Links between different analytic descriptions of constant mean curvature surfaces

Abstract

Transformations between different analytic descriptions of constant mean curvature (CMC) surfaces are established. In particular, it is demonstrated that the system \[ \begin{split} &\partial ψ_{1} = (|ψ_{1}|^{2} + |ψ_{2}|^{2}) ψ_{2} \\ &\bar{\partial} ψ_{2} =- (|ψ_{1}|^{2} + |ψ_{2}|^{2}) ψ_{1} \end{split} \] 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 $S^2\subset E^3$.

Cited by in corpus (5)

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• Surfaces in ${\mathbb{R}}^{N^2-1}$ based on harmonic maps $S^2\to C{P}^{N-1}$
• On complete integrability of the generalized Weierstrass system
• On CP1 and CP2 maps and Weierstrass representations for surfaces immersed into multi-dimensional Euclidean spaces