Dans cette thèse, nous abordons certains aspects de l'analyse locale dans les variétés presque complexes. Dans un premier temps, nous étudions le fibré cotangent qui est un outil important pour l'analyse et la géométrie complexe. Nous construisons un relevé de structure presque complexe, à l'aide d'une connexion, qui unifie les relevés complets de I.Sato et horizontaux de S.Ishihara et K.Yano. Par ailleurs, nous dégageons les principales propriétés analytiques et symplectiques du relevé ainsi construit. <br />Dans les deux études qui suivent, nous nous intéressons aux propriétés locales des domaines pseudoconvexes de type de D'Angelo fini d'une variété presque complexe de dimension réelle quatre. Nous construisons des fonctions locales pic plurisousharmoniques, généralisant des travaux de J.E.Fornaess et N.Sibony. La construction d'une telle famille de fonctions permet d'établir des propriétés d'attraction et de localisation des disques pseudoholomorphes. En particulier, elle réduit l'étude de la pseudométrique de Kobayashi à un problème purement local. Le comportement asymptotique de cette pseudométrique est relié à certaines questions fascinantes d'analyse locale dans les variétés comme les phénomènes de prolongement au bord des difféomorphismes ou encore la classification des domaines, et fournit des informations intéressantes sur les propriétés géométriques et dynamiques de la variété. Nous donnons alors des estimées locales de cette pseudométrique au voisinage du bord. De plus, dans le cas de stricte pseudoconvexité, nous obtenons des estimées très fines nous permettant d'étudier les liens entre l'hyperbolicité au sens de Kobayashi et l'hyperbolicité au sens de Gromov ; nous généralisons ainsi, au cadre presque complexe, un résultat dû à Z.M.Balogh et M.Bonk.

This thesis comprises the study of two moduli spaces of piecewise J-holomorphic curves. The main scheme is to consider a subdivision of the 2-sphere into a collection of small domains and to study collections of J-holomorphic maps into a symplectic manifold. These maps are coupled by Lagrangian boundary conditions. The work can be seen as finding a 2-dimensional analogue of the finite-dimensional path space approximation by piecewise geodesics on a Riemannian manifold (Q,g). For a nice class of target manifolds we consider tangent bundles of Riemannian manifolds and symplectizations of unit tangent bundles. Via polarization they provide a rich set of Lagrangians which can be used to define appropriate boundary value problems for the J-holomorphic pieces. The work focuses on existence theory as a pre-stage to global questions such as combinatorial refinement and the quality of the approximation. The first moduli space of lifted type is defined on a triangulation of the 2-sphere and consists of disks in the tangent bundle whose boundary projects onto geodesic triangles. The second moduli space of punctured type is defined on a circle packing domain and consists of boundary punctured disks in the symplectization of the unit tangent bundle. Their boundary components map into single fibers and at punctures the disks converge to geodesics. The coupling boundary conditions are chosen such that the piecewise problem always is Fredholm of index zero and both moduli spaces only depend on discrete data. For both spaces existence results are established for the J-holomorphic pieces which hold true on a small scale. Each proof employs a version of the implicit function theorem in a different setting. Here the argument for the moduli space of punctured type is more subtle. It rests on a connection to tropical geometry discovered by T. Ekholm for 1-jet spaces. The boundary punctured disks are constructed in the vicinity of explicit Morse flow trees which correspond to the limiting objects under degeneration of the boundary condition.

This thesis comprises the study of two moduli spaces of piecewise J-holomorphic curves. The main scheme is to consider a subdivision of the 2-sphere into a collection of small domains and to study collections of J-holomorphic maps into a symplectic manifold. These maps are coupled by Lagrangian boundary conditions. The work can be seen as finding a 2-dimensional analogue of the finite-dimensional path space approximation by piecewise geodesics on a Riemannian manifold (Q,g). For a nice class of target manifolds we consider tangent bundles of Riemannian manifolds and symplectizations of unit tangent bundles. Via polarization they provide a rich set of Lagrangians which can be used to define appropriate boundary value problems for the J-holomorphic pieces. The work focuses on existence theory as a pre-stage to global questions such as combinatorial refinement and the quality of the approximation. The first moduli space of lifted type is defined on a triangulation of the 2-sphere and consists of disks in the tangent bundle whose boundary projects onto geodesic triangles. The second moduli space of punctured type is defined on a circle packing domain and consists of boundary punctured disks in the symplectization of the unit tangent bundle. Their boundary components map into single fibers and at punctures the disks converge to geodesics. The coupling boundary conditions are chosen such that the piecewise problem always is Fredholm of index zero and both moduli spaces only depend on discrete data. For both spaces existence results are established for the J-holomorphic pieces which hold true on a small scale. Each proof employs a version of the implicit function theorem in a different setting. Here the argument for the moduli space of punctured type is more subtle. It rests on a connection to tropical geometry discovered by T. Ekholm for 1-jet spaces. The boundary punctured disks are constructed in the vicinity of explicit Morse flow trees which correspond to the limiting objects under degeneration of the boundary condition.