The study of $J$-holomorphic maps leads to the consideration of the inequations $|\frac{\partial u}{\partial \overline{z}}|\le C|u|$, and $|\frac{\partial u}{\partial \overline{z}}|\le ϵ|\frac{\partial u}{\partial z}|$. The first inequation is fairly easy to use. The second one, that is relevant to the case of rough structures, is more delicate. The case of $u$ vector valued is strikingly different from the scalar valued case. Unique continuation and isolated zeroes are the main topics under study. One of the results is that, in almost complex structures of Hölder class $\frac{1}{2}$, any $J$-holomorphic curve that is constant on...

Let $D\subset \subset {ℂ}^n,n\ge 2$, be a domain with $C^2$-boundary and $K\subset \partial D$ be a compact set such that $\partial D\setminus K$ is connected. We study univalent analytic extension of CR-functions from $\partial D\setminus K$ to parts of $D$. Call $K$ CR-convex if its $A\left(D\right)$-convex hull, $A\left(D\right)-\mathrm{hull}\left(K\right)$, satisfies $K=\partial D\cap A\left(D\right)-\mathrm{hull}\left(K\right)$ ($A\left(D\right)$ denoting the space of functions, which are holomorphic on $D$ and continuous up to $\partial D$). The main theorem of the paper gives analytic extension to $\partial D\setminus A\left(D\right)-\mathrm{hull}\left(K\right)$, if $K$ is CR- convex.

Let $M^n$ be a germ at $0\in {ℂ}^m$ of an irreducible analytic set of dimension $n$, where $n\ge 2$ and $0$ is a singular point of $M$. We study the question: when does there exist a germ of holomorphic map $\phi :({ℂ}^n,0)\to (M,0)$ such that ${\phi }^{-1}\left(0\right)=\left\{0\right\}$ ? We prove essentialy three results. In Theorem 1 we consider the case where $M$ is a quasi-homogeneous complete intersection of $k$ polynomials $F=(F_1,...,F_k)$, that is there exists a linear holomorphic vector field $X$ on ${ℂ}^m$, with eigenvalues ${\lambda }_1,...,{\lambda }_m\in {\mathbb{Q}}_{+}$ such that $X\left(F^T\right)=U·F^T$, where $U$ is a $k\times k$ matrix with entries in ${𝒪}_m$. We prove that if...

Soit $U_1$ un ouvert dans ${\mathbf{C}}^m$. Soit ${\pi }_1:\mathbf{S}\to U_1$ une famille holomorphe de structures complexes sur une surface de Riemann non-compacte $M$, avec ${\mathbf{S}}_{t_0}={\pi }_1^{-1}(t_0)=M$. ($\mathbf{S}=\mathbf{S}(M\times U_1)$ est une structure complexe sur le produit différentiable $M\times U_1$). Soit $M_1$ un domaine relativement compact dans $M$. On démontre : pour tout voisinage de Stein $U$ de $t_0$, assez petit, la famille ${\pi }_1:\mathbf{S}(M_1\times U)\to U$ est isomorphe à la famille $\pi :\Omega \to \pi \left(\Omega \right)$, où $\Omega$ est un de la variété produit $M\times {\mathbf{C}}^m$, $\pi$ étant la projection $M\times {\mathbf{C}}^m\to {\mathbf{C}}^m$. On donne aussi un résultat analogue pour le cas des variations différentiables. ...

Using recent development in Poletsky theory of discs, we prove the following result: Let $X,$ $Y$ be two complex manifolds, let $Z$ be a complex analytic space which possesses the Hartogs extension property, let $A$ (resp. $B$) be a non locally pluripolar subset of $X$ (resp. $Y$). We show that every separately holomorphic mapping $f:W:=(A\times Y)\cup (X\times B)\to Z$ extends to a holomorphic mapping $\widehat{f}$ on $\widehat{W}:=\left\{(z,w)\in X\times Y:\tilde{\omega }(z,A,X)+\tilde{\omega }(w,B,Y)\<1\right\}$ such that $\widehat{f}=f$ on $W\cap \widehat{W},$ where $\tilde{\omega }(·,A,X)$ (resp. $\tilde{\omega }(·,B,Y))$ is the plurisubharmonic measure of $A$ (resp. $B$) relative to $X$ (resp. $Y$). Generalizations...