Factorization and uniton numbers for harmonic maps into the unitary group \(U(N)\)
Any harmonic map from the Riemann sphere \(S^2\) to the unitary group \(U(N)\) can be factorized into a product of a finite number of maps into Grassmannians \(G_k(\mathbb{C}^N)\) [see \textit{K. Uhlenbeck}, J. Differ. Geom. 30, No. 1, 1-50 (1989; Zbl 0677.58020)]. These maps are called unitons. The authors prove a theorem of factorization of harmonic maps from a simply-connected domain \(\Omega\subset\mathbb{C}\cup\{\infty\}\) to the unitary group. An upper bound for the minimal uniton number of a harmonic map \(\varphi:\Omega \to U(N)\) with finite uniton number is obtained. As a corollary, it is shown that any harmonic map \(\varphi:\omega\to \mathbb{C} P^{N-1}\) with finite uniton number is isotropic.
scientific article; zbMATH DE number 936978