### 2.1 The constraints

The unknowns in the constraint equations are the initial data for the Einstein equations. These consist of a three-dimensional manifold , a Riemannian metric , and a symmetric tensor on , and initial data for any matter fields present. The equations are:
Here is the scalar curvature of the metric , and and are projections of the energy-momentum tensor. Assuming that the matter fields satisfy the dominant energy condition implies that . This means that the trivial procedure of making an arbitrary choice of and and defining and by Equations (1) is of no use for producing physically interesting solutions.

The usual method for solving the Equations (1) is the conformal method [95]. In this method parts of the data (the so-called free data) are chosen, and the constraints imply four elliptic equations for the remaining parts. The case that has been studied the most is the constant mean curvature (CMC) case, where is constant. In that case there is an important simplification. Three of the elliptic equations, which form a linear system, decouple from the remaining one. This last equation, which is nonlinear, but scalar, is called the Lichnerowicz equation. The heart of the existence theory for the constraints in the CMC case is the theory of the Lichnerowicz equation.

Solving an elliptic equation is a non-local problem and so boundary conditions or asymptotic conditions are important. For the constraints, the cases most frequently considered in the literature are that where is compact (so that no boundary conditions are needed) and that where the free data satisfy some asymptotic flatness conditions. In the CMC case the problem is well understood for both kinds of boundary conditions [76110187]. (Note that a significant mistake in the literature has been corrected in [244], section 4.) The other case that has been studied in detail is that of hyperboloidal data [9]. The kind of theorem that is obtained is that sufficiently differentiable free data, in some cases required to satisfy some global restrictions, can be completed in a unique way to a solution of the constraints. It should be noted in passing that in certain cases physically interesting free data may not be “sufficiently differentiable” in the sense it is meant here. One such case is mentioned at the end of Section 2.6. The usual kinds of differentiability conditions that are required in the study of the constraints involve the free data belonging to suitable Sobolev or Hölder spaces. Sobolev spaces have the advantage that they fit well with the theory of the evolution equations (compare the discussion in Section 2.2). The question of the minimal differentiability necessary to apply the conformal method has been studied in [243], where it was shown that the method works for metrics in the Sobolev space with . It was also shown that each of these solutions can be approximated by a sequence of smooth solutions.

Usually it is not natural to prescribe the values of solutions of the Einstein equations on a finite boundary. There is, however, one case which naturally occurs in physical problems, namely that of the boundary of a black hole. Existence of solutions of the constraints appropriate for describing black holes has been proved by solving boundary value problems in [127] and [244].

In the non-CMC case our understanding is much more limited although some results have been obtained in recent years (see [19193] and references therein). It is an important open problem to extend these so that an overview is obtained comparable to that available in the CMC case. Progress on this could also lead to a better understanding of the question of whether a spacetime that admits a compact, or asymptotically flat, Cauchy surface also admits one of constant mean curvature. Up to now there have been only isolated examples that exhibit obstructions to the existence of CMC hypersurfaces [34]. Until very recently it was not known whether there were vacuum spacetimes with a compact Cauchy surface admitting no CMC hypersurfaces. In [117] it was shown using gluing techniques (see below) that spacetimes of this type do exist and this fact restricts the applicability of CMC foliations for defining a preferred time coordinate in cosmological spacetimes. Certain limitations of the conformal method in producing non-CMC initial data sets were exhibited in [193].

It would be interesting to know whether there is a useful concept of the most general physically reasonable solutions of the constraints representing regular initial configurations. Data of this kind should not themselves contain singularities. Thus it seems reasonable to suppose at least that the metric is complete and that the length of , as measured using , is bounded. Does the existence of solutions of the constraints imply a restriction on the topology of or on the asymptotic geometry of the data? This question is largely open, and it seems that information is available only in the compact and asymptotically flat cases. In the case of compact , where there is no asymptotic regime, there is known to be no topological restriction. In the asymptotically flat case there is also no topological restriction implied by the constraints beyond that implied by the condition of asymptotic flatness itself [349189]. This shows in particular that any manifold that is obtained by deleting a point from a compact manifold admits a solution of the constraints satisfying the minimal conditions demanded above. A starting point for going beyond this could be the study of data that are asymptotically homogeneous. For instance, the Schwarzschild solution contains interesting CMC hypersurfaces that are asymptotic to the metric product of a round 2-sphere with the real line. More general data of this kind could be useful for the study of the dynamics of black hole interiors [292].

Recently techniques have been developed for gluing together solutions of the constraints (see [117] and references therein). Given two solutions of the constraints it is possible, under very general conditions, to cut a hole in each and connect the resulting pieces by a wormhole to get a new solution of the constraints. Depending on the variant of the method used, the geometry on the original pieces is changed by an arbitrarily small amount, or not at all. This gives a new flexibility in constructing solutions of the constraints with interesting properties.

To sum up, the conformal approach to solving the constraints, which has been the standard one up to now, is well understood in the compact, asymptotically flat and hyperboloidal cases under the constant mean curvature assumption, and only in these cases. For some other approaches see [3536353]. New techniques have been applied by Corvino [121] to prove the existence of regular solutions of the vacuum constraints on that are Schwarzschild outside a compact set. The latter ideas have also flowed into the gluing constructions mentioned above.