Minimal surfaces in Euclidean spaces by way of complex analysis

TL;DR

This paper surveys recent complex-analytic methods applied to conformal minimal surfaces in Euclidean spaces, highlighting new results on approximation, interpolation, and existence problems, including the Calabi-Yau problem.

Contribution

It provides a comprehensive survey with new results on minimal surface approximation, interpolation, and existence, connecting complex analysis with minimal surface theory.

Findings

New approximation and interpolation results for minimal surfaces

Existence of minimal surfaces with prescribed Gauss maps

Progress on the Calabi-Yau problem for minimal surfaces

Abstract

This is an expanded version of my plenary lecture at the 8th European Congress of Mathematics in Portoro\v{z} on 23 June 2021. The main part of the paper is a survey of recent applications of complex-analytic techniques to the theory of conformal minimal surfaces in Euclidean spaces. New results concern approximation, interpolation, and general position properties of minimal surfaces, existence of minimal surfaces with a given Gauss map, and the Calabi-Yau problem for minimal surfaces. To be accessible to a wide audience, the article includes a self-contained elementary introduction to the theory of minimal surfaces in Euclidean spaces.