Schwarz-type lemmas for solutions of ¯-inequalities and complete hyperbolicity of almost complex manifolds
Annales de l'Institut Fourier, Volume 54 (2004) no. 7, pp. 2387-2435.

The definition of the Kobayashi-Royden pseudo-metric for almost complex manifolds is similar to its definition for complex manifolds. We study the question of completeness of some domains for this metric. In particular, we study the completeness of the complement of submanifolds of co-dimension 1 or 2. The paper includes a discussion, with proofs, of basic facts in the theory of pseudo-holomorphic discs.

La pseudo-métrique de Kobayashi-Royden est définie pour les variétés presque complexes de façon similaire au cas complexe. Nous étudions quels domaines sont complets pour cette métrique, en particulier nous étudions le complément de sous variétés de co-dimension 1 ou 2. Le papier inclut une discussion, avec preuves, de faits à la base de la théorie des disques pseudo-holomorphes.

DOI: 10.5802/aif.2084
Classification: 32Q60,  32Q65,  32Q45
Keywords: Kobayashi-Royden pseudo-norm, almost complex manifolds, Schwarz Lemmas, complete hyperbolicity
Ivashkovich, Sergey 1; Rosay, Jean-Pierre 

1 Université Lille I, département de Mathématiques, 59655 Villeneuve d'Ascq Cedex (France), University of Wisconsin, department of Mathematics, Madison WI 53706 (USA)
@article{AIF_2004__54_7_2387_0,
     author = {Ivashkovich, Sergey and Rosay, Jean-Pierre},
     title = {Schwarz-type lemmas for solutions of $\bar{\partial }$-inequalities and complete hyperbolicity of almost complex manifolds},
     journal = {Annales de l'Institut Fourier},
     pages = {2387--2435},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {54},
     number = {7},
     year = {2004},
     doi = {10.5802/aif.2084},
     zbl = {1072.32007},
     mrnumber = {2139698},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2084/}
}
TY  - JOUR
AU  - Ivashkovich, Sergey
AU  - Rosay, Jean-Pierre
TI  - Schwarz-type lemmas for solutions of $\bar{\partial }$-inequalities and complete hyperbolicity of almost complex manifolds
JO  - Annales de l'Institut Fourier
PY  - 2004
DA  - 2004///
SP  - 2387
EP  - 2435
VL  - 54
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2084/
UR  - https://zbmath.org/?q=an%3A1072.32007
UR  - https://www.ams.org/mathscinet-getitem?mr=2139698
UR  - https://doi.org/10.5802/aif.2084
DO  - 10.5802/aif.2084
LA  - en
ID  - AIF_2004__54_7_2387_0
ER  - 
%0 Journal Article
%A Ivashkovich, Sergey
%A Rosay, Jean-Pierre
%T Schwarz-type lemmas for solutions of $\bar{\partial }$-inequalities and complete hyperbolicity of almost complex manifolds
%J Annales de l'Institut Fourier
%D 2004
%P 2387-2435
%V 54
%N 7
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2084
%R 10.5802/aif.2084
%G en
%F AIF_2004__54_7_2387_0
Ivashkovich, Sergey; Rosay, Jean-Pierre. Schwarz-type lemmas for solutions of $\bar{\partial }$-inequalities and complete hyperbolicity of almost complex manifolds. Annales de l'Institut Fourier, Volume 54 (2004) no. 7, pp. 2387-2435. doi : 10.5802/aif.2084. http://www.numdam.org/articles/10.5802/aif.2084/

[B-M] J-F. Barraud; E. Mazzilli Regular type of real hypersurfaces in (almost) complex manifolds (2003) (e-print, ArXiv:math.DG/0304146) | Zbl

[Ba] V. Bangert Existence of a complex line in tame almost complex tori, Duke Math. J, Volume 94 (1998) no. 1, pp. 29-40 | DOI | MR | Zbl

[Be] F. Berteloot Characterization of models in 2 by their automorphism groups, Int. J. Math, Volume 5 (1994), pp. 619-634 | DOI | MR | Zbl

[C-G] L. Carleson; T. Gamelin Complex Dynamics, UTM, Springer Verlag, 1993 | MR | Zbl

[D-I] R. Debalme; S. Ivashkovich Complete hyperbolic neighborhoods in almost complex surfaces, Int. J. Math, Volume 12 (2001), pp. 211-221 | DOI | MR | Zbl

[De-1] R. Debalme Kobayashi hyperbolicity of almost complex manifolds (1998) (e-print, math.CV/9805130)

[De-2] R. Debalme Variétés hyperboliques presque-complexes (2001) (Thèse, Université de Lille I)

[Do] D. Donaldson Symplectic submanifolds and almost-complex geometry, J. Differential Geom, Volume 44 (1996) no. 4, pp. 666-705 | MR | Zbl

[Du] J. Duval Un théorème de Green presque complexe (e-print, math.CV/0311299) | Zbl

[G-S] H. Gaussier; A. Sukhov Estimates of the Kobayashi metric on almost complex manifolds (to appear in Bull. SMF) | Numdam | Zbl

[Gr] M. Gromov Pseudoholomorphic curves in symplectic manifolds, Invent. Math, Volume 82 (1985), pp. 307-347 | DOI | EuDML | MR | Zbl

[Ha] F. Haggui Fonctions PSH sur une variété presque complexe, C. R. Acad. Sci. Paris, Sér. I, Volume 335 (2002), pp. 509-514 | MR | Zbl

[Hö] L. Hörmander The Analysis of Linear Partial Differential Operators III, Grund. der math. Wis., 274, Springer-Verlag, Berlin Heidelberg, 1985 | MR | Zbl

[I-P-R] S. Ivashkovich; S. Pinchuk; J.-P. Rosay Upper semi-continuity of the Royden-Kobayashi pseudo-norm, a counterexample for Hölderian almost complex structures (to appear in Arkiv for Mat) | Zbl

[IS-1] S. Ivashkovich; V. Shevchishin Structure of the moduli space in a neighborhood of a cusp curve and meromorphic hulls, Invent. Math, Volume 136 (1999), pp. 571-602 | DOI | MR | Zbl

[IS-2] S. Ivashkovich; V. Shevchishin Complex Curves in Almost-Complex Manifolds and Meromorphic Hulls, Publication Series of Graduiertenkollegs "Geometrie und Mathematische Physik" of the Ruhr-University Bochum (1999) no. 36, pp. 1-186

[K] B.S. Kruglikov Existence of Close Pseudoholomorphic Disks for Almost Complex Manifolds and an Application to the Kobayashi-Royden Pseudonorm, Funct. Anal. and Appl, Volume 33 (1999), pp. 38-48 | DOI | MR | Zbl

[K-O] B.S. Kruglikov; M. Overholt Pseudoholomorphic mappings and Kobayashi hyperbolicity, Differential Geom. Appl, Volume 11 (1999), pp. 265-277 | DOI | MR | Zbl

[Ki] P. Kiernan Hyperbolically Imbedded Spaces and Big Picard Theorem, Math. Ann, Volume 204 (1973), pp. 203-209 | DOI | EuDML | MR | Zbl

[M] S.G. Mikhlin Multidimensional Singular Equations and Integral Equations, Pergamon Press (1955)

[McD] D. McDuff Symplectic manifolds with contact type boundaries, Invent. Math, Volume 103 (1991), pp. 651-671 | DOI | EuDML | MR | Zbl

[McD-S] D. McDuff; D. Salamon J-holomorphic curves and quantum cohomology, Univ. Lect. Series AMS, Volume 6 (1994) | MR | Zbl

[N-W] A. Nijenhuis; W. Woolf Some integration problems in almost complex and complex manifolds, Ann. of Math, Volume 77 (1963), pp. 424-489 | DOI | MR | Zbl

[P] N. Pali Fonctions plurisousharmoniques et courants positifs de type (1,1) sur une variété presque complexe (e-print, Math. DG/0402029) | Zbl

[R] H. Royden The Extension of Regular Holomorphic Maps, Proc. A.M.S, Volume 43 (1974), pp. 306-310 | DOI | MR | Zbl

[Si] J.-C. Sikorav; eds. M. Audin and J. Lafontaine Some properties of holomorphic curves in almost complex manifolds, Holomorphic Curves in Symplectic Geometry (1994), pp. 351-361

[St] E.M Stein Singular Integrals and Differentiability Properties of Functions, Princeton U.P, 1970 | MR | Zbl

[Za] M. Zaidenberg Picard's theorem and hyperbolicity, Siberian Math. J., Volume 24 (1983), pp. 858-867 | DOI | MR | Zbl

Cited by Sources: