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Generalized gluing for Einstein constraint equations

  • Lorenzo MazzieriEmail author
Open Access
Article

Abstract

In this paper we construct a family of new (topologically distinct) solutions to the Einstein constraint equations by performing the generalized connected sum (or fiber sum) of two known compact m-dimensional constant mean curvature solutions (M 1, g 1, Π1) and (M 2, g 22) along a common isometrically embedded k-dimensional sub-manifold (K, g K ). Away from the gluing locus the metric and the second fundamental form of the new solutions can be chosen as close as desired to the ones of the original solutions. The proof is essentially based on the conformal method and the geometric construction produces a polyneck between M 1 and M 2 whose metric is modeled fiber-wise (i. e. along the slices of the normal fiber bundle of K) around a Schwarzschild metric; for these reasons the codimension n : =  m − k of K in M 1 and M 2 is required to be  ≥  3. In this sense our result is a generalization of the Isenberg–Mazzeo–Pollack gluing, which works for connected sum at points and in dimension 3. The solutions we obtain for the Einstein constraint equations can be used to produce new short time vacuum solutions of the Einstein system on a Lorentzian (m + 1)-dimensional manifold, as guaranteed by a well known result of Choquet-Bruhat.

Mathematics Subject Classification (2000)

53C21 58J60 83C05 53A30 57R65 

Notes

Acknowledgments

The author would like to thank J. Isenberg, R. Mazzeo and D. Pollack for many helpful suggestions and discussions during the preparation of this paper.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. 1.
    Aubin T.: Some nonlinear problems in Riemannian Geometry, Springer Monographs in Mathematics. Springer, Heidelberg (1998)Google Scholar
  2. 2.
    Bartnik R., Isenberg J.: The constraint equations. In: Chrusciel, P.T., Friedrich, H.(eds) The Einstein equations and the large scale behavior of gravitational fields, pp. 1–39. Birkhäuser, Basel (2004)Google Scholar
  3. 3.
    Christodoulou D., Choquet-Bruhat Y.: Elliptic systems in H s, δ spaces on manifolds which are Euclidean at infinity. Acta Math. 146, 129–150 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Choquet-Bruhat Y.: Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires. Acta Math. 88, 141–225 (1952)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Chruściel P.T., Isenberg J., Pollack D.: Initial data engineering. Comm. Math. Phys. 257, 29–42 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Giaquinta, M., Martinazzi, L.: An introduction to the regularity thoery for elliptic systems, harmonic maps and minimal graphs. Edizioni della Normale (2005)Google Scholar
  7. 7.
    Gilbarg D., Trudinger N.: Elliptic partial differential equation of second order. Springer, Heidelberg (1983)Google Scholar
  8. 8.
    Gromov M., Lawson H.B.: The classification of simply connected manifolds of positive scalar curvature. Ann. Math.(2) 111(3), 423–434 (1980)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Joyce D.: Constant scalar curvature metrics on connected sums. Int. J. Math. Math. Sci. 7, 405–450 (2003)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Isenberg J., Mazzeo R., Pollack D.: Gluing and wormholes for the Einstein constraint equations. Comm. Math. Phys. 231, 529–568 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Isenberg J., Maxwell D., Pollack D.: A gluing construction for non vacuum solutions of the Einstein constraint equations. Adv. Theor. Phys. 9, 129–172 (2005)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Lee J.M., Parker T.H.: The Yamabe Problem. Bull. AMS. 17, 37–91 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Mazzeo R.: Elliptic theory of differential edge operators. Commun. Partial Differ. Equ. 16, 1615–1664 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Mazzeo R., Pollack D., Uhlenbeck K.: Connected sums constructions for constant scalar curvature metrics. Topol. Method Nonlinear Anal. 6, 207–233 (1995)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Mazzeo R., Pacard F.: Constant scalar curvature metrics with isolated singularities. Duke Math. J. 99(3), 353–418 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Mazzeo R., Pacard F.: Constant mean curvature surfaces with Delaunay ends. Comm. Anal. Geom. 9(1), 169–237 (2001)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Mazzeo R., Pacard F., Pollack D.: Connected sums of constant mean curvature surfaces in Euclidean 3 space. J. Reine Angew. Math. 536, 115–165 (2001)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Mazzieri, L.: Generalized connected sum construction for scalar flat metrics. arXiv:math.DG/0611778 (2006)Google Scholar
  19. 19.
    Mazzieri L.: Generalized connected sum construction for nonzero constant scalar curvature metrics. Commun. Partial Differ. Equ. 33, 1–17 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Schoen R., Yau S.T.: On the structure of manifolds with positive scalar curvature. Manuscripta Math. 28(1–3), 159–183 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Taylor M.: Partial differential equations III: nonlinear equations. Appl. Math. Sci. 117. Springer, New York (1996)Google Scholar

Copyright information

© The Author(s) 2008

Authors and Affiliations

  1. 1.Scuola Normale SuperiorePisaItaly
  2. 2.Max-Plank-Instutut für Gravtationsphysik Albert-Einstein-InstitutGolmGermany

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