Annales Henri Poincaré

, Volume 16, Issue 5, pp 1231–1266 | Cite as

Initial Data Sets with Ends of Cylindrical Type: I. The Lichnerowicz Equation

  • Piotr T. ChruścielEmail author
  • Rafe Mazzeo


We construct large classes of vacuum general relativistic initial data sets, possibly with a cosmological constant \({\Lambda \in \mathbb{R}}\), containing ends of cylindrical type.


Manifold Scalar Curvature Compact Manifold Constant Scalar Curvature Complete Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Akutagawa K., Botvinnik B.: Yamabe metrics on cylindrical manifolds. Geom. Funct. Anal. 13, 259–333 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Akutagawa, K., Carron, G., and Mazzeo, R.: The Yamabe problem on stratified spaces. Geom. Funct. Anal. (2012, in press). arXiv:1210.8054
  3. 3.
    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)Google Scholar
  4. 4.
    Aviles P., McOwen R.C.: Conformal deformation to constant negative scalar curvature on noncompact Riemannian manifolds. J. Diff. Geom. 27, 225–239 (1988)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Baumgarte, T.W., Naculich, S.G.: Analytical representation of a black hole puncture solution. Phys. Rev. D 75, 067502, 4 (2007). arXiv:gr-qc/0701037
  6. 6.
    Bessières, L., Besson, G., Maillot, S.: Ricci flow on open 3-manifolds and positive scalar curvature. Geom. Topol. 15, 927–975 (2011). arXiv:1001.1458 [math.DG]
  7. 7.
    Byde A.: Gluing theorems for constant scalar curvature manifolds. Indiana Univ. Math. Jour. 52, 1147–1199 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Caffarelli L.A., Gidas B., Spruck J.: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Commun. Pure Appl. Math. 42, 271–297 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Chen C.C., Lin C.S.: Local behavior of singular positive solutions of semilinear elliptic equations with Sobolev exponent. Duke Math. Jour. 78, 315–334 (1995)CrossRefzbMATHGoogle Scholar
  10. 10.
    Choquet-Bruhat, Y., Isenberg, J., York, J.W. Jr.: Einstein constraints on asymptotically Euclidean manifolds. Phys. Rev. D 61, (20 pp.), 084034 (2000). arXiv:gr-qc/9906095
  11. 11.
    Christodoulou D., ÓMurchadha N.: The boost problem in general relativity. Commun. Math. Phys. 80, 271–300 (1981)CrossRefADSzbMATHGoogle Scholar
  12. 12.
    Chruściel, P.T., Mazzeo, R., Pocchiola, S.: Initial data sets with ends of cylindrical type: II. The vector constraint equation. Advances in Theoretical and Mathematical Physics, in press. arXiv:1203.5138 [gr-qc]
  13. 13.
    Chruściel, P.T., Pacard, F., Pollack, D.: Singular Yamabe metrics and initial data with exactly Kottler-Schwarzschild-de Sitter ends II. Generic metrics. Math. Res. Lett. 16, 157–164 (2009). arXiv:0803.1817 [gr-qc]
  14. 14.
    Chruściel, P.T., Pollack, D.: Singular Yamabe metrics and initial data with exactly Kottler–Schwarzschild–de Sitter ends. Ann. Henri Poincaré 9, 639–654 (2008). arXiv:0710.3365 [gr-qc]
  15. 15.
    Gabach Clément, M.E.: Conformally flat black hole initial data, with one cylindrical end. Class. Quantum Grav. 27, 125010 (2010). arXiv:0911.0258 [gr-qc]
  16. 16.
    Dahl, M., Gicquaud, R., Humbert, E.: A limit equation associated to the solvability of the vacuum Einstein constraint equations using the conformal method. (2010). arXiv:1012.2188 [gr-qc]
  17. 17.
    Dain, S., Gabach Clément, M.E.: Extreme Bowen-York initial data. Class. Quantum Grav. 26, 035020, 16 (2009). arXiv:0806.2180 [gr-qc]
  18. 18.
    Dain, S., Gabach Clément, M.E.: Small deformations of extreme Kerr black hole initial data. Class. Quantum Grav. 28, p. 20, 075003 (2010). arXiv:1001.0178 [gr-qc]
  19. 19.
    Delay, E.: Smooth compactly supported solutions of some underdetermined elliptic PDE, with gluing applications. Commun. Partial Diff. Eq. 37(10), 1689–1716 (2012). arXiv:1003.0535 [math.FA]
  20. 20.
    Estabrook F., Wahlquist H., Christensen S., DeWitt B., Smarr L., Tsiang E.: Maximally slicing a black hole. Phys. Rev. D7, 2814–2817 (1973)ADSGoogle Scholar
  21. 21.
    Friedrich H.: Yamabe numbers and the Brill-Cantor criterion. Ann. Henri Poincaré 12, 1019–1025 (2011)CrossRefADSzbMATHMathSciNetGoogle Scholar
  22. 22.
    Gicquaud, R., Sakovich, A.: A large class of non constant mean curvature solutions of the Einstein constraint equations on an asymptotically hyperbolic manifold. Commun. Math. Phys. 310(3), 705–763 (2012). arXiv:1012.2246 [gr-qc]
  23. 23.
    Gilbarg D., Trudinger N.S.: Elliptic partial differential equations of second order. Springer, Berlin (1983)CrossRefzbMATHGoogle Scholar
  24. 24.
    Hannam, M., Husa, S., Ó Murchadha, N.: Bowen-York trumpet data and black-hole simulations. Phys. Rev. D80, 124007 (2009). arXiv:0908.1063 [gr-qc]
  25. 25.
    Hannam M., Husa S., Pollney D., Brügmann B., Ó Murchadha N.: Geometry and regularity of moving punctures. Phys. Rev. Lett. 99, 241102–241104 (2007)CrossRefADSMathSciNetGoogle Scholar
  26. 26.
    Hebey, E.: Existence, stability and instability for Einstein–scalar field Lichnerowicz equations (2009).
  27. 27.
    Hebey, E., Pacard, F., Pollack, D.: A variational analysis of Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds. Commun. Math. Phys. 278(1), 117–132 (2008). arXiv:gr-qc/0702031
  28. 28.
    Holst, M., Nagy, G., Tsogtgerel, G.: Far-from-constant mean curvature solutions of Einstein’s constraint equations with positive Yamabe metrics. Phys. Rev. Lett. 100, 161101, 4 (2008). arXiv:0802.1031 [gr-qc]
  29. 29.
    Isenberg J.: Constant mean curvature solutions of the Einstein constraint equations on closed manifolds. Class. Quantum Grav. 12, 2249–2274 (1995)CrossRefADSzbMATHMathSciNetGoogle Scholar
  30. 30.
    Isenberg J., Mazzeo R., Pollack D.: On the topology of vacuum space times. Ann. Henri Poincaré 4, 369–383 (2003)zbMATHMathSciNetGoogle Scholar
  31. 31.
    Isenberg J., Moncrief V.: A set of nonconstant mean curvature solutions of the Einstein constraint equations on closed manifolds. Class. Quantum Gravity 13, 1819–1847 (1996)CrossRefADSzbMATHMathSciNetGoogle Scholar
  32. 32.
    Kastor, D., Traschen, J.: Cosmological multi-black-hole solutions. Phys. Rev. D (3) 47, 5370–5375 (1993). arXiv:hep-th/9212035
  33. 33.
    Komatsu, E., et al.: Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation. Astrophys. Jour. Suppl. 192, (47 pp.), 18 (2011). arXiv:1001.4538 [astr-ph.CO]
  34. 34.
    Korevaar N., Mazzeo R., Pacard F., Schoen R.: Refined asymptotics for constant scalar curvature metrics with isolated singularities. Invent. Math. 135, 233–272 (1999)CrossRefADSzbMATHMathSciNetGoogle Scholar
  35. 35.
  36. 36.
    Malec, E., Ó Murchadha, N.: Constant mean curvature slices in the extended Schwarzschild solution and the collapse of the lapse. Phys. Rev. D (3) 68, p. 16, 124019 (2003). arXiv:gr-qc/0307046
  37. 37.
    Marques, F.C.: Isolated singularities of solutions of the Yamabe equation. Calc. Var. 32, 349–371 (2008). doi: 10.1007/s00526-007-0144-3
  38. 38.
    Maxwell, D.: A class of solutions of the vacuum Einstein constraint equations with freely specified mean curvature. Math. Res. Lett. 16(4), 627–645 (2008). arXiv:0804.0874 [gr-qc]
  39. 39.
    Mazzeo R., Pacard F.: Constant scalar curvature metrics with isolated singularities. Duke Math. Jour. 99, 353–418 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Mazzeo, R., Pollack, D.: Gluing and moduli for noncompact geometric problems. Geometric theory of singular phenomena in partial differential equations (Cortona, 1995), Sympos. Math., XXXVIII, Cambridge Univ. Press, Cambridge, 1998, pp. 17–51Google Scholar
  41. 41.
    Mazzeo R., Pollack D., Uhlenbeck K.: Connected sum constructions for constant scalar curvature metrics. Topol. Methods Nonlinear Anal. 6, 207–233 (1995)zbMATHMathSciNetGoogle Scholar
  42. 42.
    Mazzeo R., Pollack D., Uhlenbeck K.: Moduli spaces of singular Yamabe metrics. J. Am. Math. Soc. 9, 303–344 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  43. 43.
    Pollack D.: Compactness results for complete metrics of constant positive scalar curvature on subdomains of S n. Indiana Univ. Math. Jour. 42, 1441–1456 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  44. 44.
    Ratzkin J.: An end to end gluing construction for metrics of constant positive scalar curvature. Indiana Univ. Math. Jour. 52, 703–726 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  45. 45.
    Riess, A.G., et al.: New Hubble Space Telescope discoveries of type Ia Supernovae at z > 1: Narrowing constraints on the early behavior of dark energy. Astroph. J. 659, 98–121 (2007). arXiv:astro-ph/0611572
  46. 46.
    Schoen R.: The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation. Comm. Pure and Appl. Math. XLI, 317–392 (1988)CrossRefMathSciNetGoogle Scholar
  47. 47.
    Stanciulescu, C.: Spherically symmetric solutions of the vacuum Einstein field equations with positive cosmological constant. 1998, Diploma Thesis, University of ViennaGoogle Scholar
  48. 48.
    Sullivan D.: Related aspects of positivity in Riemannian geometry. J. Diff. Geom. 25, 327–351 (1987)zbMATHGoogle Scholar
  49. 49.
    Waxenegger, G.: Black hole initial data with one cylindrical end. Ph.D. thesis, University of Vienna, in preparationGoogle Scholar
  50. 50.
    Waxenegger, G., Beig, R., Ó Murchadha, N.: Existence and uniqueness of Bowen-York Trumpets. Class. Quantum Grav. 28, pp. 15, 245002 (2011). arXiv:1107.3083 [gr-qc]
  51. 51.
    Wood-Vasey, W.M. et al.: Observational Constraints on the Nature of the Dark Energy: First Cosmological Results from the ESSENCE Supernova Survey. Astroph. J. 666, 694–715 (2007). arXiv:astro-ph/0701041

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.IHESBures-sur-YvetteFrance
  2. 2.Gravitational PhysicsUniversity of ViennaViennaAustria
  3. 3.Erwin Schrödinger InstituteViennaAustria
  4. 4.Department of MathematicsStanford UniversityStanfordUSA

Personalised recommendations