  We initiate the study of the norm-squared of the momentum map as a rigorous
tool in infinite dimensions. In particular, we calculate the Hessian at a
critical point, show that it is positive semi-definite along the complexified
orbit, and determine a decomposition of the stabilizer under the complexified
action. We apply these results to the action of the group of symplectomorphisms
on the spaces of compatible almost complex structures and of symplectic
connections. In the former case, we extend results of Calabi to not necessarily
integrable almost complex structures that are extremal in a relative sense. In
both cases, the momentum map is not equivariant, which gives rise to new
phenomena and opens up new avenues for interesting applications. For example,
using the prequantization construction, we obtain new central extensions of the
group of symplectomorphisms that are encoding geometric information of the
underlying finite-dimensional manifold.
