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\(\mathbb{C} P^{N-1}\) harmonic maps and the Weierstrass problem. (English) Zbl 1062.58020
Summary: A Weierstrass-type system of equations corresponding to the \(\mathbb{C} P^{N-1}\) harmonic maps is presented. The system constitutes a further generalization of our previous construction [J. Math. Phys. 44, No. 1, 328–337 (2003; Zbl 1061.53043)]. It consists of four first order equations for three complex functions which are shown to be equivalent to the \(\mathbb{C} P^{N-1}\) harmonic maps. When the harmonic maps are holomorphic (or antiholomorphic) one of the functions vanishes and the system reduces to the previously given generalization of the Weierstrass problem. We also discuss a possible interpretation of our results and show that in our new case the induced metric is proportional to the total energy density of the map and not only to its holomorphic part, as was the case in the previous generalizations.

MSC:
58E20 Harmonic maps, etc.
53C43 Differential geometric aspects of harmonic maps
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