Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves

TL;DR

This paper develops methods to construct complete minimal surfaces with Jordan boundaries in Euclidean spaces, extending classical results to higher dimensions and providing new boundary value problem solutions.

Contribution

It introduces approximate solutions to Riemann-Hilbert problems for minimal surfaces and constructs complete conformal minimal immersions with prescribed boundary properties in any dimension.

Findings

Constructed complete conformal minimal surfaces with Jordan boundaries in $ ^n$

Provided proper minimal immersions into convex domains extending continuously to boundaries

Established existence of embeddings for dimensions $n \\ge 5$

Abstract

In this paper we find approximate solutions of certain Riemann-Hilbert boundary value problems for minimal surfaces in $\mathbb{R}^n$ and null holomorphic curves in $\mathbb{C}^n$ for any $n\ge 3$. With this tool in hand we construct complete conformally immersed minimal surfaces in $\mathbb{R}^n$ which are normalized by any given bordered Riemann surface and have Jordan boundaries. We also furnish complete conformal proper minimal immersions from any given bordered Riemann surface to any smoothly bounded, strictly convex domain of $\mathbb{R}^n$ which extend continuously up to the boundary; for $n\ge 5$ we find embeddings with these properties.