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
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
Primary: 49Q05
Secondary: 30E10