Approximation theory for nonorientable minimal surfaces and applications

Alarcón, Antonio; López, Francisco J

doi:10.2140/gt.2015.19.1015

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Approximation theory for nonorientable minimal surfaces and applications

Antonio Alarcón and Francisco J López

Geometry & Topology 19 (2015) 1015–1062

DOI: 10.2140/gt.2015.19.1015

Abstract

We prove a version of the classical Runge and Mergelyan uniform approximation theorems for nonorientable minimal surfaces in Euclidean $3$ –space $ℝ^{3}$ . 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 $ℝ^{3}$ with arbitrary conformal structure, properly projecting into a plane.
An existence result for nonorientable minimal surfaces in $ℝ^{3}$ with arbitrary conformal structure and Gauss map omitting one projective direction.

Keywords

nonorientable minimal surfaces, uniform approximation

Mathematical Subject Classification 2010

Primary: 49Q05

Secondary: 30E10

References

Publication

Received: 2 October 2013

Revised: 29 May 2014

Accepted: 6 July 2014

Published: 10 April 2015

Proposed: Tobias H Colding

Seconded: Yasha Eliashberg, Simon Donaldson

Authors

Antonio Alarcón
Departamento de Geometría y Topología Universidad de Granada E-18071 Granada Spain
http://www.ugr.es/local/alarcon/
Francisco J López
Departamento de Geometría y Topología Universidad de Granada E-18071 Granada Spain
http://www.ugr.es/local/fjlopez/

Volume 19, issue 2 (2015)