Rate of Asymptotic Convergence Near an Isolated Singularity of \(\hbox {G}_2\) Manifold

Abstract

In this paper, a metric with \(\hbox {G}_2\) holonomy and slow rate of convergence to the cone metric is constructed on a ball inside the cone over the flag manifold.

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References

  1. 1.

    Adams, D., Simon, L.: Rates of asymptotic convergence near isolated singularities of geometric extrema. Indiana Univ. Math. J. 37(2), 225–254 (1988)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Behrndt, T.: On the Cauchy problem for the heat equation on Riemannian manifolds with conical singularities. Q. J. Math. 64, 981–1007 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Bryant, R.L.: Metrics with exceptional holonomy. Ann. Math. 126(3), 525–576 (1987)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Cheeger, J.: Spectral geometry of singular Riemannian spaces. J. Differ. Geom. 18(1983), 575–657 (1984)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Cheeger, J., Tian, G.: On the cone structure at infinity of Ricci flat manifolds with Euclidean volume growth and quadratic curvature decay. Invent. Math. 118(3), 493–571 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Colding, T., Minicozzi II, W.: On uniqueness of tangent cones for Einstein manifolds. Invent. Math. 196(3), 515–588 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Degeratu, A., Mazzeo, R.: Fredholm theory for elliptic operators on quasi-asymptotically conical spaces. https://arxiv.org/abs/1406.3465 (preprint) (2014)

  8. 8.

    Fernández, M., Gray, A.: Riemannian manifolds with structure group \(\text{G}_2\). Ann. Math. Pura Appl. (4) 132(1982), 19–45 (1983)

  9. 9.

    Foscolo, L.: Deformation theory of nearly Kähler manifolds. J. Lond. Math. Soc. 95, 586–612 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Hausel, T., Hunsicker, E., Mazzeo, R.: Hodge cohomology of gravitational instantons. Duke Math. J. 122(3), 485–548 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Hein, H.-J., Sun, S.: Calabi-Yau manifolds with isolated conical singularities. https://arxiv.org/abs/1607.02940 (preprint) (2016)

  12. 12.

    Joyce, D.: Compact Riemannian 7-manifolds with holonomy \(\text{ G }_2\). I. J. Differ. Geom. 43(2), 291–328 (1996)

    Article  Google Scholar 

  13. 13.

    Joyce, D.: Compact Manifolds with Special Holonomy. Oxford Mathematical Monographs. Oxford University Press, Oxford (2000)

    Google Scholar 

  14. 14.

    Joyce, D.: Special Lagrangian submanifolds with isolated conical singularities. I. Ann. Glob. Anal. Geom. 25, 201–251 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Karigiannis, S.: Desingularization of \(\text{ G }_2\) manifolds with isolated conical singularities. Geom. Topol. 13(3), 1583–1655 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Lockhart, R.B., McOwen, R.C.: Elliptic differential operators on noncompact manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12(3), 409–447 (1985)

  17. 17.

    Mazzeo, R.: Elliptic theory of edge operators. I. Commun. Partial Differ. Equ. 16, 1615–1664 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Melrose, R.B.: The Atiyah-Patodi-Singer Index Theorem. Research Notes in Mathematics, vol. 4. A K Peters, Ltd., Wellesley (1993)

  19. 19.

    Moroianu, A., Semmelmann, U.: The Hermitian Laplacian operator on nearly Kähler manifolds. Commun. Math. Phys. 294(1), 251–272 (2010)

    Article  MATH  Google Scholar 

  20. 20.

    Pacini, T.: Desingularizing isolated conical singularities. Commun. Anal. Geom. 21, 105–170 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Vertman, B.: Ricci flow on singular manifolds. https://arxiv.org/abs/1603.06545 (preprint) (2016)

  22. 22.

    Wang, Y.Q.: An elliptic theory of indicial weights and applications to non-linear geometry problems. https://arxiv.org/abs/1702.05864 (preprint) (2017)

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Acknowledgements

The author is grateful to the insightful and helpful discussions with Xiuxiong Chen, Lorenzo Foscolo, Song Sun and Yuanqi Wang. The author also thanks the referee for making this article more readable.

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Correspondence to Gao Chen.

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Chen, G. Rate of Asymptotic Convergence Near an Isolated Singularity of \(\hbox {G}_2\) Manifold. J Geom Anal 28, 3139–3170 (2018). https://doi.org/10.1007/s12220-017-9952-7

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Keywords

  • G_2 holonomy
  • Special holonomy
  • Ricci-flat

Mathematics Subject Classification

  • 53C25
  • 53C29