Volume 19, issue 2 (2015)

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Approximation theory for nonorientable minimal surfaces and applications

Antonio Alarcón and Francisco J López

Geometry & Topology 19 (2015) 1015–1062

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

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nonorientable minimal surfaces, uniform approximation
Mathematical Subject Classification 2010
Primary: 49Q05
Secondary: 30E10
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
Antonio Alarcón
Departamento de Geometría y Topología
Universidad de Granada
E-18071 Granada
Francisco J López
Departamento de Geometría y Topología
Universidad de Granada
E-18071 Granada