This talk will develop part of the foundation needed to develop a partial Morse theory for conformal harmonic maps from a Riemann surface into a Riemannian manifold. Such maps are also called parametrized minimal surfaces. A partial Morse theory for such objects should parallel the well-known Morse theory of smooth closed geodesics.

The first step needed is a bumpy metric theorem which states that when a Riemannian manifold has a generic metric, all prime minimal surfaces are free of branch points and lie on nondegenerate critical submanifolds. (A parametrized minimal surface is prime if it does not cover a parametrized minimal surface of lower energy.)

We will present such a theorem and describe some applications.