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The Hyperbolic Geometry of the Sinh-Gordon Equation

Magdalena Toda
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Abstract

This preliminary report studies immersed surfaces of constant mean curvature in $H^3$ through their {\it adjusted Gauss maps} (as harmonic maps in $S^2$) and their {\it adjusted frames} in SU(2). Lawson's correspondence between Euclidean CMC surfaces and their hyperbolic cousins is interpreted here under a different perspective: the equivalence of their Weierstrass representations (normalized potentials). This work also presents a construction algorithm for the moving frame, the adjusted frame, their Maurer-Cartan forms, and ultimately the CMC immersion.

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Magdalena Toda. 2004-09-15. The Hyperbolic Geometry of the Sinh-Gordon Equation. https://arxiv.org/abs/math/0303326

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