Loop Weierstrass Representation

TL;DR

This paper introduces a unified Loop Weierstrass Representation for minimal and constant mean curvature surfaces, applying integral methods to unify various transformations and classify examples.

Contribution

It presents a novel unified framework for representing and transforming minimal and CMC surfaces using holomorphic data and integral systems.

Findings

Unified representation for minimal and CMC surfaces.

Introduction of a simple factor dressing method.

Classification of various surface examples.

Abstract

We introduce the Loop Weierstrass Representation for minimal surfaces in Euclidean space and constant mean curvature 1 surfaces in hyperbolic space by applying integral system methods to the Weierstrass and Bryant representations. We unify associated families, dual surfaces and Goursat transformations under the same holomorphic data, we introduce a simple factor dressing for minimal surfaces, and we compute and classify various examples.