New complex analytic methods in the study of non-orientable minimal surfaces in $\mathbb{R}^n$

TL;DR

This paper develops new complex analytic methods based on Oka theory to study non-orientable minimal surfaces in Euclidean spaces, leading to significant theoretical advances and explicit examples.

Contribution

It adapts complex analytic techniques to non-orientable minimal surfaces, proving they form real analytic Banach manifolds and establishing approximation and general position theorems.

Findings

Space of conformal minimal immersions is a real analytic Banach manifold.

Constructed the first properly embedded non-orientable minimal surface in $\

Provided new existence results for non-orientable minimal surfaces with prescribed properties.

Abstract

The aim of this work is to adapt the complex analytic methods originating in modern Oka theory to the study of non-orientable conformal minimal surfaces in $\mathbb{R}^n$ for any $n\ge 3$. These methods, which we develop essentially from the first principles, enable us to prove that the space of conformal minimal immersions of a given bordered non-orientable surface to $\mathbb{R}^n$ is a real analytic Banach manifold, obtain approximation results of Runge-Mergelyan type for conformal minimal immersions from non-orientable surfaces, and show general position theorems for non-orientable conformal minimal surfaces in $\mathbb{R}^n$. We also give the first known example of a properly embedded non-orientable minimal surface in $\mathbb{R}^4$; a Mobius strip. All our new tools mentioned above apply to non-orientable minimal surfaces endowed with a fixed choice of a conformal structure. This enables us to obtain significant new applications to the global theory of non-orientable minimal surfaces. In particular, we construct proper non-orientable conformal minimal surfaces in $\mathbb{R}^n$ with any given conformal structure, complete non-orientable minimal surfaces in $\mathbb{R}^n$ with arbitrary conformal type whose generalized Gauss map is nondegenerate and omits $n$ hyperplanes of $\mathbb{CP}^{n-1}$ in general position, complete non-orientable minimal surfaces bounded by Jordan curves, and complete proper non-orientable minimal surfaces normalized by bordered surfaces in $p$-convex domains of $\mathbb{R}^n$.