Approximation theory for non-orientable minimal surfaces and applications

TL;DR

This paper extends classical approximation theorems to non-orientable minimal surfaces in 3D space and explores their geometric applications, including existence results and conformal surface properties.

Contribution

It develops approximation theorems for non-orientable minimal surfaces and applies them to establish new existence and geometric properties of such surfaces.

Findings

Proved Runge and Mergelyan type theorems for non-orientable minimal surfaces.

Established existence of non-orientable minimal surfaces with arbitrary conformal structures.

Derived geometric applications including projection and Gauss map properties.

Abstract

We prove a version of the classical Runge and Mergelyan uniform approximation theorems for non-orientable minimal surfaces in Euclidean 3-space R3. Then, we obtain some geometric applications. Among them, we emphasize the following ones: 1. A Gunning-Narasimhan type theorem for non-orientable conformal surfaces. 2. An existence theorem for non-orientable minimal surfaces in R3, with arbitrary conformal structure, properly projecting into a plane. 3. An existence result for non-orientable minimal surfaces in R3 with arbitrary conformal structure and Gauss map omitting one projective direction.