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 $(M,J)$.

Suppose that $Y$ is a complex manifold such that any holomorphic map from a compact convex set in a Euclidean space ${ℂ}^n$ to $Y$ is a uniform limit of entire maps ${ℂ}^n\to Y$. We prove that a holomorphic map $X_0\to Y$ from a closed complex subvariety $X_0$ in a Stein manifold $X$ admits a holomorphic extension $X\to Y$ provided that it admits a continuous extension. We then establish the equivalence of four Oka-type properties of a complex manifold.

The purpose of this article is twofold. The first is to show a criterion for the normality of holomorphic mappings into Abelian varieties; an extension theorem for such mappings is also given. The second is to study the convergence of meromorphic mappings into complex projective varieties. We introduce the concept of d-convergence and give a criterion of d-normality of families of meromorphic mappings.

Let f : M → M' be a CR homeomorphism between two minimal, rigid polynomial varieties of Cn without holomorphic curves. We show that f extends biholomorphically in a neighborhood of M if f extends holomorphically in a neighborghood of a point p0 ∈ M or if f is of class C1. In the other hand, in case M and M' are two algebraic hypersurfaces, we obtain the extension without supplementary conditions.

We study the extension problem for germs of holomorphic isometries $f:(D;x_0)\to (\Omega ;f\left(x_0\right))$ up to normalizing constants between bounded domains in Euclidean spaces equipped with Bergman metrics $d{s}_D^2$ on $D$ and $d{s}_{\Omega }^2$ on $\Omega$. Our main focus is on boundary extension for pairs of bounded domains $(D,\Omega )$ such that the Bergman kernel $K_D(z,w)$ extends meromorphically in $(z,\overline{w})$ to a neighborhood of $\overline{D}\times D$, and such that the analogous statement holds true for the Bergman kernel $K_{\Omega }(\varsigma ,\xi )$ on $\Omega$. Assuming that $(D;d{s}_D^2)$ and $(\Omega ;d{s}_{\Omega }^2)$ are complete Kähler manifolds, we prove that the germ...

In this paper we extend the results on analytic continuation of germs of holomorphic mappings from a real analytic hypersurface to a real algebraic hypersurface to the case when the target hypersurface is of higher dimension than the source. More precisely, we prove the following: Let $M$ be a connected smooth real analytic minimal hypersurface in $C^n$, $M^{\prime }$ be a compact strictly pseudoconvex real algebraic hypersurface in $C^N$, $1\<n\le N$. Suppose that $f$ is a germ of a holomorphic map at a point $p$ in $M$ and $f\left(M\right)$ is in...