The general relativistic constraint equations. (English) Zbl 1514.83005
Summary: We present the state-of-the-art concerning the relativistic constraints, which describe the geometry of hypersurfaces in a spacetime subject to the Einstein field equations. We review a variety of solvability results, the construction of several classes of solutions of special relevance and place results in the broader context of mathematical general relativity. Apart from providing an overview of the subject, this paper includes a selection of open questions, as well as a few complements to some significant contributions in the literature.
MSC:
| 83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
| 70H45 | Constrained dynamics, Dirac’s theory of constraints |
| 58J47 | Propagation of singularities; initial value problems on manifolds |
| 65E10 | Numerical methods in conformal mappings |
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 58H15 | Deformations of general structures on manifolds |
Keywords:
Einstein constraints; initial data sets; conformal methods; gluing schemes; scalar curvature prescription and deformation problemsReferences:
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