Planar harmonic mappings with a given Jacobian

The article is devoted to the study of the Jacobians of sense-preserving harmonic mappings in the unit disk of the complex plane. The main result is a criterion for an infinitely differentiable positive function to be a Jacobian of some sense-preserving harmonic mapping. The relationship between a Jacobian of a harmonic mapping and the Schwarzian derivative of its dilatation is revealed. The structure of the set of harmonic mappings with a given Jacobian is described. The results are illustrated by examples. In conclusion, we consider an application of the main results of the article to the construction of variational formulas in classes of harmonic mappings with a given Jacobian.