Complete minimal surfaces densely lying in arbitrary domains of ℝn

Alarcón, Antonio¹ ; Castro-Infantes, Ildefonso¹

¹ Departamento de Geometría y Topología, Universidad de Granada, Granada, Spain

Geometry & topology, Tome 22 (2018) no. 1, pp. 571-590.

Résumé

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 $M\to {ℝ}^n$ with $\overline{X\left(M\right)}={ℝ}^n$ forms a dense subset in the space of all conformal minimal immersions $M\to {ℝ}^n$ endowed with the compact-open topology. Moreover, we show that every domain in ${ℝ}^n$ 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 nonorientable minimal surfaces in ${ℝ}^n\left(n\ge 3\right)$, complex curves in ${ℂ}^n\left(n\ge 2\right)$, holomorphic null curves in ${ℂ}^n\left(n\ge 3\right)$, and holomorphic Legendrian curves in ${ℂ}^{2n+1}\left(n\in \mathbb{N}\right)$.

Détail

DOI : 10.2140/gt.2018.22.571

Classification : 49Q05, 32H02
Keywords: complete minimal surface, Riemann surface, holomorphic curve

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