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Approximation theory for nonorientable minimal surfaces and applications. (English) Zbl 1314.49026

Summary: We prove a version of the classical Runge and Mergelyan uniform approximation theorems for nonorientable minimal surfaces in Euclidean 3-space \(\mathbb{R}^3\). Then we obtain some geometric applications. Among them, we emphasize the following ones:
\(\bullet\) A Gunning-Narasimhan-type theorem for nonorientable conformal surfaces.
\(\bullet\) An existence theorem for nonorientable minimal surfaces in \(\mathbb{R}^3\) with arbitrary conformal structure, properly projecting into a plane.
\(\bullet\) An existence result for nonorientable minimal surfaces in \(\mathbb{R}^3\) with arbitrary conformal structure and Gauss map omitting one projective direction.

MSC:

49Q05 Minimal surfaces and optimization
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
30E10 Approximation in the complex plane
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