BPS Skyrme models and contact geometry

A Skyrme type energy functional for maps $\phi$ from an oriented Riemannian 3-manifold $M$ to a contact 3-manifold $N$ is defined, generalizing the BPS Skyrme energy of Ferreira and Zakrzewski. This energy has a topological lower bound, attained by solutions of a first order self-duality equation which we call (strong) Beltrami maps. In the case where $N$ is the 3-sphere, we show that the original Ferreira-Zakrzewski model (which has $N=S^3$ with the standard contact structure) can have no BPS solutions on $M=S^3$ with $\mathrm{\mid }\mathrm{d}\mathrm{e}\mathrm{g}(\phi )\mathrm{\mid }>1$ if the coupling constant has the lowest admissible value.