Darboux transforms and spectral curves of constant mean curvature surfaces revisited

TL;DR

This paper explores Darboux transforms of constant mean curvature surfaces, revealing their spectral curves and providing an algebro-geometric representation of CMC tori, connecting geometric transformations with integrable systems.

Contribution

It introduces a new algebro-geometric framework for understanding Darboux transforms of CMC surfaces and relates spectral curves from different approaches.

Findings

The space of Darboux transforms of a CMC torus forms an algebraic spectral curve.

All Darboux transforms on the spectral curve are themselves CMC tori.

The spectral curve from Darboux transforms shares the same normalization as traditional integrable systems spectral curves.

Abstract

We study the geometric properties of Darboux transforms of constant mean curvature (CMC) surfaces and use these transforms to obtain an algebro-geometric representation of constant mean curvature tori. We find that the space of all Darboux transforms of a CMC torus has a natural subset which is an algebraic curve (called the spectral curve) and that all Darboux transforms represented by points on the spectral curve are themselves CMC tori. The spectral curve obtained using Darboux transforms is not bi-rational to, but has the same normalisation as, the spectral curve obtained using a more traditional integrable systems approach.