Links between different analytic descriptions of constant mean curvature surfaces

TL;DR

This paper explores the relationships between various analytic descriptions of constant mean curvature surfaces, showing how certain systems decouple into well-known equations and linking them to sigma models and instanton solutions.

Contribution

It establishes transformations between different analytic descriptions of CMC surfaces and connects these systems to elliptic equations and sigma models, revealing new insights.

Findings

Decoupling of the system into elliptic equations

Connections with sigma model equations

Instanton solutions relate to sphere parametrizations

Abstract

Transformations between different analytic descriptions of constant mean curvature (CMC) surfaces are established. In particular, it is demonstrated that the system \[ \begin{split} &\partial \psi_{1} = (|\psi_{1}|^{2} + |\psi_{2}|^{2}) \psi_{2} \\ &\bar{\partial} \psi_{2} =- (|\psi_{1}|^{2} + |\psi_{2}|^{2}) \psi_{1} \end{split} \] descriptive of CMC surfaces within the framework of the generalized Weierstrass representation, decouples into a direct sum of the elliptic Sh-Gordon and Laplace equations. Connections of this system with the sigma model equations are established. It is pointed out, that the instanton solutions correspond to different Weierstrass parametrizations of the standard sphere $S^{2} \subset E^{3}$.