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
In this paper, we find approximate solutions of certain Riemann-Hilbert boundary value problems for minimal surfaces in $\mathbb{R}^n$ and null holomorphic curves in $\mathbb{C}^n$ for any $n \ge 3$. With this tool in hand, we construct complete conformally immersed minimal surfaces in $\mathbb{R}^n$ which are normalized by any given bordered Riemann surface and have Jordan boundaries. We also furnish complete conformal proper minimal immersions from any given bordered Riemann surface to any smoothly bounded, strictly convex domain of $\mathbb{R}^n$ which extend continuously up to the boundary; for $n \ge 5$, we find embeddings with these properties.
COBISS.SI-ID: 17458009
In 1977, P. Yang asked whether there exist complete immersed complex submanifolds $\varphi \colon M^k \to \mathbb{C}^N$ with bounded image. A positive answer is known for holomorphic curves $(k=1)$ and partial answers are known for the case when $k)1$. The principal result of the present paper is a construction of a holomorphic function on the open unit ball $\mathbb{B}_N$ of $\mathbb{C}^N$ whose real part is unbounded on every path in $\mathbb{B}_N$ of finite length that ends on $b\mathbb{B}_N$. A consequence is the existence of a complete, closed complex hypersurface in $\mathbb{B}_N$. This gives a positive answer to Yang's question in all dimensions $k$, $N$, $1 \le k ( N$, by providing properly embedded complete complex manifolds.
COBISS.SI-ID: 17459545
We study groups having the property that every non-abelian subgroup contains its centralizer. We describe various classes of infinite groups in this class, and address a problem of Berkovich regarding the classification of finite $p$-groups with the above property.
COBISS.SI-ID: 17738329
In this paper the $G^1$ interpolation of two data points and two tangent directions with spatial cubic rational PH curves is considered. It is shown that interpolants exist for any true spatial data configuration. The equations that determine the interpolants are derived by combining a closed form representation of a ten parametric family of rational PH cubics given in Kozak et al. (2014), and the Gram matrix approach. The existence of a solution is proven by using a homotopy analysis, and numerical method to compute solutions is proposed. In contrast to polynomial PH cubics for which the range of $G^1$ data admitting the existence of interpolants is limited, a switch to rationals provides an interpolation scheme with no restrictions.
COBISS.SI-ID: 17608793