## Geometry & Topology

### Approximation theory for nonorientable minimal surfaces and applications

#### Abstract

We prove a version of the classical Runge and Mergelyan uniform approximation theorems for nonorientable minimal surfaces in Euclidean $3$–space $ℝ3$. Then we obtain some geometric applications. Among them, we emphasize the following ones:

• A Gunning–Narasimhan-type theorem for nonorientable conformal surfaces.
• An existence theorem for nonorientable minimal surfaces in $ℝ3$ with arbitrary conformal structure, properly projecting into a plane.
• An existence result for nonorientable minimal surfaces in $ℝ3$ with arbitrary conformal structure and Gauss map omitting one projective direction.

#### Article information

Source
Geom. Topol., Volume 19, Number 2 (2015), 1015-1062.

Dates
Received: 2 October 2013
Revised: 29 May 2014
Accepted: 6 July 2014
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1510858718

Digital Object Identifier
doi:10.2140/gt.2015.19.1015

Mathematical Reviews number (MathSciNet)
MR3336277

Zentralblatt MATH identifier
1314.49026

Subjects
Primary: 49Q05: Minimal surfaces [See also 53A10, 58E12]
Secondary: 30E10: Approximation in the complex domain

#### Citation

Alarcón, Antonio; López, Francisco J. Approximation theory for nonorientable minimal surfaces and applications. Geom. Topol. 19 (2015), no. 2, 1015--1062. doi:10.2140/gt.2015.19.1015. https://projecteuclid.org/euclid.gt/1510858718

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