Asymptotically flat quasi-convex Riemannian metrics of nonnegative scalar curvature and the constraint equations in general relativity

Quasi-convex metrics are defined in this work to be the Riemannian metrics on R3 for which there is a global foliation by spheres of positive mean and Gauss curvature. We define M to be the set of asymptotically flat, quasi-convex metrics of nonnegative scalar curvature for which this foliation also becomes round in a prescribed way at the point of origin and at infinity. The main theorem presented here is that M is path connected. This theorem is relevant to the Cauchy problem in general relativity since it implies the path connectedness of a corresponding set of asymptotically flat initial data for the Einstein equations.; We prove the main theorem using a parabolic equation for constructing quasi-convex metrics of prescribed scalar curvature. This equation is derived in the present work and provides the most general method to date for constructing metrics of nonnegative scalar curvature. It is a generalization of an equation of Bartnik for prescribing scalar curvature within the class of metrics that admit a foliation by mean convex round spheres.