Local existence theory for a class of CMC gauges for the Einstein-non-linear scalar field equations

The purpose of this article is to develop a local existence theory for a class of CMC gauges for the Einstein-non-linear scalar field equations. We do so in the context of closed and parallelisable initial manifolds. The assumption that the initial manifold is parallelisable is not a topological restriction in the case of closed oriented 3-manifolds, but it is a restriction in higher dimensions. The results include local existence, uniqueness, a continuation criterion and Cauchy stability. Previous results concerning related gauges have been restricted to the case of n-torus spatial topology. Moreover, they have not covered the Einstein-non-linear scalar field setting. The main motivation for developing this theory is that it forms the basis for a family of past global non-linear stability results we derive in a separate article.