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
In this paper we prove that every bordered Riemann surface $M$ admits a complete proper null holomorphic embedding into a ball of the complex Euclidean 3-space $\mathbb{C}^3$. The real part of such an embedding is a complete conformal minimal immersion $M \to \mathbb{R}^3$ with bounded image. For any such $M$ we also construct proper null holomorphic embeddings $M \hookrightarrow \mathbb{C}^3$ with a bounded coordinate function; these give rise to properly embedded null curves $M \hookrightarrow SL_2(\mathbb{C})$ and to properly immersed Bryant surfaces $M \to \mathbb{H}^3$ in the hyperbolic 3-space. In particular, we give the first examples of proper Bryant surfaces with finite topology and of hyperbolic conformal type. The main novelty when compared to the existing results in the literature is that we work with a fixed conformal structure on $M$. This is accomplished by introducing a conceptually new method based on complex analytic techniques. One of our main tools is an approximate solution to the Riemann-Hilbert boundary value problem for null curves in $\mathbb{C}^3$, developed in Sect. 3.
COBISS.SI-ID: 17242457
We provide sufficient conditions on a manifold $X$ and a domain $W$ in $X$ which imply that the largest plurisubharmonic subextension of an upper-semicontinuous function on $W$ to $X$ can be represented by a disc formula.
COBISS.SI-ID: 16930905
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
The main result of this paper is a characterization of the minimal surface hull of a compact set $K$ in $\mathbb R^3$ by sequences of conformal minimal discs whose boundaries converge to $K$ in the measure theoretic sense, and also by $2$-dimensional minimal currents which are limits of Green currents supported by conformal minimal discs. Analogous results are obtained for the null hull of a compact subset of $\mathbb C^3$. We also prove a null hull analogue of the Alexander-Stolzenberg-Wermer theorem on polynomial hulls of compact sets of finite linear measure, and a polynomial hull version of classical Bochner's tube theorem.
COBISS.SI-ID: 17543769