The Weierstrass Representation always gives a minimal surface

TL;DR

This paper provides a straightforward proof that the Weierstrass Representation always produces minimal surfaces, simplifying the understanding of their relationship without addressing the converse.

Contribution

It offers a simple, direct proof of the fact that the Weierstrass Representation always yields minimal surfaces, avoiding complex converse arguments.

Findings

Weierstrass Representation always produces minimal surfaces.

The proof simplifies understanding of minimal surface generation.

The paper clarifies the relationship without the converse.

Abstract

We give a simple, direct proof of the easy fact about the Weierstrass Representation, namely, that it always gives a minimal surface. Most presentations include the much harder converse that every simply connected minimal surface is given by the Weierstrass Representation.