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

Abstract

For each set of (freely chosen) seed data, the conformal method reduces the Einstein constraint equations to a system of elliptic equations, the conformal constraint equations. We prove an admissibility criterion, based on a (conformal) prescribed scalar curvature problem, which provides a necessary condition on the seed data for the conformal constraint equations to (possibly) admit a solution. We then consider sets of asymptotically Euclidean (AE) seed data for which solutions of the conformal constraint equations exist, and examine the blowup properties of these solutions as the seed data sets approach sets for which no solutions exist. We also prove that there are AE seed data sets which include a Yamabe nonpositive metric and lead to solutions of the conformal constraints. These data sets allow the mean curvature function to have zeroes.

The Einstein constraint equations on compact manifolds with boundary

“…Finally, we require inf τ > 0, where we assume τ > 0 rather than τ < 0 without loss of generality. This is similar to [DGH12,GS12,DGI14].…”

“…We will extend the results of Holst, Nagy, Tsogtgerel and Maxwell in finding far-from-CMC solutions (cf [HNT08,Max09]), as well as extend their methods, such as proving a global sub/supersolution existence theorem and using a Green's function to show that only a supersolution is actually needed in many cases. We also make partial progress in proving the results of previous 'limit equation' papers such as [DGH12,GS12,DGI14]. We were unable, however, to complete the final step in deriving the limit equation.…”

“…Since then much progress has been made, both in considering other types of manifolds and in loosening the restriction on the mean curvature. Hyperbolic [IP97,GS12], asymptotically Euclidean [CBIY00,DGI14], asymptotically cylindrical [CM12, CMP12, Lea13] and compact with boundary [HT13] manifolds have now been considered. The case when the mean curvature is near constant (i.e., the near-CMC condition) is well understood for closed manifolds (see [IM96, I ÓM04, ACI08]), and progress has been made in other cases as well (such as in this paper or [IP97,GS12,DGI14,Lea13]).…”

“…Hyperbolic [IP97,GS12], asymptotically Euclidean [CBIY00,DGI14], asymptotically cylindrical [CM12, CMP12, Lea13] and compact with boundary [HT13] manifolds have now been considered. The case when the mean curvature is near constant (i.e., the near-CMC condition) is well understood for closed manifolds (see [IM96, I ÓM04, ACI08]), and progress has been made in other cases as well (such as in this paper or [IP97,GS12,DGI14,Lea13]). The far-from-CMC case resists analysis, but limited results have been achieved, originally by Holst, Nagy and Tsogtgerel in [HNT08] and extended by Maxwell in [Max09].…”

A new point of view on the solutions to the Einstein constraint equations with arbitrary mean curvature and small TT-tensor

“…In particular, strong results are obtained for negatively curved manifolds; see [12, proposition 6.2 and remark 6.3]. The case of asymptotically Euclidean manifolds and compact manifolds with boundary is currently work in progress [6,11]. New difficulties show up in these cases.…”