Mappings that minimize area in their homotopy classes

Abstract

Let /: M -> X be a continuous map from a compact connected oriented m-dimensional manifold M into a compact Riemannian manifold X. In this paper we consider the problem: does there exist a lipschitz map g: M -> X that minimizes m-dimensional mapping area (or some other parametric elliptic functional) subject to the condition that g be homotopic to/? If so, what is the minimum area attained? And, if not, what is the infimum? It has long been known that in each homology class of X, there is an integral current that minimizes area (in that class). In this paper we show that, for m > 3, the homotopy problem reduces to the homology problem. For instance, if X is simply connected, the infimum area of mappings homotopic to / is equal to the minimum area among integral currents homologous to/ # ([M]) (where [M] e 3? m (M) is the m-dimensional integral cycle orienting M). Furthermore, if the current solution T is sufficiently regular, then the infimum is attained by a map whose image is the support of T together with a lower-dimensional singular set.