Rough Solutions of the Einstein Constraint Equations on Asymptotically Flat Manifolds without Near-cmc Conditions

In this article we consider the conformal decomposition of Einstein’s constraint equations introduced by Lichnerowicz, Choquet-Bruhat, and York, on asymptotically flat (AF) manifolds. Using the non-CMC fixed-point framework developed in 2009 by Holst, Nagy, and Tsogtgerel and by Maxwell, we combine a priori estimates for the individual Hamiltonian and momentum constraints, barrier constructions and fixed-point techniques for the Hamiltonian constraint, Riesz-Schauder theory for the momentum constraint, together with a topological fixed-point argument for the coupled system, to establish existence of coupled non-CMC weak solutions for AF manifolds. As was the case with the 2009 rough solution results for closed manifolds, and for the more recent 2014 results of Holst, Meier, and Tsogtgerel for rough solutions on compact manifolds with boundary, our results here avoid the near-CMC assumption by assuming that the freely specifiable part of the data given by the traceless-transverse part of the rescaled extrinsic curvature and the matter fields are sufficiently small. Using a coupled topological fixed-point argument that avoids near-CMC conditions, we establish existence of coupled non-CMC weak solutions for AF manifolds of class W s,p δ (or H s,p δ ) where p ∈ (1,∞), s ∈ (1 + 3 p ,∞), −1 < δ < 0, with metric in the positive Yamabe class. The non-CMC rough solutions results here for AF manifolds may be viewed as an extension of the 2009 and 2014 results on rough far-from-CMC positive Yamabe solutions for closed and compact manifolds with boundary to the case of AF manifolds. Similarly, our results may be viewed as extending the recent 2014 results for AF manifolds of Dilts, Isenberg, Mazzeo and Meier; while their results are restricted to smoother background metrics and data, the results here allow the regularity to be extended down to the minimum regularity allowed by the background metric and the matter, further completing the rough solution program initiated by Maxwell and Choquet-Bruhat in 2004.