Bumpy metrics and closed parametrized minimal surfaces in Riemannian manifolds

The purpose of this article is to study conformal harmonic maps f : Sigma -> M, where Sigma is a closed Riemann surface and M is a compact Riemannian manifold of dimension at least four. Such maps de. ne parametrized minimal surfaces, possibly with branch points. We show that when the ambient manifold M is given a generic metric, all prime closed parametrized minimal surfaces are free of branch points, and are as Morse nondegenerate as allowed by the group of automorphisms of Sigma. They are Morse nondegenerate in the usual sense if Sigma has genus at least two, lie on two-dimensional nondegenerate critical submanifolds if Sigma has genus one, and on six-dimensional nondegenerate critical submanifolds if Sigma has genus zero.