We discuss an example of an open subset of a torus which admits a dense entire curve, but no dense Brody curve.

A complex analytic space is said to have the D*-extension property if and only if any holomorphic map from the punctured disk to the given space extends to a holomorphic map from the whole disk to the same space. A Hartogs domain H over the base X (a complex space) is a subset of X x C where all the fibers over X are disks centered at the origin, possibly of infinite radius. Denote by φ the function giving the logarithm of the reciprocal of the radius of the fibers, so that, when X is pseudoconvex,...

Let ${\Omega }_1$ and ${\Omega }_2$ be $\mathrm{strongly}\mathrm{pseudoconvex}$ domains in ${ℂ}^n$ and $f:{\Omega }_1\to {\Omega }_2$ an isometry for the Kobayashi or Carathéodory metrics. Suppose that $f$ extends as a $C^1$ map to ${\overline{\Omega }}_1$. We then prove that ${f|}_{\partial {\Omega }_1}:\partial {\Omega }_1\to \partial {\Omega }_2$ is a CR or anti-CR diffeomorphism. It follows that ${\Omega }_1$ and ${\Omega }_2$ must be biholomorphic or anti-biholomorphic.

This article considers C¹-smooth isometries of the Kobayashi and Carathéodory metrics on domains in ℂⁿ and the extent to which they behave like holomorphic mappings. First we provide an example which suggests that 𝔹ⁿ cannot be mapped isometrically onto a product domain. In addition, we prove several results on continuous extension of C⁰-isometries f : D₁ → D₂ to the closures under purely local assumptions on the boundaries. As an application, we show that there is no C⁰-isometry between a strongly...

This article gives a description, by means of functorial intrinsic fibrations, of the geometric structure (and conjecturally also of the Kobayashi pseudometric, as well as of the arithmetic in the projective case) of compact Kähler manifolds. We first define special manifolds as being the compact Kähler manifolds with no meromorphic map onto an orbifold of general type, the orbifold structure on the base being given by the divisor of multiple fibres. We next show that rationally connected Kähler...

For any compact Kähler manifold $X$ and for any equivalence relation generated by a symmetric binary relation with compact analytic graph in $X\times X$, the existence of a meromorphic quotient is known from Inv. Math. 63 (1981). We give here a simplified and detailed proof of the existence of such quotients, following the approach of that paper. These quotients are used in one of the two constructions of the core of $X$ given in the previous paper of this fascicule, as well as in many other questions.