The article under review handles the registration problem modeled as a Riemannian gradient flow on Sobolev diffeomorphisms generated by vector fields on a manifold. The authors handle it by using a gradient method regularized on the space of Riemannian metrics. The effect of transforming a template to a target is quantified through an energy functional, which penalizes non-isometric deformations. The problem is treated first in an abstract context, by considering gradient flows on Lie groups, then, for a generic right-invariant Riemannian metric, the energy functional is obtained via lifting a group action, which generates a momentum map (compare with Proposition 1.2), which allows a geometric description of the gradient. The authors initially obtain abstract global existence results (see e.g. Proposition 2.3, Theorem 2.7 and lemas 3.1 and 3.2). By considering the special case, when the group consists of Sobolev diffeomorphisms of a compact Riemannian manifold, and by overcoming technical difficulties, such as the fact that the space of Sobolev diffeomorphisms of a Hilbert manifold do not form a Lie group with respect to its manifold structure, among other hindernesses, the authors construct, through several lemmas, the proof of their main result, an existence theorem assuring that the modelling gradient flow is well-posed (Theorem 1.1). The technicalities involved in the paper are surpassed by the clear and detailed style of the authors.