Some global minimizers of a symplectic Dirichlet energy

The variational problem for the functional $F=frac12|phi^*omega|_{L^2}^2$ is considered, where $phi:(M,g)to (N,omega)$ maps a Riemannian manifold to a symplectic manifold. This functional arises in theoretical physics as the strong coupling limit of the Faddeev-Hopf energy, and may be regarded as a symplectic analogue of the Dirichlet energy familiar from harmonic map theory. The Hopf fibration $pi:S^3to S^2$ is known to be a locally stable critical point of $F$. It is proved here that $pi$ in fact minimizes $F$ in its homotopy class and this result is extended to the case where $S^3$ is given the metric of the Bergers sphere. It is proved that if $phi^*omega$ is coclosed then $phi$ is a critical point of $F$ and minimizes $F$ in its homotopy class. If $M$ is a compact Riemann surface, it is proved that every critical point of $F$ has $phi^*omega$ coclosed. A family of holomorphic homogeneous projections into Hermitian symmetric spaces is constructed and it is proved that these too minimize $F$ in their homotopy class.