SOME GLOBAL MINIMIZERS OF A SYMPLECTIC DIRICHLET ENERGY

The variational problem for the functional F = 1/2 integral(M) parallel to phi*omega parallel to(2) is considered, where phi : (M, g) -> (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-3 -> 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 Berger's sphere. It is proved that if phi*omega is coclosed, F is 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.