Gauge fixing of one Killing field reductions of canonical gravity: The case of asymptotically flat induced two-geometry

We consider one spacelike Killing vector field reductions of four-dimensional (4D) vacuum general relativity. We restrict attention to cases in which the manifold of the orbits of the Killing field is ${\mathit{R}}^3$. The reduced Einstein equations are equivalent to those for Lorentzian 3D gravity coupled to an SO(2,1) nonlinear σ model on this manifold. We examine the theory in terms of a Hamiltonian formulation obtained via a 2+1 split of the 3D manifold. We restrict attention to geometries which are asymptotically flat in a 2D sense defined recently. We attempt to pass to a reduced Hamiltonian description in terms of the true degrees of freedom of the theory via gauge-fixing conditions of 2D conformal flatness and maximal slicing. We explicitly solve the diffeomorphism constraints and relate the Hamiltonian constraint to the prescribed negative curvature equation in ${\mathit{R}}^2$ studied by mathematicians. We partially address issues of existence and/or uniqueness of solutions to the various elliptic partial differential equations encountered.