Optimal mass transportation in the Heisenberg group

In this paper we consider the problem of optimal transportation of absolutely continuous masses in the Heisenberg group H_n, in the case when the cost function is either the square of the Carnot-Carathéodory distance or the square of the Korányi norm. In both cases we show existence and uniqueness of an optimal transport map. In the former case the proof requires a delicate analysis of minimizing geodesics of the group and of the differentiability properties of the squared distance function. In the latter case the proof requires some fine properties of BV functions in the Heisenberg group.