COMPACTIFICATION OF MODULI SPACES OF EXTREMALS OF 2-DIMENSIONAL CONFORMALLY INVARIANT VARIATIONAL PROBLEMS

An extension L-O'(M) of the space Omega(2)(M)' of De Rham currents on a manifold M better adapted to the study of conformally invariant variational problems, is introduced. This extension is the dual of the space of conformally invariant first-order Lagrangian densities for maps from C to M. A map front the moduli space of maps from a Riemann surface (Sigma, j) to M to L-O(M)', is defined, and its restriction to the moduli of embeddings is proved to be injective. A general result of compactness on L-O(M)' is stated and used to obtain compactifications of subsets of the moduli space. In the particular case of J-holomorphic curves such a compactification is compared with Gromov's compactification.