Minimal two-spheres of constant curvature in a quaternion projective space

We obtain a large number of new minimal 2-spheres of constant curvature in the quaternion projective space $${\mathbb{H}}$$ Pn-1. Using the twistor fibration, we reduce the problem to constructing horizontal holomorphic spheres in the complex projective space $$\mathbb{C}P^{{2n - 1}}$$ . We prove that the set of such horizontal spheres is bijective to a closed disk consisting of certain anti-symmetric matrices modulo the action of $$U(1)\times SU(2)$$ . From this characterization, we deduce a lower bound on the dimension. Our method relies upon the singular decomposition analysis for the planes spanned by the spheres. Finally by checking the squared normal of the first $$\partial$$ -return, we illustrate that the generic ones are not homogeneous, and thus not those that are classified.