Yamabe classification and prescribed scalar curvature in the asymptotically Euclidean setting

Abstract

We prove a necessary and sufficient condition for an asymptotically Euclidean manifold to be conformally related to one with specified nonpositive scalar curvature: the zero set of the desired scalar curvature must have a positive Yamabe invariant, as defined in the article. We show additionally how the sign of the Yamabe invariant of a measurable set can be computed from the sign of certain generalized "weighted" eigenvalues of the conformal Laplacian. Using the prescribed scalar curvature result we give a characterization of the Yamabe classes of asymptotically Euclidean manifolds. We also show that the Yamabe class of an asymptotically Euclidean manifold is the same as the Yamabe class of its conformal compactification.In particular, λ g ′ ,δ (Z) ≥ 0, and λ g ′ ,δ (Z) = 0 only if u is constant. But Z has positive measure, and therefore A(Z) does not contain any constants. Hence λ g ′ ,δ (Z) > 0, and Proposition 3.10 implies Z is Yamabe positive.This completes the proof of Theorem 4.1. Turning to the compact case (Theorem 4.2) recall that we started the AE argument with the following inessential simplifying hypotheses:1. The prescribed scalar curvature R ′ is bounded.

Existence and blowup results for asymptotically Euclidean initial data sets generated by the conformal method

“…While the analytical simplifications of the conformal constraint equations which result from working with either CMC or near-CMC data sets occur regardless of whether the data is specified on a closed manifold or is asymptotically Euclidean (or is asymptotically hyperbolic), the Yamabe classification of AE metrics is more complicated (and less intuitive) than the Yamabe classification of metrics on closed manifolds. For example, it has been shown (see [DM15]) that an asymptotically Euclidean metric can be conformally deformed to an AE metric with zero scalar curvature if and only if the metric is Yamabe positive (as defined via the Yamabe invariant (29) below). As well, it has been shown that an AE metric is Yamabe null if and only if it can be conformally deformed to a metric with scalar curvature R for every function R ≤ 0 except R ≡ 0.…”

“…The very recent advances in our understanding of the Yamabe classes of metrics for asymptotically Euclidean metrics (see [DM15]), besides indicating some of the difficulties of the analysis of the conformal constraint equations for AE data, also provide information which is useful in handling these difficulties. In this paper, after a brief review (in Section 2) of asymptotically Euclidean geometries and their analytic features (properties of Fredholm operators on AE geometries, the various AE maximum principles and sub and supersolution theorems, and the AE Yamabe classes), we discuss a number of new results concerning solutions of the conformal constraint equations for various classes of AE seed data.…”

“…Many of the new insights obtained in [DM15] concerning the Yamabe classification of asymptotically Euclidean metrics involve the prescribed scalar curvature problem for conformal deformation of AE metrics. Since this problem plays a major role in the Admissibility Corollary, we are immediately led to simple results regarding the existence and nonexistence of solutions to (3)-(4) for various classes of seed data.…”

On the Positive Energy Theorem for Stationary Solutions to Fourth-Order Gravity

“…Finally, g = v It is worth noticing that, looking more carefully to the above proof, we can weaken the hypotheses of the previous theorem and still get interesting results. Indeed, in the positivity part, we only use the fact that the scalar curvature is positive at infinity, while for the rigidity statement, we only use the fact that we can make the scalar curvature flat via a conformal transformation, which is ensured by assuming that Y ([g]) > 0 by theorem 5.1 of [13], and that Q g is non-negative. Furthermore, under this last condition, equations ( 23) and ( 23) also provide proofs of positivity.…”

The seed-to-solution method for the Einstein equations and the asymptotic localization problem

“…For establishing the existence of solutions, the standard technique goes back to pioneering work by Lichnerowicz and followers who developped the so-called conformal method. For a detailed bibliography we refer the reader to the papers, and the the references therein, [3]- [14], [18], [24], and [25]. Specifically, we build here upon the recent work by Carlotto and Schoen [4] about the localization problem (see Section 1.3 below), which stems from Corvino and Schoen's earlier work [11].…”