On a compact manifold, the scalar curvature map at generic metrics is a local surjection [F-M]. We show that this result may be localized to compact subdomains in an arbitrary Riemannian manifold. The method is extended to establish the existence of asymptotically flat, scalar-flat metrics on R n (n ≥ 3) which are spherically symmetric, hence Schwarzschild, at infinity, i.e. outside a compact set. Such metrics provide Cauchy data for the Einstein vacuum equations which evolve into nontrivial vacuum spacetimes which are identically Schwarzschild near spatial infinity.

Given asymptotically flat initial data on M 3 for the vacuum Einstein field equation, and given a bounded domain in M , we construct solutions of the vacuum constraint equations which agree with the original data inside the given domain, and are identical to that of a suitable Kerr slice (or identical to a member of some other admissible family of solutions) outside a large ball in a given end. The data for which this construction works is shown to be dense in an appropriate topology on the space of asymptotically flat solutions of the vacuum constraints. This construction generalizes work in [C], where the time-symmetric case was studied.

No abstract

We study the stability of the Penrose compactification for solutions of the vacuum Einstein equation, in the context of the time-symmetric initialvalue problem. The initial data (R 3 , g) must satisfy the Hamiltonian constraint R(g) = 0, and we consider perturbations about the Euclidean metric arising from tensors h satisfying the equation L(h) = 0, where L is the linearization of the scalar curvature operator at the Euclidean metric. We show that each member h of a large family of compactly supported solutions of the linearized problem is tangent to a curveḡ of solutions to the nonlinear constraint, so that each metricḡ along the curve evolves under the vacuum Einstein equation to a spacetime which is asymptotically simple in the sense of Penrose.

We consider localized deformation for initial data sets of the Einstein field equations with the dominant energy condition. Deformation results with the weak inequality need to be handled delicately. We introduce a modified constraint operator to absorb the first order change of the metric in the dominant energy condition. By establishing the local surjectivity theorem, we can promote the dominant energy condition to the strict inequality by compactly supported variations and obtain new gluing results with the dominant energy condition. The proof of local surjectivity is a modification of the earlier work for the usual constraint map by the first named author and R. Schoen [5] and by P. Chruściel and E. Delay [2], with some refined analysis.