We study the ramification of the Gauss map of complete minimal surfaces in ${ℝ}^m$ on annular ends. This is a continuation of previous work of Dethloff-Ha (2014), which we extend here to targets of higher dimension.

We will classify $n$-dimensional real submanifolds in ${ℂ}^n$ which have a set of parabolic complex tangents of real dimension $n-1$. All such submanifolds are equivalent under formal biholomorphisms. We will show that the equivalence classes under convergent local biholomorphisms form a moduli space of infinite dimension. We will also show that there exists an $n$-dimensional submanifold $M$ in ${ℂ}^n$ such that its images under biholomorphisms $(z_1,\cdots ,z_n)\mapsto (r{z}_1,\cdots ,r{z}_{n-1},r^2{z}_n)$, $r\>1$, are not equivalent to $M$ via any local volume-preserving holomorphic...

Every $n$-dimensional complex manifold (connected, paracompact and Hausdorff) is the image of the unit ball in $C^n$ under a finite holomorphic map that is locally biholomorphic.

We prove a theorem on the boundary regularity of a purely p-dimensional complex subvariety of a relatively compact, strictly pseudoconvex domain in a Stein manifold. Some applications describing the structure of the polynomial hull of closed curves in Cn are also given.

 Abstract. The characterization of hyperbolic embeddability of relatively compact subspaces of a complex space in terms of extension of holomorphic maps from the punctured disc and of limit complex lines is given.

We prove the existence of stationary discs in the ball for small almost complex deformations of the standard structure. We define a local analogue of the Riemann map and establish its main properties. These constructions are applied to study the local geometry of almost complex manifolds and their morphisms.

We show that any open Riemann surface can be properly immersed in any Stein manifold with the (Volume) Density property and of dimension at least 2. If the dimension is at least 3, we can actually choose this immersion to be an embedding. As an application, we show that Stein manifolds with the (Volume) Density property and of dimension at least 3, are characterized among all other complex manifolds by their semigroup of holomorphic endomorphisms.

We study a class of typical Hartogs domains which is called a generalized Fock-Bargmann-Hartogs domain $D_{n,m}^p\left(\mu \right)$. The generalized Fock-Bargmann-Hartogs domain is defined by inequality ${\mathrm{e}}^{{\mu \parallel z\parallel }^2}{\sum }_{j=1}^m{\left|{\omega }_j\right|}^{2p}<1$, where $(z,\omega )\in {ℂ}^n\times {ℂ}^m$. In this paper, we will establish a rigidity of its holomorphic automorphism group. Our results imply that a holomorphic self-mapping of the generalized Fock-Bargmann-Hartogs domain $D_{n,m}^p\left(\mu \right)$ becomes a holomorphic automorphism if and only if it keeps the function ${\sum }_{j=1}^m{\left|{\omega }_j\right|}^{2p}{\mathrm{e}}^{{\mu \parallel z\parallel }^2}$ invariant.