New complex analytic methods in the theory of minimal surfaces: a survey

TL;DR

This survey reviews recent complex analytic techniques applied to the global theory of minimal surfaces, highlighting advances in existence, construction, and classification results using modern methods like Oka theory and Riemann-Hilbert problems.

Contribution

It compiles recent developments in applying complex analytic methods to minimal surface theory, emphasizing global results and new construction techniques.

Findings

Solutions to the Calabi-Yau problem for minimal surfaces

Construction of properly immersed and embedded minimal surfaces in Euclidean spaces

Analysis of the homotopy type of conformal minimal immersions

Abstract

In this paper we survey recent developments in the classical theory of minimal surfaces in Euclidean spaces which have been obtained as applications of both classical and modern complex analytic methods; in particular, Oka theory, period dominating holomorphic sprays, gluing methods for holomorphic maps, and the Riemann-Hilbert boundary value problem. Emphasis is on results pertaining to the global theory of minimal surfaces, in particular, the Calabi-Yau problem, constructions of properly immersed and embedded minimal surfaces in $\mathbb{R}^n$ and in minimally convex domains of $\mathbb{R}^n$, results on the complex Gauss map, isotopies of conformal minimal immersions, and the analysis of the homotopy type of the space of all conformal minimal immersions from a given open Riemann surface.