We study holomorphic immersions of open Riemann surfaces into ${\mathbb C}^n$ whose derivative lies in a conical algebraic subvariety $A$ of ${\mathbb C}^n$ that is smooth away from the origin. Classical examples of such $A$-immersions include null curves in ${\mathbb C}^3$ which are closely related to minimal surfaces in ${\mathbb R}^3$, and null curves in $SL_2(\mathbb{C})$ that are related to Bryant surfaces. We establish a basic structure theorem for the set of all $A$-immersions of a bordered Riemann surface, and we prove several approximation and desingularization theorems. Assuming that $A$ is irreducible and is not contained in any hyperplane, we show that every $A$-immersion can be approximated by $A$-embeddings; this holds in particular for null curves in ${\mathbb C}^3$. If in addition $A \setminus \{0\}$ is an Oka manifold, then $A$-immersions are shown to satisfy the Oka principle, including the Runge and the Mergelyan approximation theorems. Another version of the Oka principle holds when $A$ admits a smooth Oka hyperplane section. This lets us prove in particular that every open Riemann surface is biholomorphic to a properly embedded null curve in ${\mathbb C}^3$.
COBISS.SI-ID: 16655705
We study the class of all groups in which the centralizer of each element is a subnormal subgroup. In particular, we focus on the case when the defect of every centralizer is at most 2. We show that a group without involutions satisfies this property if and only if it is 3-Engel.
COBISS.SI-ID: 16556889
In this paper, a particular shape preserving parametric polynomial approximation of conic sections is studied. The approach is based upon the parametric approximation of implicitly defined planar curves. Polynomial approximants derived are given in a closed form and provide the highest possible approximation order.
COBISS.SI-ID: 16716121