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We shall discuss the class of surfaces with holomorphic right Gauss maps in non-compact duals of compact semi-simple Lie groups (e.g. ), which contains minimal surfaces in and constant mean curvature surfaces in . A Weierstrass type representation formula and a Chern-Osserman type inequality for such surfaces are given.

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2.
We show there is a global correspondence between branched constant mean curvature (i.e. CMC-) immersions in and pairs of non-conformal harmonic maps into in the same associated family. Furthermore, we give two applications.

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