Exotic Minimal Surfaces

TL;DR

This paper introduces a fusion theorem for minimal surfaces in three-dimensional space, enabling the construction of universal and space-filling minimal surfaces with complex, exotic geometries and arbitrary genus.

Contribution

It presents a general fusion theorem for complete orientable minimal surfaces with finite total curvature, allowing the creation of surfaces with exotic geometries and universal properties.

Findings

Constructed universal minimal surfaces from which all others can be derived

Produced space-filling minimal surfaces with arbitrary genus and no symmetries

Extended the class of known minimal surfaces with exotic geometries

Abstract

We prove a general fusion theorem for complete orientable minimal surfaces in $\mathbb{R}^3$ with finite total curvature. As a consequence, complete orientable minimal surfaces of weak finite total curvature with exotic geometry are produced. More specifically, universal surfaces (i.e., surfaces from which all minimal surfaces can be recovered) and space-filling surfaces with arbitrary genus and no symmetries.