Compactification of moduli spaces of extremals of 2-dimensional conformally invariant variational problems

Abstract

An extension ℒ ′ O ( M ) of the space Ω 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 from the moduli space of maps from a Riemann surface (Σ, 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.