Some global minimizers of a symplectic Dirichlet energy

The variational problem for the functional $F=\frac12\|ϕ^*ω\|_{L^2}^2$ is considered, where $ϕ:(M,g)\to (N,ω)$ 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 $π:S^3\to S^2$ 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 $S^3$ 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.

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