Far-from-Constant Mean Curvature Solutions of Einstein’s Constraint Equations with Positive Yamabe Metrics

Abstract

We establish new existence results for the Einstein constraint equations for mean extrinsic curvature arbitrarily far from constant. The results hold for rescaled background metric in the positive Yamabe class, with freely specifiable parts of the data sufficiently small, and with matter energy density not identically zero. Two technical advances make these results possible: A new topological fixed-point argument without smallness conditions on spatial derivatives of the mean extrinsic curvature, and a new global supersolution construction for the Hamiltonian constraint that is similarly free of such conditions. The results are presented for strong solutions on closed manifolds, but also hold for weak solutions and for compact manifolds with boundary. These results are apparently the first that do not require smallness conditions on spatial derivatives of the mean extrinsic curvature. Introduction. The question of existence of solutions to the Lichnerowicz-York conformally rescaled Einstein's constraint equations, for an arbitrarily prescribed mean extrinsic curvature, has remained an open problem for more than thirty years [1]. The rescaled equations, which are a coupled nonlinear elliptic system consisting of the scalar Hamiltonian constraint coupled to the vector momentum constraint, have been studied almost exclusively in the setting of constant mean extrinsic curvature, known as the CMC case. In the CMC case the equations decouple, and it has long been known how to establish existence of solutions. The case of CMC data on closed (compact without boundary) manifolds was completely resolved by several authors over the last twenty years, with the last remaining subcases resolved and summarized by Isenberg in [2]. Over the last ten years, other CMC cases were studied and resolved; see the survey [3].

Solutions to the Einstein-scalar field constraint equations with a small TT-tensor

“…In this paper, we prove a far-from-CMC result similar to [15,20,21,29] for the conformal Einstein-scalar field constraint equations on compact Riemannian manifolds with positive (modified) Yamabe invariant. …”

“…This assumption however will turn out to be very important in our analysis and plays a role analog to the assumption that the metric g has positive Yamabe invariant in [20,21,29]. Another important assumption we will need is that (M, g) has no non-zero conformal Killing vector field.…”

“…In this section, we adapt the method of [20,21,29] to our context. The first step is to prove an existence result for solutions to the Lichnerowicz equation.…”

“…It was only in 2008 that Holst, Nagy and Tsogtgerel found a method to construct solutions to the equations of the conformal method with arbitrarily prescribed mean curvature. See [20,21]. The method was then extended by Maxwell [29] to the vacuum case.…”

On the stability of solutions of the Lichnerowicz–York equation

“…One motivation for that study was to examine the uniqueness of solutions in the far-fromconstant mean curvature regime now that existence results for this case exist (see [8], [14]). He found interesting non-existence and non-uniqueness results showing that the constraints are ill-posed beyond the non-uniqueness that is introduced when one couples the lapse fixing equation to the four constraint equations, as in the extended conformal thin sandwich (XCTS) formulation (see [15], [2], [19]) and some constrained evolution schemes (see [16], [6] for a resolution of this scaling problem).…”