Null curves and directed immersions of open Riemann surfaces

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.

Inventiones Math., in press

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• Approximation theory for non-orientable minimal surfaces and applications
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• Every meromorphic function is the Gauss map of a conformal minimal surface
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• Complete CMC-1 surfaces in hyperbolic space with arbitrary complex structure
• Immersions of open Riemann surfaces into the Riemann sphere
• Proper superminimal surfaces of given conformal types in the hyperbolic four-space
• Interpolation by complete minimal surfaces whose Gauss map misses two points
• Families of proper holomorphic maps
• Minimal surfaces with symmetries