Lagrangian harmonic two-spheres in \(\mathbb{C} P^ n\) with few higher order singularities (Q1924527)

Using properties of the directrix curve \(\psi_0\) or the harmonic sequence \(\psi_0, \dots, \psi_n\), one studies harmonic maps \(\psi: S^2\to \mathbb{C} P^n\) which are Lagrangian [see \textit{X. H. Mo}, Sci. China Ser. A 38, No. 5, 524-532 (1995; Zbl 0827.53048); \textit{J. Eells} and \textit{J. C. Wood}, Adv. Math. 49, No. 3, 217-263 (1983; Zbl 0528.58007), and \textit{J. Bolton}, \textit{W. M. Oxbury}, \textit{L. Vrancken}, and \textit{L. M. Woodward}, Lect. Notes Math. 1481, 18-27 (1991; Zbl 0745.53033)]. With the help of existence and uniqueness theorems the following result is obtained: Let \(\psi\) be a linearly full harmonic map which is \(k\)-point ramified for \(k\leq 2\) and factors through \(\mathbb{R} P^2\). Then \(\psi\) is a Lagrangian if and only if every higher order singularity of \(\psi\) (if any) has the same type.