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Quelques problèmes d'analyse géométrique dans les variétés presque complexes à bord.

Abstract : We study the real analyticity of a CR mapping between two hypersurfaces in almost complex manifolds. We prove that a CR mapping defined on the boundary of a model domain is real analytic. We also prove that a CR mapping is real analytic when the almost complex structure of the codomain is a deformation of a model structure and when the hypersurfaces are small deformation of the hypersurface ∂h defined by ∂h={z∈Cⁿ,RE(zₙ)+|z'|²=0}. We make use of a method of prolongation for the tangential Cauchy-Riemann equations and a result about complete systems. Then, we use the previous result to extend the Poincaré-Alexander Theorem in the almost complex case. The Poincaré-Alexander Theorem states that holomorphic mappings defined on an open subset of the unit ball of Cⁿ may, under certain conditions, be extended to a biholomorphism of the unit ball. In a complex manifold, every strongly pseudoconvex homogeneous domain is biholomorphic to the unit ball. In an almost complex manifold, the unit ball is not the only strongly pseudoconvex homogeneous domain. A strongly pseudoconvex homogeneous domain is biholomorphic to a model domain. We extend the Poincaré-Alexander Theorem theorem to model domains. Finally, we define J-quasiconformal mappings and we prove that open sets and totally real submanifolds of the boundary are unicity sets for J-quasiconformal mappings. We also prove that a J-quasiconformal mapping admitting zero limits at every point of a totally real submanifold of the boundary is identically zero.
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Marianne Peyron. Quelques problèmes d'analyse géométrique dans les variétés presque complexes à bord.. Mathématiques générales [math.GM]. Université de Grenoble, 2013. Français. ⟨NNT : 2013GRENM028⟩. ⟨tel-00989815⟩

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