Extremal symplectic connections on surfaces

M. Cahen and S. Gutt found the moment map for the action of the symplectomorphism group on the symplectic affine space of symplectic connections on a symplectic manifold. This paper studies extremal and moment constant volume-preserving connections on surfaces, where a symplectic connection is extremal if it is critical for the $L^{2}$ norm of the moment map with respect to arbitrary variations. A symplectic connection is extremal if and only if the Hamiltonian vector field generated by its moment map image acts on it as an infinitesimal automorphism. A preferred symplectic connection is extremal. On a compact surface the Levi-Civita connection of a K\ahler structure is extremal symplectic if and only if it has constant curvature, and, more generally, an extremal connection is either projectively flat or is somehow inherently nonmetric; if the genus is at least two an extremal symplectic connection is moment flat. On the plane there are constructed examples of extremal symplectic connections that are neither moment constant nor preferred. Similar examples are found on the sphere, but they are not smooth at the poles. On any compact surface there are moment flat connections that are not projectively flat; the map associating to an orbit of the connected component of the identity of the symplectomorphism group acting in the space of moment flat symplectic connections the cohomology class of the one-form obtained by contracting with the symplectic form the projective Cotton tensor of a representative connection is surjective onto the first cohomology of the surface. W. Goldman showed that the projective Cotton tensor is a moment map for the action of the diffeomorphism group on the space of strictly convex flat real projective structures on a closed surface. The close relation between the Cahen-Gutt and Goldman moment maps for a surface equipped with a volume form is explained.