Here
*R*
is the scalar curvature of the metric
and
and
are projections of the energy-momentum tensor. Assuming matter
fields which 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) and (2) is of no use for producing physically interesting
solutions.

The usual method for solving the equations (1) and (2) is the conformal method [41]. 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 which has been studied 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
*S*
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 [32,
55,
96]. The other case which has been studied in detail is that of
hyperboloidal data [2]. The kind of theorem which 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.

In the non-CMC case our understanding is much more limited although some results have been obtained in recent years (see [99, 40] 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, when a spacetime which admits a compact, or asymptotically flat, Cauchy surface also admits one of constant mean curvature. Up to now there are only isolated examples which exhibit obstructions to the existence of CMC hypersurfaces [13].

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
*S*
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
*S*, 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 [166] This shows in particular that any manifold which 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
which are asymptotically homogeneous. For instance, the
Schwarzschild solution contains interesting CMC hypersurfaces
which are asymptotic to the product of a 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 [144].

To sum up, the conformal approach to solving the constraints, which is 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 [14], [15] and [169].

Local and Global Existence Theorems for the Einstein
Equations
Alan D. Rendall
http://www.livingreviews.org/lrr-2000-1
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