Some global minimizers of a symplectic Dirichlet energy
Abstract
The variational problem for the functional F=12\|φ*ω\|L22 is considered, where φ:(M,g) (N,ω) 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 π:S3 S2 is known to be a locally stable critical point of F. It is proved here that π in fact minimizes F in its homotopy class and this result is extended to the case where S3 is given the metric of the Berger's sphere. It is proved that if φ*ω is coclosed then φ 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 φ*ω 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.