On proper discs in complex manifolds

[Sur les disques holomorphes propres dans les variétés analytiques complexes]

Drinovec Drnovšek, Barbara

Annales de l'Institut Fourier, Tome 57 (2007), p. 1521-1535 / Harvested from Numdam

Résumé

Soit $X$ une variété analytique complexe de dimension au moins $2$ qui possède une fonction d’exhaustion telle que sa forme de Levi possède au moins $2$ valeurs propres strictement positives en tout point de $X$. On construit les disques holomorphes dans $X$ par n’importe quel point donné et dans n’importe quelle direction donnée.

Abstract

Let $X$ be a complex manifold of dimension at least $2$ which has an exhaustion function whose Levi form has at each point at least $2$ strictly positive eigenvalues. We construct proper holomorphic discs in $X$ through any given point and in any given direction.

Détail

Publié le : 2007-01-01

MR 2364140 | Zbl pre05214649

DOI : https://doi.org/10.5802/aif.2304
Classification:  32H35,  32C25
Mots clés: variété analytique complexe, disque holomorphe propre

Bibliographie

[1] Colţoiu, Mihnea $q$-convexity. A survey, Complex analysis and geometry (Trento, 1995), Longman, Harlow (Pitman Res. Notes Math. Ser.) Tome 366 (1997), pp. 83-93 | MR 1477441 | Zbl 0883.32016

[2] Dor, Avner A domain in ${\bf C}^m$ not containing any proper image of the unit disc, Math. Z., Tome 222 (1996), pp. 615-625 | Article | MR 1406270 | Zbl 0864.32018

[3] Drinovec-Drnovšek, Barbara; Forstnerič, Franc Holomorphic curves in complex spaces (to appear in Duke Math. J.) | Zbl 1133.32002

[4] Forstnerič, Franc; Globevnik, Josip Discs in pseudoconvex domains, Comment. Math. Helv., Tome 67 (1992), pp. 129-145 | Article | MR 1144617 | Zbl 0779.32016

[5] Forstnerič, Franc; Globevnik, Josip Proper holomorphic discs in $\mathbb{C}^2$, Math. Res. Lett., Tome 8 (2001), pp. 257-274 | MR 1839476 | Zbl 1027.32018

[6] Globevnik, Josip Discs in Stein manifolds, Indiana Univ. Math. J., Tome 49 (2000), pp. 553-574 | Article | MR 1793682 | Zbl 0974.32017

[7] Grauert, Hans Theory of $q$-convexity and $q$-concavity, Several complex variables, VII, Springer, Berlin (Encyclopaedia Math. Sci.) Tome 74 (1994), pp. 259-284 | MR 1326623 | Zbl 0806.32004

[8] Greene, Robert Everist; Wu, Hung Hsi Embedding of open Riemannian manifolds by harmonic functions, Ann. Inst. Fourier (Grenoble), Tome 25 (1975), pp. 215-235 | Article | Numdam | MR 382701 | Zbl 0307.31003

[9] Henkin, Gennadi M.; Leiterer, Jürgen Andreotti-Grauert theory by integral formulas, Birkhäuser Boston Inc., Boston, MA, Progress in Mathematics, Tome 74 (1988) | MR 986248 | Zbl 0654.32002

[10] Hörmander, Lars An introduction to complex analysis in several variables, North-Holland Publishing Co., Amsterdam (1973) (North-Holland Mathematical Library, Vol. 7) | MR 344507 | Zbl 0271.32001

[11] Lárusson, Finnur; Sigurdsson, Ragnar Plurisubharmonic functions and analytic discs on manifolds, J. Reine Angew. Math., Tome 501 (1998), pp. 1-39 | MR 1637837 | Zbl 0901.31004

[12] Rosay, Jean-Pierre Approximation of non-holomorphic maps, and Poletsky theory of discs, J. Korean Math. Soc., Tome 40 (2003), pp. 423-434 | Article | MR 1973910 | Zbl 1040.32015

[13] Rosay, Jean-Pierre Poletsky theory of disks on holomorphic manifolds, Indiana Univ. Math. J., Tome 52 (2003), pp. 157-169 | Article | MR 1970025 | Zbl 1033.31006

[14] Royden, H. L. The extension of regular holomorphic maps, Proc. Amer. Math. Soc., Tome 43 (1974), pp. 306-310 | Article | MR 335851 | Zbl 0292.32019

[15] Rudin, Walter Real and complex analysis, McGraw-Hill Book Co., New York (1987) | MR 924157 | Zbl 0925.00005