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Harmonic maps of \(S^ 2\) into a complex Grassmann manifold. (English) Zbl 0601.58023
The authors study harmonic maps from the Riemann sphere \(S^ 2\) to the complex Grassmann manifold \(G_{2,n}\) of complex 2-planes in \(C^ n\). Two processes are described called ”crossing” and ”turning” which, when applied repeatedly, change a harmonic map \(\phi\) : \(S^ 2\to G_{2,n}\) into a harmonic map \(\phi_ 0: S^ 2\to G_{2,n}\) describable in terms of holomorphic maps. These processes depend for their success on the automatic vanishing of holomorphic differentials on \(S^ 2\). The original harmonic map can then be obtained from \(\phi_ 0\) by inverse procedures which involve choosing arbitrary holomorphic sections of \({\mathbb{C}}P^ 1\) bundles. The methods are those of moving frames; full details appear in part II of this paper [Preprint; per revr.]. For an alternative treatment and some further results see work of F. E. Burstall and the reviewer [J. Differ. Geom. 23, 255-297 (1986; Zbl 0588.58018)].
{Reviewer’s remark: In the statement of Theorem 6 it seems that there should be a third possibility (called a mixed pair in Burstall and the reviewer’s paper [op. cit., cf. Theorem 3.3]) where the orthogonal harmonic points in the double harmonic flag \(w_ 0\) are, respectively, a holomorpic and an antiholomorphic map h,g: \(S^ 2\to P_{n-1}\) into complex projective space with \(\partial h\perp g\) (mixed pairs are always harmonic).}

MSC:
58E20 Harmonic maps, etc.
14M15 Grassmannians, Schubert varieties, flag manifolds
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