We study pseudoholomorphic discs with boundaries attached to a real hypersurface E in an almost complex manifold of dimension 2. We prove that if E contains no discs, then they fill a one sided neighborhood of E. MSC: 32H02, 53C15. Key words: almost complex manifold, Bishop disc. * 1 , then P + and P − are bounded operators C α (ID) −→ C 1−β (ID) for every 0 < α < 1, 0 < β < 1.

Abstract. -We establish a lower estimate for the Kobayashi-Royden infinitesimal pseudometric on an almost complex manifold (M, J) admitting a bounded strictly plurisubharmonic function. We apply this result to study the boundary behaviour of the metric on a strictly pseudoconvex domain in M and to give a sufficient condition for the complete hyperbolicity of a domain in (M, J). Résumé (Estimées de la métrique de Kobayashi-Royden dans les variétés presque complexes)Nousétablissons une estimée inférieure pour la métrique de Kobayashi-Royden sur une variété presque complexe (M, J) admettant une fonction bornée strictement pluri-sous-harmonique. Nous appliquons ce résultatà l'étude du comportement de la métrique au bord d'un domaine strictement pseudoconvexe dans M et donnons une condition suffisante d'hyperbolicité complète d'un domaine dans (M, J).

We give a necessary and sufficient condition for the smooth extension of a diffeomorphism between smooth strictly pseudoconvex domains in four real dimensional almost complex manifolds (see Theorem 1.1). The proof is mainly based on a reflection principle for pseudoholomorphic discs, on precise estimates of the Kobayashi-Royden infinitesimal pseudometric and on the scaling method in almost complex manifolds. (2000): 32H02, 53C15 Mathematics Subject Classification

We study some special almost complex structures on strictly pseudoconvex domains in R 2n . They appear naturally as limits under a nonisotropic scaling procedure and play a role of model objects in the geometry of almost complex manifolds with boundary. We determine explicitely some geometric invariants of these model structures and derive necessary and sufficient conditions for their integrability. As applications we prove a boundary extension and a compactness principle for some elliptic diffeomorphisms between relatively compact domains.

We show that biholomorphic automorphisms of a real analytic hypersurface in C n+1 can be considered as (pointwise) Lie symmetries of a holomorphic completely overdetermined involutive second order PDE system defining its Segre family. Using the classical S.Lie method we obtain a complete description of infinitesimal symmetries of such a system and give a new proof of some well known results of CR geometry.