We study the $\bar{\partial }$-equation with Hölder estimates in $q$-convex wedges of ${ℂ}^n$ by means of integral formulas. If $D\subset {ℂ}^n$ is defined by some inequalities $\{{\rho }_i\le 0\}$, where the real hypersurfaces $\{{\rho }_i=0\}$ are transversal and any nonzero linear combination with nonnegative coefficients of the Levi form of the ${\rho }_i$’s have at least $(q+1)$ positive eigenvalues, we solve the equation $\bar{\partial }f=g$ for each continuous $(n,r)$-closed form $g$ in $D$, $n-q\le r\le n$, with the following estimates: if $d$ denotes the distance to the boundary of $D$ and if $d^{\beta }g$ is bounded, then for all $ϵ\>0$,...

We show the variation formula for the Schiffer span s(t) for moving Riemann surfaces R(t) with $t\in B=t\in ℂ\left|\right|t|<\rho$, and apply it to show the simultaneous uniformization of moving planar Riemann surfaces of class $O_{AD}$.

We study the analytic structure of the leaves of a holomorphic foliation by curves on a compact complex manifold. We show that if every leaf is a hyperbolic surface then they can be simultaneously uniformized in a continuous manner. In case the manifold is complex projective space a sufficient condition is that there are no algebraic leaf.

We survey results on unique determination of local $CR$-automorphisms of smooth $CR$-manifolds and of local biholomorphisms of real-analytic $CR$-submanifolds of complex spaces by their jets of finite order at a given point. Examples generalizing [28] are given showing that the required jet order may be arbitrarily high.

We prove a uniqueness result for Coleff-Herrera currents which in particular means that if $f=(f_1,...,f_m)$ defines a complete intersection, then the classical Coleff-Herrera product associated to $f$ is the unique Coleff-Herrera current that is cohomologous to $1$ with respect to the operator ${\delta }_f-\overline{\partial }$, where ${\delta }_f$ is interior multiplication with $f$. From the uniqueness result we deduce that any Coleff-Herrera current on a variety $Z$ is a finite sum of products of residue currents with support on $Z$ and holomorphic forms.

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 a non-empty...