New Complex Analytic Methods in the Study of Non-Orientable Minimal Surfaces in $ MathbbR^{n} - Alarcón, Antonio; Forstnerič, Franc; López, Francisco J (American Mathematical Society, 2020)
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 the authors develop essentially from the first principles, enable them 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. The authors also give the first known example of a properly embedded non-orientable minimal surface in \mathbb{R}^4; a Möbius strip. All the new tools mentioned above apply to non-orientable minimal surfaces endowed with a fixed choice of a conformal structure. This enables the authors to obtain significant new applications to the global theory of non-orientable minimal surfaces. In particular, they 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.