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**On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications. (Sur les propriétés de l’opérateur de contraintes relativistes dans des espaces à poids, et applications).**
*(English.
French summary)*
Zbl 1058.83007

This interesting paper is devoted to the study of the mapping properties of the general relativistic constraints operator, acting in weighted function spaces. The general relativistic constraints equations have the form
\[
\begin{pmatrix} J\cr \rho \end{pmatrix} (K,g)\equiv \begin{pmatrix} 2(-\nabla^jK_{ij}+\nabla_i \text{tr} \;K)\cr R(g)-| K| ^2+( \text{tr} \;K)^2\end{pmatrix}=\begin{pmatrix} 0\cr 0\end{pmatrix},\tag{\(*\)}
\]
where \((K,g)\) are the initial data for the vacuum Einstein equations. These initial data belong to the zero level set of the constraints map \((*)\). The authors introduce the linearization of the constraints map at \((K,g)\) as an operator \(P(Q,h)\), where \(h=\delta g\), \(Q=\delta K\). The order of the differential operators that appear in \(P\) is \( \left(\begin{smallmatrix} 1 & 1 \cr 0 & 2 \end{smallmatrix} \right)\).

J. Corvino (2000) presented a new gluing construction of scalar flat metrics, leading to the existence of non-trivial scalar flat metrics, which are exactly Schwarzschildian at large distances. It turns out that the methods introduced by J. Corvino and R. Schoen can be applied to obtain new classes of solutions of the general relativistic constraint equations. Here the authors present an abstract version of the arguments of Corvino and Schoen in a large class of weighted Sobolev spaces. A general theory of mapping properties of the solutions of the linearized constraint operator in a class of weighted Sobolev spaces is developed. The class of these spaces includes those of Christodoulou-Choquet-Bruhat (1981). The distance-weighted spaces near a boundary and an exponentially weighted version are of interest. These classes are relevant near a compact boundary, or in an asymptotically hyperboloidal context. Very important estimates in all the spaces are established. An appropriate version of the inverse function theorem allows new classes of solutions to be obtained. An application of the techniques developed here concerns the construction of initial data, which are exactly Kerrian outside of a compact set. A second application is the construction of initial data, which are stationary to high asymptotic order. Another noteworthy application is the construction of initial data containing black-hole regions with exactly Kerrian geometry both near the apparent horizons, and in the asymptotic region. This leads to the result of the existence of “many Schwarzschild” black holes.

J. Corvino (2000) presented a new gluing construction of scalar flat metrics, leading to the existence of non-trivial scalar flat metrics, which are exactly Schwarzschildian at large distances. It turns out that the methods introduced by J. Corvino and R. Schoen can be applied to obtain new classes of solutions of the general relativistic constraint equations. Here the authors present an abstract version of the arguments of Corvino and Schoen in a large class of weighted Sobolev spaces. A general theory of mapping properties of the solutions of the linearized constraint operator in a class of weighted Sobolev spaces is developed. The class of these spaces includes those of Christodoulou-Choquet-Bruhat (1981). The distance-weighted spaces near a boundary and an exponentially weighted version are of interest. These classes are relevant near a compact boundary, or in an asymptotically hyperboloidal context. Very important estimates in all the spaces are established. An appropriate version of the inverse function theorem allows new classes of solutions to be obtained. An application of the techniques developed here concerns the construction of initial data, which are exactly Kerrian outside of a compact set. A second application is the construction of initial data, which are stationary to high asymptotic order. Another noteworthy application is the construction of initial data containing black-hole regions with exactly Kerrian geometry both near the apparent horizons, and in the asymptotic region. This leads to the result of the existence of “many Schwarzschild” black holes.

Reviewer: Dimitar A. Kolev (Sofia)

### MSC:

83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |

37N20 | Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics) |

83C57 | Black holes |