A far-from-CMC existence result for the constraint equations on manifolds with ends of cylindrical type

We extend the study of the vacuum Einstein constraint equations on manifolds with ends of cylindrical type initiated by Chru'sciel and Mazzeo by finding a class of solutions to the fully coupled system on such manifolds. We show that given a Yamabe positive metric g, which is conformally asymptotically cylindrical on each end, and a 2-tensor K such that (tr K)^2 is bounded below away from zero and asymptotically constant, then we may find an initial data set (g’,K’) such that g’ lies in the conformal class of g.