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Abstract. Let M be a closed orientable surface of genus larger than zero and N a compact Riemannian manifold. If u: M -> N is a continuous map, such that the map induced by it between the fundamental groups of M and N contains no nontrivial element represented by a simple closed curve in its kernel, then there exists a conformai branched minimal immersion s: M -N having least area among all branched immersions with the same action on it¡(M) as u. Uniqueness within the homotopy class of u fails in general: It is shown that for certain 3-manifolds which fiber over the circle there are at least two geometrically distinct conformai branched minimal immersions within the homotopy class of any inclusion map of the fiber. There is also a topological discussion of those 3-manifolds for which uniqueness fails.Introduction. In a previous paper we obtained results on minimal immersions of the two-sphere into compact Riemannian manifolds [18]. Here we extend some of these results to surfaces of higher topological type. The main idea is to reduce the minimal area problem for such surfaces to a variational problem on a moduli space for conformai structures on the surface. The main technical difficulty to overcome is the fact that the moduli spaces are not compact. We overcome this by using a standard compactification of the Riemann moduli space (see Bers [5, 6] and Abikoff [2]) and controlling the behaviour of our variational problem near the boundary points.Our main result, Theorem 4.4, proves the existence of a conformai branched minimal immersion of a surface M with genus larger than zero corresponding to every homotopy class of maps u: M -* N which induces an injection u^: ttx(M) -» 77,(jV). Here N is any compact Riemannian manifold of dimension larger than two.

In this article we announce a series of results on the existence of harmonic maps from surfaces to Riemannian manifolds and, as corollaries of these results, obtain theorems on the existence of minimal immersions of 2-spheres.Let N be a compact connected Riemannian manifold and, for convenience, assume that N is isometrically imbedded in R k for some sufficiently large k. Let M be a closed Riemann surface with any metric compatible with its conformaiHarmonic maps satisfy an Euler-Lagrange equationAs + A(s)(ds f ds) = 0 in a weak sense, where A is the second fundamental form of the imbedding N C R fc . It then follows from regularity theorems that harmonic maps are C°°. If s is harmonic and a conformai immersion, it is also an extremal for the area integral.Proving the existence of harmonic maps of M into N by direct methods from global analysis such as Morse theory or Ljusternik-Schnirelman theory applied to E defined on some function space manifold is difficult, because E is invariant under the conformai group of M y and the extremal maps of E form a noncompact set when M = S 2 . In particular, E does not satisfy condition C of Palais-Smale. However, for a > 1, a slightly different integral, AMS (MOS) subject classifications (1970). Primary 53A10, 58E05.