Harmonic sequences and harmonic maps of surfaces into complex Grassmannian manifolds (Q1101750)

Let \(G(k,n)\) be the Grassmann manifold of all k-dimensional subspaces \({\mathbb{C}}^ k\) in complex space \({\mathbb{C}}^ n.\) The author describes all the harmonic maps of the two-sphere \(S^ 2\) into \(G(k,n)\) in terms of holomorphic data and all the harmonic maps of the torus \(T^ 2\) into \(G(k,n)\) in terms of holomorphic data and degree zero harmonic maps. The classification depends on a construction called the \(\partial\)-transform (or \({\bar \partial}\)-transform) defined as follows. Let \(f: M\to G(k,n)\) be a map of a Riemann surface into \(G(k,n)\). For \(x\in M\) the space \(f(x)\) has an orthogonal space \(f(x)^{\perp}\) of dimension \(n-k\). Denoteby \([f(x)]\) and \([f(x)^{\perp}]\) their corresponding projective spaces. For a vector \(Z(x)\in f(x)\) the orthogonal projection of \(\partial Z\) in \(f(x)^{\perp}\) (after suitable identifications) defines a point of \(f(x)^{\perp}\). This defines a fundamental collineation \(\partial: [f(x)]\to [f(x)^{\perp}]\). The mapping defined by sending \(x\in M\) to the image of \([f(x)]\) under \(\partial\) is called the \(\partial\)-transform of \(f\). It \(f\) is now a harmonic map, successive applications of the \(\partial\)-transform (or \({\bar \partial}\)-transform) yields a sequence of harmonic maps called a harmonic sequence. A reverse procedure is possible called returning. The author shows that a harmonic map \(f: S^ 2\to G(k,n)\) can be constructed from holomorphic or antiholomorphic curves \(S^ 2\to G(\ell,n)\), \(1\leq \ell \leq k\), using \(\partial\)- and \({\bar \partial}\)-transforms and returnings. A similar procedure applies to the case when \(M\) has genus 1. The description of harmonic at harmonic maps of \(S^ 2\) into \(G(k,n)\) has also been obtained by \textit{K. Uhlenbeck} [J. Differ. Geom. 30, 1-50 (1989; Zbl 0677.58020)] by using quite different methods.