Annals of PDE

, 4:2 | Cite as

An Extension Procedure for the Constraint Equations

  • Stefan Czimek


Let \((\bar{g}, \bar{k})\) be a solution to the maximal constraint equations of general relativity on the unit ball \(B_1\) of \({\mathbb R}^3\). We prove that if \((\bar{g},\bar{k})\) is sufficiently close to the initial data for Minkowski space, then there exists an asymptotically flat solution (gk) on \({\mathbb R}^3\) that extends \((\bar{g}, \bar{k})\). Moreover, (gk) is bounded by \((\bar{g}, \bar{k})\) and has the same regularity. Our proof uses a new method of solving the prescribed divergence equation for a tracefree symmetric 2-tensor, and a geometric variant of the conformal method to solve the prescribed scalar curvature equation for a metric. Both methods are based on the implicit function theorem and an expansion of tensors based on spherical harmonics. They are combined to define an iterative scheme that is shown to converge to a global solution (gk) of the maximal constraint equations which extends \((\bar{g},\bar{k})\).


General relativity Constraint equations Analysis of PDE Differential geometry 



This work forms part of my Ph.D. thesis and I am grateful to my Ph.D. advisor Jérémie Szeftel for his kind supervision and careful guidance. Furthermore, I am grateful to the RDM-IdF for financial support.


  1. 1.
    Adams, R., Fournier, J.: Sobolev Spaces, 2nd edn. Pure and Applied Mathematics, vol. 140. Elsevier/Academic Press, Amsterdam (2003)Google Scholar
  2. 2.
    Anderson, M.T., Khuri, M.: On the Bartnik extension problem for the static vacuum Einstein equations. Class. Quantum Gravity 30(12), 125005 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Anderson, M.T.: On the Bartnik conjecture for the static vacuum Einstein equations. Class. Quantum Gravity 33(1), 015001 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bartnik, R.: Existence of maximal surfaces in asymptotically flat spacetimes. Commun. Math. Phys. 94(2), 155–175 (1984)ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bartnik, R.: The mass of an asymptotically flat manifold. Commun. Pure Appl. Math. 39, 661–693 (1986)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bartnik, R.: Quasi-spherical metrics and prescribed scalar curvature. J. Differ. Geom. 37(1), 31–71 (1993)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bartnik, R.: Energy in General Relativity. Tsing Hua Lectures on Geometry & Analysis (Hsinchu, 1990–1991), pp. 5–27. International Press, Cambridge, MA (1997)Google Scholar
  8. 8.
    Bartnik, R.: Mass and 3-metrics of non-negative scalar curvature. In: Proceedings of the International Congress of Mathematicians, vol. II, pp. 231–240. Higher Education Press, Beijing (2002)Google Scholar
  9. 9.
    Choquet-Bruhat, Y., Christodoulou, D.: Elliptic systems in \(H_{s,\delta }\)-spaces on manifolds which are Euclidean at infinity. Acta Math. 146(1–2), 129–150 (1981)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Christodoulou, D., Klainerman, S.: The Global Nonlinear Stability of the Minkowski Space. Princeton Mathematical Series, vol. 41. Princeton University Press, Princeton (1993)MATHGoogle Scholar
  11. 11.
    Chruściel, P.T., Delay, E.: On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications. Mém. Soc. Math. Fr. 94, 1–103 (2003)MathSciNetMATHGoogle Scholar
  12. 12.
    Chruściel, P.T., Isenberg, J., Pollack, D.: Initial data engineering. Commun. Math. Phys. 257(1), 29–42 (2005)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Corvino, J.: Scalar curvature deformation and a gluing construction for the Einstein constraint equations. Commun. Math. Phys. 214(1), 137–189 (2000)ADSMathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Corvino, J., Schoen, R.: On the asymptotics for the vacuum Einstein constraint equations. J. Differ. Geom. 73(2), 185–217 (2006)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. I. Interscience Publishers, Inc., London (1953)MATHGoogle Scholar
  16. 16.
    Czimek, S.: Boundary harmonic coordinates and the localised bounded \(L^2\) curvature theorem. arXiv:1708.01667
  17. 17.
    Delay, E.: Smooth compactly supported solutions of some underdetermined elliptic PDE, with gluing applications. Commun. Partial Differ. Equ. 37(10), 1689–1716 (2012)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Fischer, A., Marsden, J.: Deformations of the scalar curvature. Duke Math. J. 42(3), 519–547 (1975)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Reprint of the 1998 edition. Classics in Mathematics. Springer (2001)Google Scholar
  20. 20.
    Hill, E.: The theory of vector spherical harmonics. Am. J. Phys. 22, 211–214 (1954)ADSMathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Huisken, G., lmanen, T.: The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differ. Geom. 59, 353–437 (2001)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Isenberg, J., Mazzeo, R., Pollack, D.: On the topology of vacuum spacetimes. Ann. Henri Poincaré 4(2), 369–383 (2003)ADSMathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Isett, P., Oh, S.-J.: On nonperiodic euler flows with Hölder regularity. arXiv:1402.2305v2
  24. 24.
    Klainerman, S., Rodnianski, I., Szeftel, J.: The bounded \(L^2\) curvature conjecture. Invent. Math. 202(1), 91–216 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Maxwell, D.: Rough solutions of the Einstein constraint equations. J. Reine Angew. Math. 590, 1–29 (2006)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Maxwell, D.: Initial data for black holes and rough spacetimes. Dissertation, University of Washington (2004)Google Scholar
  27. 27.
    Miao, P.: On existence of static metric extensions in general relativity. Commun. Math. Phys. 241(1), 27–46 (2003)ADSMathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Oh, S.-J., Tataru, D.: Local well-posedness of the (4+1)-dimensional Maxwell–Klein–Gordon equation at energy regularity. Ann. PDE 2(1) (2016)Google Scholar
  29. 29.
    Ratiu, T., Abraham, R., Marsden, J.E.: Manifolds, Tensor Analysis, and Applications. Global Analysis. Addison-Wesley, Reading (1983)MATHGoogle Scholar
  30. 30.
    Sandberg, V.: Tensor spherical harmonics on \(S^2\) and \(S^3\) as eigenvalue problems. J. Math. Phys. 19(12), 2441–2446 (1978)ADSMathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Sharples, J.: Local existence of quasi-spherical space-time initial data. J. Math. Phys. 46(5), 052501 (2005). 28 ppADSMathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Shi, Y., Tam, L.-F.: Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature. J. Differ. Geom. 62(1), 79–125 (2002)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Shi, Y., Tam, L.-F.: Quasi-local mass and the existence of horizons. Commun. Math. Phys. 274(2), 277–295 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Smith, B., Weinstein, G.: Quasiconvex foliations and asymptotically flat metrics on non-negative scalar curvature. Commun. Anal. Geom. 12(3), 511–551 (2004)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Smith, B., Weinstein, G.: On the connectedness of the space of initial data for the Einstein equations. Electron. Res. Announc. Am. Math. Soc. 6, 52–63 (2000)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Stein, E.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton (1971)CrossRefGoogle Scholar
  37. 37.
    Taylor, M.: Partial Differential Equations I: Basic Theory. Springer, New York (1999). (2nd printing)Google Scholar
  38. 38.
    Taylor, M.: Partial Differential Equations III: Nonlinear Equations. Springer, New York (2011). (2nd printing)CrossRefMATHGoogle Scholar
  39. 39.
    Wald, R.: General Relativity. University of Chicago Press, Chicago, IL (1984)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Laboratoire Jacques-Louis LionsUniversité Pierre et Marie Curie (Paris 6)ParisFrance

Personalised recommendations