Quasiconvex Foliations and Asymptotically Flat Metrics of Non-negative Scalar Curvature

Abstract

We prove that a broad subset of the space of asymptotically flat Riemannian metrics of nonnegative scalar curvature on R 3 is connected using a new method for prescribing scalar curvature that generalizes a method developed by Bartnik for quasi-spherical metrics.

Deforming three-manifolds with positive scalar curvature

“…In [41], the authors proved path-connectedness of the space of metrics in M 2 that admit a quasi-convex global foliation. Once we have established Theorem 1.2, Theorem 1.3 will follow by the conformal method.…”

Black hole initial data with a horizon of prescribed geometry

“…Of course, this requires some pointwise a priori bounds on the solution u, but in the case that u is positive and bounded initially, these are easily obtained by the maximum principle under appropriate assumptions on the free data. The result of this is contained in the next lemma, whose proof can be found in [16,17], and is a slight generalization of a result contained in [2]. The proof is repeated here since it yields simple but important bounds on the components of the constructed metric.…”

“…By using the maximum principle (see [17]) one can bound a solution of this equation with initial data w 1 from below by a solution of the ordinary differential equation…”

On the Existence and Stability of the Penrose Compactification

“…We note that the weighting conventions are not universal, and that we have chosen to follow those in [3] and [39] (cf. [8,11,33]).…”

“…Since we restrict to weights 0 < β, τ < (n − 2), and p > 1, the Laplacian ∆ : X k 0 → X k −2 is an isomorphism [3,39]. Thus we see that L * :…”