The Mittag-Leffler theorem for proper minimal surfaces and directed meromorphic curves

Abstract

We establish a Mittag-Leffler-type theorem with approximation and interpolation for meromorphic curves $M\to {\mathbb{C}}^n$ ($n\ge 3$) directed by Oka cones in ${\mathbb{C}}^n$ on any open Riemann surface $M$. We derive a result of the same type for proper conformal minimal immersions $M\to {\mathbb{R}}^n$. This includes interpolation in the poles and approximation by embeddings, the latter if $n\ge 5$ in the case of minimal surfaces. As applications, we show that complete minimal ends of finite total curvature in ${\mathbb{R}}^5$ are generically embedded, and characterize those open Riemann surfaces which are the complex structure of a proper minimal surface in ${\mathbb{R}}^3$ of weak finite total curvature.

arXiv commentTo appear in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)