Null curves and directed immersions of open Riemann surfaces

TL;DR

This paper investigates holomorphic immersions of open Riemann surfaces into complex spaces with derivatives in specific algebraic varieties, establishing structure, approximation, and embedding theorems, including applications to null curves and minimal surfaces.

Contribution

It provides a comprehensive structure theorem and approximation results for A-immersions, extending the Oka principle to null curves and related immersions in complex Euclidean spaces.

Findings

Every A-immersion can be approximated by A-embeddings under certain conditions.

Null curves in C^3 can be properly embedded in C^3.

The Oka principle applies to A-immersions when A is an Oka manifold.

Abstract

In this paper we study holomorphic immersions of open Riemann surfaces into C^n whose derivative lies in a conical algebraic subvariety A of C^n that is smooth away from the origin. Classical examples of such A-immersions include null curves in C^3 which are closely related to minimal surfaces in R^3, and null curves in SL_2(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 C^3. If in addition A-{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 C^3.