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The Volume Growth of Hyper-Kähler Manifolds of Type A ∞

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Abstract

We study the volume growth of hyper-Kähler manifolds of type A ∞ constructed by Anderson–Kronheimer–LeBrun (Commun. Math. Phys. 125:637–642, 1989) and Goto (Geom. Funct. Anal. 4(4):424–454, 1994). These are noncompact complete 4-dimensional hyper-Kähler manifolds of infinite topological type. These manifolds have the same topology, but the hyper-Kähler metrics depend on the choice of parameters. By taking a certain parameter, we show that there exists a hyper-Kähler manifold of type A ∞ whose volume growth is r α for each 3<α<4.

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References

  1. Anderson, T., Kronheimer, P., LeBrun, C.: Complete Ricci-flat Kähler manifolds of infinite topological type. Commun. Math. Phys. 125, 637–642 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bando, S., Kobayaslai, R.: Ricci-flat Kähler metrics on affine algebraic manifolds. In: Sunada, T. (ed.) Geometry and Analysis on Manifolds. Lect. Notes Math., vol. 1339, pp. 20–31. Springer, Berlin, (1988)

    Chapter  Google Scholar 

  3. Bando, S., Kobayaslai, R.: Ricci-flat Kähler metrics on affine algebraic manifolds II. Math. Ann. 287, 175–180 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bielawski, R., Dancer, A.S.: The geometry and topology of toric hyperKähler manifolds. Commun. Anal. Geom. 8, 727–760 (2000)

    MathSciNet  MATH  Google Scholar 

  5. Bielawski, R.: Complete hyperKähler 4n-manifolds with n commuting tri-Hamiltonian vector fields. Math. Ann. 314, 505–528 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bishop, R.L., Crittenden, R.J.: Geometry on Manifolds. Academic Press, New York (1964)

    Google Scholar 

  7. Cherkis, S., Hitchin, N.: Gravitational instantons of type D k . Commun. Math. Phys. 260(2), 299–317 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Eguchi, T., Hanson, A.J.: Asymptotically flat selfdual solutions to Euclidean gravity. Phys. Lett. B 74(3), 249–251 (1978)

    Article  Google Scholar 

  9. Gibbons, G.W., Hawking, S.W.: Gravitational multi-instantons. Phys. Lett. B 78(4), 430–432 (1978)

    Article  Google Scholar 

  10. Goto, R.: On hyper-Kähler manifolds of type A ∞. Geom. Funct. Anal. 4(4), 424–454 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Goto, R.: On hyper-Kähler manifolds of type A ∞ and D ∞. Commun. Math. Phys. 198, 469–491 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gromov, M., Lafontaine, J., Pansu, P.: Structures métriques pour les variétés riemanniennes. Cédic, Fernand Nathan, Paris (1981)

    MATH  Google Scholar 

  13. Hawking, S.W.: Gravitational instantons. Phys. Lett. A 60, 81 (1977)

    Article  MathSciNet  Google Scholar 

  14. Hitchin, N.J., Karlhede, A., Lindström, U., Roček, M.: Hyper-Kähler metrics and supersymmetry. Commun. Math. Phys. 108(4), 535–589 (1987)

    Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  16. Newman, E., Tamburino, L., Unti, T.: Empty-space generalization of the Schwarzschild metric. J. Math. Phys. 4, 915 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  17. Pederson, H., Poon, Y.S.: Hyper-Kähler metrics and a generalization of the Bogomolny equations. Commun. Math. Phys. 117, 569–580 (1988)

    Article  Google Scholar 

  18. Taub, A.H.: Empty space-times admitting a three parameter group of motions. Ann. Math. 53(3), 472–490 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  19. Tian, G., Yau, S.T.: Complete Kähler manifolds with zero Ricci curvature I. J. Am. Math. Soc. 3(3), 579–609 (1990)

    MathSciNet  MATH  Google Scholar 

  20. Tian, G., Yau, S.T.: Complete Kähler manifolds with zero Ricci curvature II. Invent. Math. 106, 27–60 (1991)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914, Japan

    Kota Hattori

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  1. Kota Hattori
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Correspondence to Kota Hattori.

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Communicated by Jiaping Wang.

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Hattori, K. The Volume Growth of Hyper-Kähler Manifolds of Type A ∞ . J Geom Anal 21, 920–949 (2011). https://doi.org/10.1007/s12220-010-9173-9

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  • DOI: https://doi.org/10.1007/s12220-010-9173-9

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