Minimizing conformal energies in homotopy classes

Abstract

We consider a minimizing sequence for the conformal energy in a given homotopy class of maps between two compact Riemannian manifolds M and N. In general this sequence will fail to be (strongly) convergent in the natural Sobolev class, but will have a weak limit which is not a priori in the original homotopy class. We prove a topological decomposition theorem: the homotopy class of the original map is given as the composition (in an appropriate sense) of the homotopy class of the weak limit with a finite number of free homotopy classes of maps from the sphere (with dimension that of the manifold M) into N. The method of proof shows that the weak limit is a minimizer in its homotopy class, and also shows that the homotopy classes of maps from the sphere occurring in the decomposition can be represented by minimizers in their respective classes.