Plurisubharmonic exhaustion functions and almost complex Stein structures

We prove that a relatively compact pseudoconvex domain with smooth boundary in an almost complex manifold admits a bounded strictly plurisubharmonic exhaustion function. We use this result for the study of convexity and hyperbolicity properties of these domains and the contact geometry of their boundaries.

[1]  A. Tumanov,et al.  Filling hypersurfaces by discs in almost complex manifolds of dimension 2 , 2005, math/0505459.

[2]  A. Sukhov,et al.  Pseudoholomorphic discs attached to CR-submanifolds of almost complex spaces , 2004 .

[3]  J. Barraud,et al.  Regular type of real hyper-surfaces in (almost) complex manifolds , 2003, math/0304146.

[4]  B. Kruglikov Existence of close pseudoholomorphic disks for almost complex manifolds and an application to the Kobayashi-Royden pseudonorm , 2000, math/0002190.

[5]  B. Kruglikov,et al.  Pseudoholomorphic mappings and Kobayashi hyperbolicity , 1999 .

[6]  R. Hind Filling by Holomorphic Disks with Weakly Pseudoconvex Boundary Conditions , 1997 .

[7]  D. Mcduff Symplectic manifolds with contact type boundaries , 1991 .

[8]  Norberto Kerzman,et al.  Fonctions plurisouscharmoniques d'exhaustion bornées et domaines taut , 1981 .

[9]  J. Fornæss,et al.  Complex manifolds in pseudoconvex boundaries , 1981 .

[10]  R. Range A REMARK ON BOUNDED STRICTLY PLURISUBHARMONIC EXHAUSTION FUNCTIONS , 1981 .

[11]  N. Sibony A CLASS OF HYPERBOLIC MANIFOLDS , 1981 .

[12]  K. Diederich,et al.  Strictly pseudohyperbolic domains , 1978 .

[13]  K. Diederich,et al.  Pseudoconvex domains: Bounded strictly plurisubharmonic exhaustion functions , 1977 .

[14]  H. Skoda Fibrés holomorphes à base et à fibre de Stein , 1977 .

[15]  A. Nijenhuis,et al.  SOME INTEGRATION PROBLEMS IN ALMOST-COMPLEX AND COMPLEX MANIFOLDS , 1963 .

[16]  Olli Lehto,et al.  Boundary behaviour and normal meromorphic functions , 1957 .

[17]  H. Gaussier,et al.  Estimates of the Kobayashi-Royden metric in almost complex manifolds , 2005 .

[18]  Dusa McDuff,et al.  J-Holomorphic Curves and Symplectic Topology , 2004 .

[19]  N. Levenberg,et al.  Function theory in several complex variables , 2001 .

[20]  J. Kohn Quantitative Estimates for Global Regularity , 1999 .

[21]  J. Sikorav Some properties of holomorphic curves in almost complex manifolds , 1994 .

[22]  J. Demailly Différents exemples de fibrés holomorphes non de Stein , 1978 .

[23]  Jean-Luc Stehlé Fonctions plurisousharmoniques et convexite holomorphe de certains fibres analytiques , 1975 .

[24]  矢野 健太郎,et al.  Tangent and cotangent bundles : differential geometry , 1973 .

[25]  H. Royden Remarks on the Kobayashi metric , 1971 .