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Science China Mathematics

, Volume 62, Issue 11, pp 2155–2194 | Cite as

Limiting behavior of a class of Hermitian Yang-Mills metrics, I

  • Jixiang FuEmail author
Articles
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Abstract

This paper begins to study the limiting behavior of a family of Hermitian Yang-Mills (HYM for brevity) metrics on a class of rank two slope stable vector bundles over a product of two elliptic curves with Kähler metrics ωε when ε → 0. Here, ωε are flat and have areas ε and ε-1 on the two elliptic curves, respectively. A family of Hermitian metrics on the vector bundle are explicitly constructed and with respect to them, the HYM metrics are normalized. We then compare the family of normalized HYM metrics with the family of constructed Hermitian metrics by doing estimates. We get the higher order estimates as long as the C0-estimate is provided. We also get the estimate of the lower bound of the C0-norm. If the desired estimate of the upper bound of the C0-norm can be obtained, then it would be shown that these two families of metrics are close to arbitrary order in ε in any Ck norms.

Keywords

Hermitian Yang-Mills metric stable vector bundle Ck-estimates 

Notes

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11871016, 11421061 and 11025103). The draft of the first six sections was finished in 2002 with the help of Professor Jun Li. The author thanks Jun Li for discussions on algebraic geometry and Professors Jiaxing Hong and Shing-Tung Yau on PDEs. The author also thanks Professors Guofang Wang, Qingxue Wang, Weiping Zhang and Xi Zhang for helpful discussions.

References

  1. 1.
    Croke C. Some isoperimetric inequalities and eigenvalue estimates. Ann Sei École Norm Sup (4), 1980, 13: 419–435MathSciNetCrossRefGoogle Scholar
  2. 2.
    Donaldson S K. Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles. Proc Lond Math Soc (3), 1985, 50: 1–26MathSciNetCrossRefGoogle Scholar
  3. 3.
    Donaldson S K. Infinite determinants, stable bundles and curvature. Duke Math J, 1987, 54: 231–247MathSciNetCrossRefGoogle Scholar
  4. 4.
    Friedman R. Rank two vector bundles over regular elliptic surfaces. Invent Math, 1989, 96: 283–332MathSciNetCrossRefGoogle Scholar
  5. 5.
    Friedman R, Morgan J, Witten E. Vector bundles and F theory. Comm Math Phys, 1997, 187: 679–743MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fukaya K. Mirror symmetry of abelian varieties and multi-theta functions. J Algebraic Geom, 2002, 11: 393–512MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fukaya K. Multivalued Morse theory, asymptotic analysis and mirror symmetry. In: Graphs and Patterns in Mathe matics and Theoretical Physics. Proceedings of Symposia in Pure Mathematics, vol. 73. Providence: Amer Math Soc, 2005, 205–278MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gidas B, Ni W-M, Nirenberg L. Symmetry and related properties via the maximum principle. Comm Math Phys, 1979, 68: 209–243MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gilbarg D, Trudinger N S. Elliptic Partial Differential Equations of Second Order. Berlin: Springer-Verlag, 2001zbMATHGoogle Scholar
  10. 10.
    Griffiths P, Harris J. Principles of Algebraic Geometry. New York: John Wiley & Sons, 1994CrossRefGoogle Scholar
  11. 11.
    Gross M, Tosatti V, Zhang Y. Collapsing of abelian fibred Calabi-Yau manifolds. Duke Math J, 2013: 162, 517–551MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gross M, Wilson P M H. Large complex structure limites of K3 surfaces. J Differential Geom, 2000, 55: 475–546MathSciNetCrossRefGoogle Scholar
  13. 13.
    Jost J. Partial Differential Equations. Graduate Texts in Mathematics, vol. 214. New York: Springer, 2007CrossRefGoogle Scholar
  14. 14.
    Kobayashi S. Differential Geometry of Complex Vector Bundles. Princeton: Princeton University Press, 1987CrossRefGoogle Scholar
  15. 15.
    Kontsevich M. Homological algebra of mirror symmetry. In: Proceedings of the International Congress of Mathematicians, vol. 1, 2. Basel: Birkhäuser, 1995, 120–139MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kontsevich M, Soibelman Y. Homological mirror symmetry and torus fibrations. In: Symplectic Geometry and Mirror Symmetry. River Edge: World Sei Publ, 2001, 203–263CrossRefGoogle Scholar
  17. 17.
    Lang S. Fundamentals of Diophantine Geometry. New York: Springer-Verlag, 1983CrossRefGoogle Scholar
  18. 18.
    Leung N C. Geometric aspects of mirror symmetry (with SYZ for rigid CY manifolds). In: Second International Congress of Chinese Mathematicians. New Studies in Advanced Mathematics, vol. 4. Somerville: Int Press, 2004, 305–342MathSciNetzbMATHGoogle Scholar
  19. 19.
    Leung N C, Yau S-T, Zaslow E. From special Lagrangian to Hermitian-Yang-Mills via Fourier-Mukai transform. Adv Theor Math Phys, 2000, 4: 1319–1341MathSciNetCrossRefGoogle Scholar
  20. 20.
    Li P. On the Sobolev constant and the p-spectrum of a compact Riemannian manifold. Ann Sei École Norm Sup (4), 1980, 13: 451–468MathSciNetCrossRefGoogle Scholar
  21. 21.
    Loftin J, Yau S-T, Zaslow E. Affine manifolds, SYZ geometry and the “Y” vertex. J Differential Geom, 2005, 71: 129–158MathSciNetCrossRefGoogle Scholar
  22. 22.
    Mazzeo R, Swoboda J, Weiss H, et al. Ends of the moduli space of Higgs bundles. Duke Math J, 2016, 165: 2227–2271MathSciNetCrossRefGoogle Scholar
  23. 23.
    Nirenberg L. On elliptic partial differential equations. Ann Scuola Norm Sup Pisa (3), 1959, 13: 115–162MathSciNetzbMATHGoogle Scholar
  24. 24.
    Ruan W-D. Zhang Y. Convergence of Calabi-Yau manifolds. Adv Math, 2011, 228: 1543–1589MathSciNetCrossRefGoogle Scholar
  25. 25.
    Schryer N L. Solution of monotone nonlinear elliptic boundary value problems. Numer Math, 1971, 18: 336–344MathSciNetCrossRefGoogle Scholar
  26. 26.
    Simpson C T. Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization. J Amer Math Soc, 1988, 1: 867–918MathSciNetCrossRefGoogle Scholar
  27. 27.
    Siu Y T. Lectures on Hermitian-Einstein Metrics for Stable Bundles and Kähler-Einstein Metrics. DMV Seminar, 8. Basel: Birkhäuser, 1987Google Scholar
  28. 28.
    Strominger A, Yau S-T, Zaslow E. Mirror symmetry is T-duality. Nuclear Phys B, 1996, 479: 243–259MathSciNetCrossRefGoogle Scholar
  29. 29.
    Thomas R P. Moment maps, monodromy and mirror manifolds. In: Symplectic Geometry and Mirror Symmetry. River Edge: World Sci Publ, 2001, 467–498CrossRefGoogle Scholar
  30. 30.
    Thomas R P, Yau S-T. Special Lagrangians, stable bundles and mean curvature flow. Comm Anal Geom, 2002, 10: 1075–1113MathSciNetCrossRefGoogle Scholar
  31. 31.
    Tosatti V. Adiabatic limits of Ricci-flat Kahler metrics. J Differential Geom, 2010, 84: 427–453MathSciNetCrossRefGoogle Scholar
  32. 32.
    Uhlenbeck K K, Yau S-T. On the existence of Hermitian-Yang-Mills connections in stable vector bundles. Comm Pure Appl Math, 1986, 39: S257–S293MathSciNetCrossRefGoogle Scholar
  33. 33.
    Wilson P M H. Metric limits of Calabi-Yau manifolds. In: The Fano Conference. Turin: University of Torino, 2004, 793–804Google Scholar
  34. 34.
    Witten E. Mirror symmetry, Hitchin’s equations, and Langlands duality. In: The Many Facets of Geometry. Oxford: Oxford Univiversity Press, 2010, 113–128CrossRefGoogle Scholar
  35. 35.
    Yau S-T. On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I. Comm Pure Appl Math, 1978, 31: 339–411MathSciNetCrossRefGoogle Scholar
  36. 36.
    Zharkov I. Limiting behavior of local Calabi-Yau metrics. Adv Theor Math Phys, 2004, 8: 395–420MathSciNetCrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of MathematicsFudan UniversityShanghaiChina

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