The Hyperbolic Geometry of the Sinh-Gordon Equation

Details The Hyperbolic Geometry of the Sinh-Gordon Equation The Hyperbolic Geometry of the Sinh-Gordon Equation

2003-03-26

arxiv.org/abs/math/0303326v2

Mathematics

Differential Geometry

19 pages

Scientific paper

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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