Weierstrass type representation of harmonic maps into symmetric spaces. (English) Zbl 0932.58018

The paper is concerned with describing the harmonic maps between a Riemannian surface \(M\) and a compact symmetric space \(G/K\). The situation of simply connected \(M\) is treated, in particular \(M\) can be thought of as the universal cover of some compact Riemann surface of genus \(\geq 1\). In this case, a systematic scheme for the construction of all harmonic maps \(M\to G/K\) is developed.
To do so, a Weierstrass type representation is used which relates harmonic maps to certain holomorphic 1-forms taking values in a subspace of a twisted loop algebra. Integration of this form over \(M\) and some factorization according to loop group decompositions are tools for the construction of a harmonic map. If \(M\) is not the Riemann sphere \(S^2\), every harmonic map \(M\to G/K\) can be obtained by such a construction.


58E20 Harmonic maps, etc.
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