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 prove the existence of a maximal plurisubharmonic minorant of a given upper semicontinous function $f$ on an almost complex manifold $(M, J)$ of complex dimension two. Such maximal function is obtained as the pointwise minimum of averages of $f$ over the boundaries of all $J$-complex discs in $M$ centered at the given point. This result is an almost complex analogue of classical results of Poletsky, Larusson and Sigurdsson, Rosay and others in the case when the almost complex structure $J$ is integrable.
COBISS.SI-ID: 16617561
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