Conformal minimal surfaces immersed into \({\mathbb {H}}P^{n}\)

In this paper, we want to construct conformal minimal surfaces and conformal minimal two-spheres in \({\mathbb {H}}P^{n}\) by the twistor map \(\pi : {\mathbb {C}}P^{2n+1} \rightarrow {\mathbb {H}}P^{n}\). The construction is due to Eells and Wood’s conclusion about the composition of a horizontal harmonic map in 1983. Firstly, we give a characterization of horizontal holomorphic surfaces in \({\mathbb {C}}P^{5}\). Under this characterization, we construct eight families of conformal minimal surfaces in \({\mathbb {H}}P^{2}\). Then, we study horizontal Veronese sequences in \({\mathbb {C}}P^{4}\) and \({\mathbb {C}}P^{5}\), and we transform the construction into solving a quadratic equation. Based on this, we get some examples of conformal minimal two-spheres in \({\mathbb {H}}P^{2}\) with constant curvature \(\frac{4}{5}\) and \(\frac{4}{13}\).