  A Hilbert manifold structure is described for the ADM phase space of
asymptotically flat initial data $(g,\pi)$ with local $H^2\times H^1$ Sobolev
regularity. Solutions of the constraint equations form a Hilbert submanifold. A
regularized RT Hamiltonian is defined and smooth on the full phase space and
generates the Einstein evolution for any lapse-shift asymptotic to a (time)
translation at infinity. Critical points for the total (ADM) mass, considered
as a function on the Hilbert manifold of constraint solutions, arise precisely
at initial data generating stationary vacuum spacetimes.

General Relativity and Quantum Cosmology