Communications in Mathematical Physics

, Volume 253, Issue 3, pp 561–583 | Cite as

Solutions of the Einstein Constraint Equations with Apparent Horizon Boundaries

  • David Maxwell


We construct asymptotically Euclidean solutions of the vacuum Einstein constraint equations with an apparent horizon boundary condition. Specifically, we give sufficient conditions for the constant mean curvature conformal method to generate such solutions. The method of proof is based on the barrier method used by Isenberg for compact manifolds without boundary, suitably extended to accommodate semilinear boundary conditions and low regularity metrics. As a consequence of our results for manifolds with boundary, we also obtain improvements to the theory of the constraint equations on asymptotically Euclidean manifolds without boundary.


Boundary Condition Neural Network Manifold Complex System Nonlinear Dynamics 
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  1. 1.
    Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Commun. Pure Appl. Math. 17, 35–92 (1964)zbMATHGoogle Scholar
  2. 2.
    Bartnik, R.: The mass of an asymptotically flat manifold. Commun. Pure Appl. Math. 39, 661–693 (1986)zbMATHGoogle Scholar
  3. 3.
    Brill, D., Lindquist, R.W.: Interaction energy in geometrostatics. Phys. Rev. (2) 131, 471–476 (1963)Google Scholar
  4. 4.
    Cantor, M.: Some problems of global analysis on asymptotically simple manifolds. Compositio Math. 38, 3–35 (1979)zbMATHGoogle Scholar
  5. 5.
    Cantor, M., Brill, D.: The Laplacian on asymptotically flat manifolds and the specification of scalar curvature. Compositio Math. 43(3), 317–330 (1981)zbMATHGoogle Scholar
  6. 6.
    Choquet-Bruhat, Y.: Einstein constraints on compact n-dimensional manifolds. Class. Quantum Grav. 21, S127–S151 (2004)Google Scholar
  7. 7.
    Choquet-Bruhat, Y., Christodoulou, D.: Elliptic systems in Hsδ spaces on manifolds which are Euclidean at infinity. Acta. Math. 146, 129–150 (1981)zbMATHGoogle Scholar
  8. 8.
    Choquet-Bruhat, Y. Isenberg, J., York, Jr J.W.: Einstein constraints on asymptotically Euclidean manifolds. Phys. Rev. D 61, 1–20 (2000)Google Scholar
  9. 9.
    Choquet-Bruhat, Y., York, Jr J.W.: The Cauchy problem. In: Held, A., (ed.), General Relativity and Gravitation. New York: Plenum, 1980Google Scholar
  10. 10.
    Cook, G.B.: Initial data for numerical relativity. Living Rev. 5, 2000 []
  11. 11.
    Cook, G.B.: Corotating and irrotatinal binary black holes in quasi-circular orbits. Phys. Rev. D 65, 084003 (2002)CrossRefGoogle Scholar
  12. 12.
    Christodoulou, D., O’Murchadha, N.: The Boost Problem in General Relativity. Commun. Math. Phys. 80, 271–300 (1981)zbMATHGoogle Scholar
  13. 13.
    Dain, S.: Initial data for black hole collisions In: L. Gutierrez, J. Alberto, (eds.), Gravitational and Cosmology. Proc. of Spanish Relativity Meeting ERE-2002, Barcelona: Univ. de Barcelona, 2003Google Scholar
  14. 14.
    Dain, S.: Trapped Surfaces as Boundaries for the Constraint Equations Class. Quantom Grav. 21, 555–573 (2004)CrossRefzbMATHGoogle Scholar
  15. 15.
    Eardley, D.M.: Black hole boundary conditions and coordinate conditions. Phys. Rev. D 57(4), 2299–2304 (1998)CrossRefGoogle Scholar
  16. 16.
    Escobar, J.F.: The Yambe problem on manifolds with boundary. J. Differ. Geom. 35, 21–84 (1992)zbMATHGoogle Scholar
  17. 17.
    Hörmander, L.: Analysis of linear partial differential operators. Vol. III, Berlin: Springer-Verlag, 1985Google Scholar
  18. 18.
    Isenberg, J.: Constant mean curvature solutions of the Einstein constraint equations on closed manifolds. Class. Quantum Grav. 12, 2249–2274 (1995)CrossRefzbMATHGoogle Scholar
  19. 19.
    Kulkarni, R.S., Pinkall, U.: Conformal Geometry. Wiesbaden: Friedr. Vieweg & Sohn, 1988Google Scholar
  20. 20.
    Klainerman, S., Rodnianski, I.: Rough solutions of the Einstein vacuum equations C. R. Acad. Sci. Paris Sér. I Math. 334, 125–130 (2002)zbMATHGoogle Scholar
  21. 21.
    Lichernowicz, A.: Sur l’intégration des équations d’Einstein. J. Math. Pures Appl. 23, 26–63 (1944)Google Scholar
  22. 22.
    Lockhart, R.B., McOwen, R.C.: Elliptic differential operators on noncompact manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 12(4), 409–447 (1985)zbMATHGoogle Scholar
  23. 23.
    McOwen, R.C.: The behavior of the Laplacian on weighted Sobolev spaces. Commun. Pure Appl. Math. 32, 783–795 (1979)zbMATHGoogle Scholar
  24. 24.
    Misner, C.: The method of images in geometrostatics. Ann. Phys. 24, 102–117 (1963)CrossRefzbMATHGoogle Scholar
  25. 25.
    Moncrief, V., Isenberg, J.: A set of nonconstant mean curvature solutions of the Einstein constraint equations on closed manifolds. Class. Qunatum Grav. 13, 1819–1847 (1996)CrossRefzbMATHGoogle Scholar
  26. 26.
    Schechter, M.: Principles of functional analysis. Providence, Rhode Island: Americal Mathematical Society, 2002Google Scholar
  27. 27.
    Smith, H., Tataru, D.: Sharp local well posedness results for the nonlinear wave equation. To appear Ann. Math.Google Scholar
  28. 28.
    Thornburg, J.: Coordinates and boundary conditions for the general relativistic initial data problem. Class. Quantum Grav. 4, 1119–1131 (1987)CrossRefGoogle Scholar
  29. 29.
    Trudinger, N.: Linear elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 27, 265–308 (1973)zbMATHGoogle Scholar
  30. 30.
    Wald, R.M.: General relativity. Chicago: The University of Chicago Press, 1984Google Scholar
  31. 31.
    York, Jr J.W., Bowen, J.M.: Time-asymmetric initial data for black holes and black hole collisions. Phys. Rev. D 24(8), 2047–2056 (1980)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • David Maxwell
    • 1
  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

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