Embedded minimal surfaces in ${\mathbb{R}}^n$

Abstract

In this paper, we prove that every confomal minimal immersion of an open Riemann surface into ${\mathbb{R}}^n$ for $n\ge 5$ can be approximated uniformly on compacts by conformal minimal embeddings. Furthermore, we show that every open Riemann surface carries a proper conformal minimal embedding into ${\mathbb{R}}^5$. One of our main tools is a Mergelyan approximation theorem for conformal minimal immersions to ${\mathbb{R}}^n$ for any $n\ge 3$ which is also proved in the paper.

Math. Z., in press. The official version is available on Springerlink at http://link.springer.com/article/10.1007%2Fs00209-015-1586-5

References in corpus (4)

• Bordered Riemann surfaces in C^2
• Null curves and directed immersions of open Riemann surfaces
• Embeddings of infinitely connected planar domains into C^2
• Approximation of holomorphic mappings on strongly pseudoconvex domains

Cited by in corpus (9)

• New complex analytic methods in the theory of minimal surfaces: a survey
• Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves
• Holomorphic Legendrian curves
• Complete minimal surfaces densely lying in arbitrary domains of ${\mathbb{R}}^n$
• Carleman approximation by conformal minimal immersions and directed holomorphic curves
• Every meromorphic function is the Gauss map of a conformal minimal surface
• Holomorphic Legendrian curves in ${\mathbb{CP}}^3$ and superminimal surfaces in ${\mathbb{S}}^4$
• The Oka principle for holomorphic Legendrian curves in ${\mathbb{C}}^{2n+1}$
• Interpolation by complete minimal surfaces whose Gauss map misses two points