Abstract
We first show that the connected sum along submanifolds introduced by the second author for compact initial data sets of the vacuum Einstein system can be adapted to the asymptotically Euclidean and to the asymptotically hyperbolic context. Then, we prove that in every case, and generically, the gluing procedure can be localized, in order to obtain new solutions which coincide with the original ones outside of a neighborhood of the gluing locus.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Andersson, L., Chruściel, P.T.: On asymptotic behavior of solutions of the constraint equations in general relativity with “hyperboloidal boundary conditions”. Dissert. Math. 355, 1–100 (1996)
Bartnik, R.: The mass of an asymptotically flat manifold. Comm. Pure Appl. Math. 39(5), 661–693 (1986)
Bartnik, R., Isenberg, J.: The Einstein equationsand the large scale behavior of gravitational fields. In: Chrusciel, P.T., Friedrich, H. (eds.) The Constraint Equations, pp. 1–39. Basel, Birkhäuser (2004)
Beig, R., Chruściel, P.T., Schoen, R.: KIDs are non-generic. Ann. Henri Poincaré 6(1), 155–194 (2005)
Christodoulou, D., O’Murchada, N.: The boost problem in general relativity. Commun. Math. Phys. 80, 171–300 (1981)
Choquet-Bruhat, Y.: Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires. Acta Math. 88, 141–225 (1952)
Chruściel, P.T., Delay, E.: On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications. Mémoires de la S.M.F., n\(^0\)94, 103p. (2003) gr-qc/0301073
Corvino, J., Schoen, R.: On the asymptotics for the vacuum Einstein constraint equations. J. Differential Geom. 73(2), 185–217 (2006). gr-qc/0301071
Giaquinta, M., Martinazzi, L.: An introduction to the regularity thoery for elliptic systems, harmonic maps and minimal graphs. Edizioni della Normale (2005)
Gicquaud, R.: De l’équation de prescription de courbure scalaire aux équations de contrainte en relativité générale sur une variété asymptotiquement hyperbolique. Journal de Mathmatiques Pures et Appliqus, 94(2), 200–227 (2010). arXiv:0802.3279
Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equation of Second Order. Springer, Berlin (1997)
Isenberg, J., Maxwell, D., Pollack, D.: A gluing construction for non vacuum solutions of the Einstein constraint. Adv. Theor. Phys. 9, 129–172 (2005)
Isenberg, J., Mazzeo, R., Pollack, D.: Gluing and wormholes for the Einstein constraint equations. Commun. Math. Phys. 231, 529–568 (2002)
Lee, J.M.: Fredholm operators and Einstein metrics on conformally compact manifolds. Memoirs AMS 183(864), (2006) math.DG/0105046
Mazzieri, L.: Generalized connected sum construction for nonzero constant scalar curvature metrics. Commun. Part. Differ. Eqs. 33, 1–17 (2008)
Mazzieri, L.: Generalized connected sum construction for scalar flat metrics. Manuscr. Math. 129, 137–168 (2009)
Mazzieri, L.: Generalized gluing for Einstein constraint equations. Calc. Var. Partial Differ. Eqs. 34, 453–473 (2009)
Acknowledgments
The first author is partially supported by the french project ANR-10-BLAN 0105 ACG and the ANR SIMI-1-003-01 GR-A-G. The second author is supported by the Italian project FIRB–IDEAS ‘Analysis and Beyond’ and he is grateful to the University of Avignon for the hospitality during the preparation of this work.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Delay, E., Mazzieri, L. Refined gluing for vacuum Einstein constraint equations. Geom Dedicata 173, 393–415 (2014). https://doi.org/10.1007/s10711-013-9948-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-013-9948-9