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 ${ℝ}^3$. 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.