Abstract
This chapter reviews complex structures on a manifold, then gives an elementary exposition of the complex structure induced on a surface by a Riemannian metric. In this way a complex structure is induced on any surface immersed into one of the space forms. Surfaces immersed into Möbius space inherit a complex structure. In all cases we use this structure to define a reduction of a moving frame to a unique frame associated to a given complex coordinate. Umbilic points do not hinder the existence of these frames, in contrast to the obstruction they can pose for the existence of second order frame fields. The Hopf invariant and the Hopf quadratic differential play a prominent role in the space forms as well as in Möbius geometry. Using the structure equations of the Hopf invariant h, the conformal factor e u, and the mean curvature H of such frames, we give an elementary description of the Lawson correspondence between minimal surfaces in Euclidean geometry and constant mean curvature equal to one (CMC 1) surfaces in hyperbolic geometry; and between minimal surfaces in spherical geometry and CMC surfaces in Euclidean geometry.
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Jensen, G.R., Musso, E., Nicolodi, L. (2016). Complex Structure. In: Surfaces in Classical Geometries. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-27076-0_7
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DOI: https://doi.org/10.1007/978-3-319-27076-0_7
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