The existence of minimal immersions of two-spheres

Abstract

In this paper we develop an existence theory for minimal 2-spheres in compact Riemannian manifolds. The spheres we obtain are conformally immersed minimal surfaces except at a finite number of isolated points, where the structure is that of a branch point. We obtain an existence theory for harmonic maps of orientable surfaces into Riemannian manifolds via a complete existence theory for a perturbed variational problem. Convergence of the critical maps of the perturbed problem is sufficient to produce at least one harmonic map of the sphere into the Riemannian manifold. A harmonic map from a sphere is in fact a conformal branched minimal immersion. We prove the existence of minimizing harmonic maps in two cases. If N is a compact Riemannian manifold with j2(N) = 0, then every homotopy class of maps from a closed orientable surface M to N contains a minimiz