Compact complete minimal immersions in R^3

TL;DR

This paper constructs compact, complete minimal immersions in R^3 for any finite topological type, extending to the boundary with a 1-dimensional image, and shows their density among minimal surfaces spanning finite curves.

Contribution

It introduces a method to produce compact complete minimal surfaces of arbitrary finite topological type with boundary embeddings and Hausdorff dimension one, and proves their density in the space of minimal surfaces.

Findings

Existence of compact complete minimal immersions with boundary embeddings.

Boundary images have Hausdorff dimension one.

Density of complete minimal surfaces in the space of surfaces spanning finite curves.

Abstract

In this paper we find, for any arbitrary finite topological type, a compact Riemann surface $\mathcal{M},$ an open domain $M\subset\mathcal{M}$ with the fixed topological type, and a conformal complete minimal immersion $X:M\to\R^3$ which can be extended to a continuous map $X:\bar{M}\to\R^3,$ such that $X_{|\partial M}$ is an embedding and the Hausdorff dimension of $X(\partial M)$ is $1.$ We also prove that complete minimal surfaces are dense in the space of minimal surfaces spanning a finite set of closed curves in $\R^3$, endowed with the topology of the Hausdorff distance.