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Désingularisation de métriques d’Einstein. I

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Abstract

We find a new obstruction for a real Einstein 4-orbifold with an A 1-singularity to be a limit of smooth Einstein 4-manifolds. The obstruction is a curvature condition at the singular point.

For asymptotically hyperbolic metrics, with boundary at infinity a conformal metric, we prove that if the obstruction vanishes, one can desingularize Einstein orbifolds with such singularities.

The Dirichlet problem consists in finding Einstein metrics with given conformal infinity on the boundary: we prove that our obstruction defines a wall in the space of conformal metrics on the boundary, and that all the Einstein metrics must have their conformal infinity on one side of the wall.

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Bibliographie

  1. Anderson, M.T.: Ricci curvature bounds and Einstein metrics on compact manifolds. J. Am. Math. Soc. 2(3), 455–490 (1989)

    Article  MATH  Google Scholar 

  2. Anderson, M.T.: Geometric aspects of the AdS/CFT correspondence. In: Biquard, O. (ed.) AdS/CFT Correspondence: Einstein Metrics and Their Conformal Boundaries. IRMA Lectures in Mathematics and Theoretical Physics, vol. 8, pp. 1–31. Eur. Math. Soc., Zurich (2005)

    Chapter  Google Scholar 

  3. Anderson, M.T.: Einstein metrics with prescribed conformal infinity on 4-manifolds. Geom. Funct. Anal. 18(2), 305–366 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Anderson, M.T., Herzlich, M.: Unique continuation results for Ricci curvature and applications. J. Geom. Phys. 58(2), 179–207 (2008); Erratum (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bando, S., Kasue, A., Nakajima, H.: On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth. Invent. Math. 97(2), 313–349 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  6. Biquard, O.: Métriques d’Einstein asymptotiquement symétriques. Astérisque, vol. 265 (2000), vi+109 pp. English translation: SMF/AMS Texts and Monographs, vol. 13 (2006)

  7. Biquard, O.: Continuation unique à partir de l’infini conforme pour les métriques d’Einstein. Math. Res. Lett. 15(6), 1091–1099 (2008)

    MathSciNet  MATH  Google Scholar 

  8. Biquard, O., Rollin, Y.: Wormholes in ACH Einstein manifolds. Trans. Am. Math. Soc. 361(4), 2021–2046 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kronheimer, P.B.: The construction of ALE spaces as hyper-Kähler quotients. J. Differ. Geom. 29(3), 665–683 (1989)

    MathSciNet  MATH  Google Scholar 

  10. Kronheimer, P.B.: A Torelli-type theorem for gravitational instantons. J. Differ. Geom. 29(3), 685–697 (1989)

    MathSciNet  MATH  Google Scholar 

  11. Mazzeo, R.: Unique continuation at infinity and embedded eigenvalues for asymptotically hyperbolic manifolds. Am. J. Math. 113, 25–45 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  12. Mazzeo, R., Singer, M.: Some remarks on conic degeneration and bending of Poincaré-Einstein metrics (2007). arXiv:0709.1498

  13. Page, D.N., Pope, C.: Inhomogeneous Einstein metrics on complex line bundles. Class. Quantum Gravity 4, 213–225 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  14. Pedersen, H.: Einstein metrics, spinning top motions and monopoles. Math. Ann. 274(1), 35–59 (1986)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. UPMC Université Paris 6 et École Normale Supérieure, UMR 8553 du CNRS, Paris, France

    Olivier Biquard

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  1. Olivier Biquard
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Correspondence to Olivier Biquard.

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Biquard, O. Désingularisation de métriques d’Einstein. I. Invent. math. 192, 197–252 (2013). https://doi.org/10.1007/s00222-012-0410-7

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  • DOI: https://doi.org/10.1007/s00222-012-0410-7

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