Harmonic maps of the two-sphere into the complex hyperquadric
\textit{S. S. Chern} and the author have studied harmonic maps of \(S^ 2\) into a complex Grassmann manifold [cf. Proc. Natl. Acad. Sci. USA 82, 2217-2219 (1985; Zbl 0601.58023)], in which the degenerate harmonic maps arise. In this paper the author discusses the minimal immersions or harmonic maps of the two-sphere \(S^ 2\) into the complex hyperquadric \(Q_{n-2}\). To each harmonic map \(f: S^ 2\to Q_{n-2}^ a \)directrix curve \(\Delta_ f: S^ 2\to G(2,n;C)\) can be associated which is either a holomorphic curve or a degenerate harmonic map. The main theorem of this paper is as follows: Let \(f: S^ 2\to G(2,n;C)\) be a real nondegenerate harmonic map. Then there is associated to f either (i) a unique r-null holomorphic curve \(\Delta_ f: S^ 2\to G(2,n;C)\), \(r\geq 0\), or (ii) a unique r-null degenerate harmonic map \(\Delta_ f: S^ 2\to G(2,n;C)\), \(r\geq 0\). f can be recovered from \(\Delta_ f\) via recrossing and the \(\partial\) and \({\bar \partial}\) transforms. Since degenerate harmonic maps can be constructed from holomorphic curves \(S^ 2\to P_{n-1}\) using returnings, this theorem provides a description of the harmonic maps \(S^ 2\to Q_{n-2}\).
scientific article; zbMATH DE number 3984163
scientific article