Manifold structures for sets of solutions of the general relativistic constraint equations (Q557257)

A natural question that arises in general relativity is whether some sets of solutions of the, say vacuum, constraint equations carry a manifold structure. For example, it is useful to have a Banach manifold structure on the set of asymptotically flat solutions of the constraint equations when trying to minimize the ADM mass. The purpose of this note is to show that a Banach manifold structure can be obtained by a variation of the Fischer-Marsden-Moncrief-Corvino-Schoen method. It turns out that a direct application of the standard a priori estimates for the streamlined construction does not seem to lead to a manifold structure based on Sobolev spaces, which would have been more natural for the evolution problem, and which would have led to a Hilbert manifold structure. Instead, e.g. on compact manifolds without boundary, the authors obtain a manifold modeled on (a subset of) the space \(C^{k,\alpha}\times C^{k,\alpha}\), \(k\geq 4\), \(\alpha\in (0, 1)\) of symmetric tensors. This appears somewhat surprising at first sight, as a natural set-up for the evolution problem (regardless of the Sobolev versus Hölder space issue) might seem to be one where the differentiability of the extrinsic curvature tensor \(K\) is one order less than that of the metric \(g\). On the other hand, since \(K\)'s can be thought of as variations of \(g\)'s, from a manifold structure point of view it seems natural that the \(K\)'s live in a space with the same differentiability as \(g\). Whatever the natural space is, the \(C^{k,\alpha}\times C^{k,\alpha}\) topology or weighted versions thereof are the ones which are obtained by the method here; this is a rather unexpected consequence of the analysis in this paper. The manifolds of initial data obtained here exhibit more structure than what is obtained by the conformal method and its variations. In the construction of the manifold structure, the authors use a smoothing device to recover the loss of regularity inherent to the Fischer-Marsden-Moncrief approach. This allows one to work consistently in spaces with finite differentiability, leading to the Banach manifold structure described above. The authors use a general approach of weighted spaces, as in their previous paper [On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications, Mém. Soc. Math. Fr. 94, 1--103 (2003; Zbl 1058.83007), see also gr-qc/0301073v2], which allows a simultaneous treatment of the compact case with or without boundary, of the asymptotically flat case, and of the conformally compactifiable case, with families of different topologies. All the results obtained remain valid in the time-symmetric setting, \(K=0\). This implies that all the manifold structures presented here have their obvious counterparts for the set of Riemannian metrics with prescribed scalar curvature. This paper is organized as follows: The authors show how to define a local Hilbert manifold of solutions near a given solution; the resulting manifolds cannot be patched together in general because of insufficient regularity of the operators involved. It is presented the basic regularization procedure, which turns out to still be insufficient to provide a (global) Hilbert manifold structure. The authors therefore pass to an analysis in weighted Hölder spaces and prove there that (the KID-free part of) the level sets of the constraints map are, globally, embedded submanifolds in a Banach space, see Theorem 5.2 of the paper, under very general conditions on the weights; this is the main result of the paper. In fact, the authors prove that the level sets of the constraint map foliate (in some sense) the KID-free part of the space of all \((K,g)\)'s. It is shown that the hypotheses made in Theorem~ 5.2 are fulfilled on compact manifolds with or without boundary, or on asymptotically compactifiable manifolds, or on asymptotically flat manifolds. The authors prove a lemma which provides a submanifold structure in Banach spaces under rather general conditions, as well as a foliation result and present two regularization procedures in weighted spaces, as needed in applications of the submanifold Theorem~ 5.2.