I prove that the only minimal surfaces of constant positive Gaussian curvature in the n-sphere are, up to rigid motion, the Boruvka spheres, i.e., the 2-dimensional orbits of an irreducible representation of SO(3) into SO(m) for some m less than or equal to n+1. I rederive Ejiri's classification of the minimal surfaces of zero Gaussian curvature in the n-sphere and prove that there are no minimal surfaces of constant negative curvature in the n-sphere. (Partial results had been obtained by Ejiri.) I also prove that the only minimal surfaces of constant curvature in the hyperbolic n-ball are the totally geodesic surfaces. (That the only minimal surfaces of constant curvature in flat space are the planes is due to Pinl.)
The methods are purely local and depend on analysing the overdetermined system for minimal isometric embedding by organizing the integrability conditions into managable form, so that one can actually differentiate them many times and still have some control over the resulting relations.
The actual results are stronger than these theorems suggest. What I do is classify the harmonic maps with constant energy density from a surface of constant Gauss curvature to the n-sphere. In this form, I have recently generalized these results in On extremals with prescribed Lagrangian densities.
Reprints are available.