Characterizing the image space of a shape-dependent operator for a potential flow problem

We study the reachability of a shape-dependent operator based on a potential flow and give a complete characterization of the image space. We draw a connection between the structure of the image space and the set of stagnation points, i.e. the set of surface points where the tangential velocity vanishes. We use conformal pull-back to a reference domain and reduce the problem to the question of whether there exists a diffeomorphism which pulls back one top-dimensional differential form to another. For volume forms this question has been answered by Moser 1965, but since we do allow singularities we have to prove a modified version. This leads to a volume condition, which must be fulfilled on every connected component of the nonzero set of the form.