Some global minimizers of a symplectic Dirichlet energy

TL;DR

This paper studies the minimizers of a symplectic Dirichlet energy functional, proving the minimality of certain maps like the Hopf fibration and exploring conditions for critical points and minimizers in various geometric contexts.

Contribution

It establishes that the Hopf fibration minimizes the symplectic Dirichlet energy in its homotopy class and extends this result to Berger's spheres, also characterizing critical points with coclosed pullback forms.

Findings

Hopf fibration minimizes the energy in its homotopy class

Coclosed pullback forms imply criticality and minimality

Holomorphic projections into Hermitian symmetric spaces also minimize energy

Abstract

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