A gluing construction for non-vacuum solutions of the Einstein-constraint equations

Abstract

We extend the conformal gluing construction of [18] by establishing an analogous gluing result for field theories obtained by minimally coupling Einstein's gravitational theory with matter fields. We treat classical fields such as perfect fluids and the Yang-Mills equations as well as the Einstein-Vlasov system, which is an important example coming from kinetic theory. In carrying out these extensions, we extend the conformal gluing technique to higher dimensions and codify it in such a way as to make more transparent where it can, and can not, be applied. In particular, we show exactly what criteria need to be met in order to apply the construction, in its present form, to any other non-vacuum field theory.

Gluing Initial Data Sets for General Relativity

“…This deformation is done so that we are in the setting in which a generalization of the gluing theorem of [2] to compact manifolds with boundary (and to include matter fields) may be applied. This constitutes the third step in the construction and is essentially done by repeating the arguments of [2,10] in this new setting. We thus obtain a one parameter family of initial data which satisfies the constraint equations, and which contains a neck connecting the spheres S(p a , r).…”

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

“…[1] A justification for this way of specifying initial data for perfect fluids is presented in Section 4.1 of [32].…”

Initial Data Engineering

“…Theorems 1.1 and 1.4 are established in Section 4. The proofs are a mixture of gluing techniques developed in [17][18][19] and those of [12][13][14]. In fact, the proof proceeds via a generalisation of the analysis in [18,19] to compact manifolds with boundary; this is carried through in Section 2 in vacuum with cosmological constant Λ = 0, and in Section 3 with matter and Λ ∈ R. These results may be of independent interest.…”

“…Although the paper [18] only treats the case n = 3, since the generalization to higher dimensions is not difficult (the necessary modifications are discussed in [17]), we work here in general dimension n ≥ 3. We begin with an initial data set (M ,γ,K) whereM has non-empty smooth boundary ∂M and we assume first thatK has constant traceτ = trγK (i.e., these are constant mean curvature, or CMC, initial data sets).…”