Parametrization of the Space of Solutions of Einstein's Equations

An explicit parametrization is obtained of the set of all space-time solutions of Einstein's equations which are globally hyperbolic and contain a compact spatial hypersurface with constant mean curvature. This parametrization is based upon the conformal treatment of the initial-value problem for Einstein's equations, which is studied by the method of sub and super solutions for quasilinear elliptic partial differential equations. The Yamabe-Aubin-Trudinger-Schoen classification of conformal classes of Riemannian metrics plays a key role.