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

For maps $u\colon M\to M'$ of closed Riemannian manifolds a study is made of the quasi-linearly perturbed harmonic-map equation
$$ \tau(u)(x)=\mathsf G(x,u(x))\cdot du(x)+\mathsf g(x,u(x)), \qquad x\in M. $$
In the case of a non-positively curved manifold $M'$ and a small linear part of the perturbation $\mathsf G$ 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 $C^1$-norms of $\mathsf G$ and $\mathsf g$. The crux of this analysis is an inequality called the monotonicity property.