### A boundary rigidity problem for holomorphic mappings.

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In this survey we give geometric interpretations of some standard results on boundary behaviour of holomorphic self-maps in the unit disc of ℂ and generalize them to holomorphic self-maps of some particular domains of ℂⁿ.

We prove a boundary uniqueness theorem for harmonic functions with respect to Bergman metric in the unit ball of Cn and give an application to a Runge type approximation theorem for such functions.

Let $M$ be a two dimensional totally real submanifold of class ${C}^{2}$ in ${\mathbf{C}}^{2}$. A continuous map $F:\stackrel{\u203e}{\Delta}\to {\mathbf{C}}^{2}$ of the closed unit disk $\stackrel{\u203e}{\Delta}\subset \mathbf{C}$ into ${\mathbf{C}}^{2}$ that is holomorphic on the open disk $\Delta $ and maps its boundary $b\Delta $ into $M$ is called an analytic disk with boundary in $M$. Given an initial immersed analytic disk ${F}^{0}$ with boundary in $M$, we describe the existence and behavior of analytic disks near ${F}^{0}$ with boundaries in small perturbations of $M$ in terms of the homology class of the closed curve ${F}^{0}\left(b\Delta \right)$ in $M$. We also prove a regularity theorem...

Boundary values of zero-smooth Besov analytic functions in the unit ball of ${\u2102}^{n}$ are investigated. Bounded Besov functions with prescribed lower semicontinuous modulus are constructed. Correction theorems for continuous Besov functions are proved. An approximation problem on great circles is studied.

Let Ω be a bounded pseudo-convex domain in Cn with a C∞ boundary, and let S be the set of strictly pseudo-convex points of ∂Ω. In this paper, we study the asymptotic behaviour of holomorphic functions along normals arising from points of S. We extend results obtained by M. Ortel and W. Schneider in the unit disc and those of A. Iordan and Y. Dupain in the unit ball of Cn. We establish the existence of holomorphic functions of given growth having a "prescribed behaviour" in almost all normals arising...

Let Ω be a domain in Cn. It is known that a holomorphic function on Ω behaves better in complex tangential directions. When Ω is of finite type, the best possible improvement is quantified at each point by the distance to the boundary in the complex tangential directions (see the papers on the geometry of finite type domains of Catlin, Nagel-Stein and Wainger for precise definition). We show that this improvement is characteristic: for a holomorphic function, a regularity in complex tangential directions...

We study the boundary behaviour of holomorphic functions in the Hardy-Sobolev spaces ${\mathscr{H}}^{p,k}\left(\right)$, where is a smooth, bounded convex domain of finite type in ℂⁿ, by describing the approach regions for such functions. In particular, we extend a phenomenon first discovered by Nagel-Rudin and Shapiro in the case of the unit disk, and later extended by Sueiro to the case of strongly pseudoconvex domains.

We solve the Dirichlet problem for line integrals of holomorphic functions in the unit ball: For a function $u$ which is lower semi-continuous on $\partial {\mathbb{B}}^{n}$ we give necessary and sufficient conditions in order that there exists a holomorphic function $f\in \mathbb{O}\left({\mathbb{B}}^{n}\right)$ such that $$u\left(z\right)={\int}_{\left|\lambda \right|<1}{\left|f\left(\lambda z\right)\right|}^{2}\mathrm{d}{\U0001d50f}^{2}\left(\lambda \right).$$