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Then we find relationships with almost complex maps from a surface into the 6-sphere; this enables us to construct examples of nilconformal harmonic maps into $G_2/{\\mathrm SO}(4)$ which are not of finite uniton number, and which have lifts into any of the three twistor spaces. 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Wood, Martin Svensson","submitted_at":"2013-03-28T16:53:37Z","abstract_excerpt":"We show that a harmonic map from a Riemann surface into the exceptional symmetric space $G_2/{\\mathrm SO}(4)$ has a $J_2$-holomorphic twistor lift into one of the three flag manifolds of $G_2$ if and only if it is `nilconformal', i.e., has nilpotent derivative. Then we find relationships with almost complex maps from a surface into the 6-sphere; this enables us to construct examples of nilconformal harmonic maps into $G_2/{\\mathrm SO}(4)$ which are not of finite uniton number, and which have lifts into any of the three twistor spaces. 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