Some Global Minimizers of a Symplectic Dirichlet Energy

Preprint number: CP3-Origins-2010-10
Authors: J.M. Speight (University of Leeds) and M. Svensson (CP3-Origins)
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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 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.