Complete minimal surfaces densely lying in arbitrary domains of ${\mathbb{R}}^n$

Abstract

In this paper we prove that, given an open Riemann surface $M$ and an integer $n\ge 3$, the set of complete conformal minimal immersions $M\to {\mathbb{R}}^n$ with $\overline{X(M)}={\mathbb{R}}^n$ forms a dense subset in the space of all conformal minimal immersions $M\to {\mathbb{R}}^n$ endowed with the compact-open topology. Moreover, we show that every domain in ${\mathbb{R}}^n$ contains complete minimal surfaces which are dense on it and have arbitrary orientable topology (possibly infinite); we also provide such surfaces whose complex structure is any given bordered Riemann surface. Our method of proof can be adapted to give analogous results for non-orientable minimal surfaces in ${\mathbb{R}}^n$ $(n\ge 3)$, complex curves in ${\mathbb{C}}^n$ $(n\ge 2)$, holomorphic null curves in ${\mathbb{C}}^n$ $(n\ge 3)$, and holomorphic Legendrian curves in ${\mathbb{C}}^{2n+1}$ $(n\in \mathbb{N})$.

15 pages, 1 figure

References in corpus (4)

• Bordered Riemann surfaces in C^2
• Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves
• Holomorphic Legendrian curves
• Holomorphic discs with dense images

Cited by in corpus (7)

• New complex analytic methods in the theory of minimal surfaces: a survey
• Interpolation by conformal minimal surfaces and directed holomorphic curves
• Carleman approximation by conformal minimal immersions and directed holomorphic curves
• Holomorphic Legendrian curves in ${\mathbb{CP}}^3$ and superminimal surfaces in ${\mathbb{S}}^4$
• Complete CMC-1 surfaces in hyperbolic space with arbitrary complex structure
• The halfspace theorem for minimal hypersurfaces in regions bounded by minimal cones
• Interpolation by complete minimal surfaces whose Gauss map misses two points