOI | MR | Zbl

[12] Forster, O. Lectures on Riemann surfaces, Springer, 1999 | MR | Zbl

[13] Forster, O.; Ramspott, K. J. Analytische Modulgarben und Endromisbündel, Invent. Math., Volume 2 (1966) no. 2, pp. 145-170 | DOI | MR | Zbl

[14] Forstnerič, F. Noncritical Holomorphic Functions on Stein Manifolds, Acta. Math., Volume 191 (2003), pp. 143-189 | DOI | MR | Zbl

[15] Forstnerič, F. Extending holomorphic mappings from subvarieties in Stein manifolds, Ann. Inst. Fourier (Grenoble), Volume 55 (2005), pp. 733-751 | DOI | Numdam | MR | Zbl

[16] Forstnerič, F. Runge approximation on convex sets implies the Oka property, Ann. Math. (2), Volume 163 (2006), pp. 689-707 | DOI | MR | Zbl

[17] Forstnerič, F. Oka manifolds, C. R. Math. Acad. Sci. Paris, Volume 347 (2009), pp. 1017-1020 | DOI | MR | Zbl

[18] Forstnerič, F. Stein manifolds and holomorphic mappings. The homotopy principle in Complex Analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer, 2011 | MR | Zbl

[19] Forstnerič, F.; Globevnik, J. Proper holomorphic discs in $ℂ^{2}$ , Math. Res. Letters, Volume 8 (2001), pp. 257-274 | DOI | MR | Zbl

[20] Forstnerič, F.; Rosay, J.-P. Approximation of biholomorphic mappings by automorphisms of $ℂ^{n}$ , Invent. Math., Volume 112 (1993) no. 2, pp. 323-349 | DOI | MR | Zbl

[21] Gilligan, B.; Huckleberry, A.T. Complex homogeneous manifolds with two ends, Mich. J. Math., Volume 28 (1981), pp. 183-198 | DOI | MR | Zbl

[22] Gromov, M. Oka’s principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc., Volume 2 (1989), pp. 851-897 | MR | Zbl

[23] Henkin, G.; Leiterer, J. The Oka-Grauert principle without induction over the base dimension, Math. Ann., Volume 311 (1998) no. 2, pp. 71-93 | DOI | MR | Zbl

[24] Kaliman, Sh.; Kutschebauch, F. On the present state of the Andersén-Lempert theory, Affine algebraic geometry (CRM Proc. Lecture Notes), Volume 54, Amer. Math. Soc., Providence, RI, 2011, pp. 85-122 | Zbl

[25] Narasimhan, R. Imbedding of holomorphically complete complex spaces, Amer. J. Math., Volume 82 (1960), pp. 917-934 | DOI | MR | Zbl

[26] Remmert, R. Sur les espaces analytiques holomorphiquement séparables et holomorphiquement convexes, C. R. Acad. Sci. Paris, Volume 243 (1956), pp. 118-121 | MR | Zbl

[27] Schoen, R.; Yau, S. Lectures on harmonic maps, Conference Proceedings and Lecture Notes in Geometry and Topology, International Press, Cambridge MA, 1997 | MR | Zbl

[28] Schreier, J. Über Abbildungen einer abstrakten Menge auf ihre Teilmengen, Fund. Math, Volume 28 (1937), pp. 261-264 | Zbl

[29] Serre, J.-P. Quelques problèmes globaux relatifs aux varietés de Stein, Colloque sur les fonctions de plusieurs variables, Bruxelles 1953, Masson, Paris, 1953, pp. 57-68 | Zbl

[30] Siu, Y. T. Every Stein subvariety admits a Stein neighborhood, Invent. Math., Volume 38 (1976) no. 1, pp. 89-100 | DOI | MR | Zbl

[31] Varolin, D. The Density Property for Complex Manifolds and Geometric Structures II, Internat. J. Math., Volume 11 (2000) no. 6, pp. 837-847 | DOI | MR | Zbl

[32] Varolin, D. The Density Property for Complex Manifolds and Geometric Structures, J. Geom. Anal., Volume 11 (2001), pp. 135-160 | DOI | MR | Zbl

Cité par Sources :