Homotopy classes in Sobolev spaces and the existence of energy minimizing maps

Abstract

Consider the problem of finding maps f: M--->N that are stationary for an energy functional such as fM IDfl p (where M and N are compact riemannian manifolds and p~>l). Such maps may be found by minimizing the functional, but if we minimize among all maps from M to N, then the minimum is 0 and is attained only by constant maps. Thus in order to find nontrivial stationary maps, we would like to use the topology of M and N to define classes of maps from M to N in which we can minimize the functional. For instance one could try to minimize among maps in a given homotopy class, but this is not possible in general (unless p>dimM), since a minimizing sequence of mappings in one homotopy class can converge (in the appropriate weak topology) to a map in another homotopy class. However, in this paper we show that it is possible to minimize among maps fwhose restrictions to a lower dimensional skeleton of (a triangulation of) M belong to a given homotopy class. To state the results precisely we need to refer to certain Sobolev spaces and norms. We will assume without loss of generality that M and N are submanifolds of euclidean spaces R m and R ~, respectively, and we let Lip (M, N) denote the space of lipschitz maps from M to N. We define LI'p(M, R) to be the space of all functions fE LP(M, R n) such that there exist functions