Regularity and estimates for JJ-holomorphic discs attached to a maximal totally real submanifold

We prove that pseudo-holomorphic discs attached to a maximal totally real submanifold inherit their regularity from the regularity of the submanifold and of the almost complex structure. The proof is based on the computation of an explicit lower bound for the Kobayashi metric in almost complex manifolds, which also yields explicit estimates of H\"olderian norms of such discs.Comment: 20 pages, 2 figure

Stationary discs and finite jet determination for non-degenerate generic real submanifolds

In case M is Levi non-degenerate in the sense Tumanov, we construct stationary discs for $M$. If furthermore M satisfies an additional non-degeneracy condition, we apply the method of stationary discs to obtain 2-jet determination of CR automorphisms of M

Disques analytiques et problèmes au bord en géométries complexe et presque complexe

This thesis deals with the study of analytic discs attached to some submanifold.In the first part, we obtain an explicit parametrization of some special family of analytic discs attached to different types of non-degenerate real hypersurfaces in \C^n. These discs are invariant under the action of biholomorphisms. We use this parametrization to construct a circular representation of the hypersurface, and we also get some properties of uniqueness for biholomorphisms.In the second part of this thesis, we consider proper pseudo-holomorphic maps between two strictly pseudoconvex bounded domains in almost complex manifolds. We prove that such a map extends up to the boundary. We establish the link between the regularity of the extension and the regularity of the amost complex structures, and we give explicit estimates for the Hölderian norms.Cette thèse est centrée sur l'étude des disques analytiques attachés à une sous-variété. Dans une première partie, nous obtenons une paramétrisation explicite d'une famille particulière de disques holomorphes attachés à différents types d'hypersurfaces réelles non-dégénérée de \C^n. Ces disques sont invariants sous l'action des biholomorphismes. Nous utilisons cette paramétrisation pour construire une représentation circulaire de l'hypersurface, ce qui donne également des propriétés d'unicité pour les biholomorphismes.Dans une seconde partie, nous considérons les applications pseudo-holomorphes propres entre domaines bornés strictement pseudoconvexes de variétés presque complexes. Nous montrons qu'une telle application se prolonge au bord. Nous établissons le lien entre la régularité hölderienne de l'application au bord et la régularité des structures presque complexes, et nous donnons des estimations explicites des normes hölderiennes

Metrical and dynamical aspects in complex analysis

The central theme of this reference book is the metric geometry of complex analysis in several variables. Bridging a gap in the current literature, the text focuses on the fine behavior of the Kobayashi metric of complex manifolds and its relationships to dynamical systems, hyperbolicity in the sense of Gromov and operator theory, all very active areas of research. The modern points of view expressed in these notes, collected here for the first time, will be of interest to academics working in the fields of several complex variables and metric geometry. The different topics are treated coherently and include expository presentations of the relevant tools, techniques and objects, which will be particularly useful for graduate and PhD students specializing in the area