Abstract:
The Kohn-Nireberg domains are unbounded domains in the complex Euclidean space of dimension 2 upon which many outstanding questions are yet to be explored. The primary aim of this article is to demonstrate that the Bergman and Caratheodory metrics of any Kohn-Nirenberg domains are positive and complete.

Abstract:
In this paper we extend the notion of the Kobayashi-Royden pseudonorm for almost complex manifolds. Its basic properties known from the complex analysis are preserved in the nonintegrable case as well. The main theorem on coincidence of the pseudodistance induced by this pseudonorm with the Kobayashi pseudodistance for the almost complex manifold is equivalent to the possibility of deforming slightly a pseudoholomorphic disk in an almost complex manifold. We also describe the result in terms of h-principle and consider a geometric application for moduli spaces of pseudoholomorphic curves.

Abstract:
This article considers isometries of the Kobayashi and Carath\'{e}od-ory metrics on domains in $ \mathbf{C}^n $ and the extent to which they behave like holomorphic mappings. First we prove a metric version of Poincar\'{e}'s theorem about biholomorphic inequivalence of $ \mathbf{B}^n $, the unit ball in $ \mathbf{C}^n $ and $ \Delta^n $, the unit polydisc in $ \mathbf{C}^n $ and then provide few examples which \textit{suggest} that $ \mathbf{B}^n $ cannot be mapped isometrically onto a product domain. In addition, we prove several results on continuous extension of isometries $ f : D_1 \rightarrow D_2 $ to the closures under purely local assumptions on the boundaries. As an application, we show that there is no isometry between a strongly pseudoconvex domain in $ \mathbf{C}^2 $ and certain classes of weakly pseudoconvex finite type domains in $ \mathbf{C}^2 $.

Abstract:
The purpose of this article is to consider two themes both of which emanate from and involve the Kobayashi and the Carath\'eodory metric. First we study the biholomorphic invariant introduced by B. Fridman on strongly pseudoconvex domains, on weakly pseudoconvex domains of finite type in $\mathbf C^2$ and on convex finite type domains in $\mathbf C^n$ using the scaling method. Applications include an alternate proof of the Wong-Rosay theorem, a characterization of analytic polyhedra with noncompact automorphism group when the orbit accumulates at a singular boundary point and a description of the Kobayashi balls on weakly pseudoconvex domains of finite type in $ \mathbf{C}^2$ and convex finite type domains in $ \mathbf{C}^n $ in terms of Euclidean parameters. Second a version of Vitushkin's theorem about the uniform extendability of a compact subgroup of automorphisms of a real analytic strongly pseudoconvex domain is proved for $C^1$-isometries of the Kobayashi and Carath\'eodory metrics on a smoothly bounded strongly pseudoconvex domain.

Abstract:
We extend the definition of the Kobayashi pseudodistance to almost complex manifolds and show that its familliar properties are for the most part preserved. We also study the automorphism group of an almost complex manifold and finish with some examples.

Abstract:
We study the Carath\'{e}odory and Kobayashi metrics by way of the method of dual extremal problems in functional analysis. Particularly incisive results are obtained for convex domains.

Abstract:
We show, using the Kobayashi and Caratheodory metrics on special holomorphic disks in the universal Teichmuller space, that a wide class of holomorphic functionals on the space of univalent functions in the disk is maximized by the Koebe function or by its root transforms; their extremality is forced by hyperbolic features. As consequences, this implies the proofs of the famous Zalcman and Bieberbach conjectures.

Abstract:
In this paper we define Kobayashi-Royden pseudonorm for almost complex manifolds. Its basic properties known from the complex analysis are preserved in the nonintegrable case as well. We prove that the pseudodistance induced by this pseudonorm coincides with the Kobayashi pseudodistance defined for the almost complex case earlier. We also consider a geometric application for moduli spaces of pseudoholomorphic curves.

Abstract:
We establish a lower estimate for the Kobayashi-Royden infinitesimal pseudometric on an almost complex manifold $(M,J)$ admitting a bounded strictly plurisubharmonic function. We apply this result to study the boundary behaviour of the metric on a strictly pseudoconvex domain in $M$ and to give a sufficient condition for the complete hyperbolicity of a domain in $(M,J)$. Finally we obtain the regularity up to the bounday of $J$-holomorphic discs attached to a totally real submanifold in $M$.

Abstract:
In the paper we give some necessary conditions for a mapping to be a $\kappa$-geodesic in non-convex complex ellipsoids. Using these results we calculate explicitly the Kobayashi metric in the ellipsoids $\{|z_1|^2+|z_2|^{2m}<1\}\subset\bold C^2$, where $m<\frac12$.