The property of compactness of the quasi-linearly perturbed harmonic-map equation

For maps u: M ? M' of closed Riemannian manifolds a study is made of the quasi-linearly perturbed harmonic-map equation t(u)(x) = G(x, u(x)) · du(x) + g(x, u(x)), x ? M. In the case of a non-positively curved manifold M' and a small linear part of the perturbation G it is proved that the space of classical solutions in a fixed homotopy class is compact. The proof is based on a uniform estimate for the norm of the differential of a solution of the perturbed equation in terms of its energy and the C¹norms of G and g. The crux of this analysis is an inequality called the monotonicity property.