Preprint number: CP3-Origins-2010-10

Authors: J.M. Speight (University of Leeds) and M. Svensson (CP3-Origins)

External link: arXiv.org

The variational problem for the functional *F=½ ∫*_{M}||*φ*ω*||*²* 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.