Critical symplectic connections on surfaces

Abstract

The space of symplectic connections on a symplectic manifold is a symplectic affine space. M. Cahen and S. Gutt showed that the action of the group of Hamiltonian diffeomorphisms on this space is Hamiltonian and calculated the moment map. This is analogous to, but distinct from, the action of Hamiltonian diffeomorphisms on the space of compatible almost complex structures that motivates study of extremal K\"ahler metrics. In particular, moment constant connections are critical, where a symplectic connection is \textit{critical} if it is critical, with respect to arbitrary variations, for the $L^{2}$-norm of the Cahen-Gutt moment map. This occurs if and only if the Hamiltonian vector field generated by its moment map image is an infinitesimal automorphism of the symplectic connection. This paper develops the study of moment constant and critical symplectic connections, following, to the extent possible, the analogy with the similar, but different, setting of constant scalar curvature and extremal K\"ahler metrics. It focuses on the special context of critical symplectic connections on surfaces, for which general structural results are obtained, although some results about the higher-dimensional case are included as well. For surfaces, projectively flat and preferred symplectic connections are critical, and the relations between these and other related conditions are examined in detail. The relation between the Cahen-Gutt moment map and the Goldman moment map for projective structures is explained.