Harmonic parametrization of surfaces of arbitrary genus

TL;DR

This paper explores the use of the Weierstrass representation to construct harmonic surfaces of any genus, overcoming topological limitations of minimal surfaces and enabling complex geometric modeling.

Contribution

It demonstrates how the Weierstrass representation can be applied to harmonic surfaces to create embedded surfaces with arbitrary genus and complicated ends.

Findings

Constructed harmonic surfaces of arbitrary genus

Combined embedded harmonic ends successfully

Extended the applicability of Weierstrass representation

Abstract

The Weierstrass representation for minimal surfaces in $\mathbb{R}^3$ provides a flexible method for constructing minimal surfaces of arbitrary genus. The topological limitations of minimal surfaces interfere with this providing a more general geometric modeling tool. Minimal surfaces lie in the larger class of harmonic surfaces, which in general don't have the same topological limitations of minimal surfaces and can have complicated embedded ends. In this paper we demonstrate the flexibility of using the Weierstrass representation for harmonic surfaces to combine embedded harmonic ends together to construct embedded harmonic surfaces of arbitrary genus.