Skip to main content
SpringerLink
Log in

Local behaviour of sequences of three-dimensional generalised monopoles

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

The purpose of this paper is to study the behaviour of sequences of generalised monopoles with a uniform bound on a certain \(L^2\)-norm. We focus on the case that the target hyperKähler manifolds are Swann bundles. In 3-dimensional case, suppose that there exists an open submanifold \(Y'\) such that the hyperKähler potential along the monopoles has a uniform lower bound over \(Y'\). Then we show that there exist convergent subsequences of generalised monopoles over any compact subset of \(Y'\). Under similar assumptions, the same conclusion holds for the generalised harmonic spinors in dimension four.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alekseevskii, D.V.: Riemannian spaces with exceptional holonomy groups. Funct. Anal. Appl. 2, 97–105 (1968)

    Article  Google Scholar 

  2. Alekseevskii, D.V.: Compact quaternion spaces. Funkcional. Funct. Anal. Appl. 2, 106–114 (1968)

    Article  MathSciNet  Google Scholar 

  3. Biswas, I., Thakre, V.: Generalised monopole equations on Kähler surfaces. J. Math. Phys. 59(4), 043503 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  4. Callies, M.: Dimensional Reduction for the Generalized Seiberg–Witten Equations and the Chern–Simons–Dirac Functional. Diplom in mathematics, Georg–August–Universität, Göttingen (2010). http://webdoc.sub.gwdg.de/ebook/serien/e/mathematica-gottingensis/mg.2010.03.pdf

  5. Callies, M.: Permuting Actions, Moment Maps and the Generalized Seiberg–Witten Equations. PhD thesis, Georg–August–Universität, Göttingen (2015). https://ediss.uni-goettingen.de/bitstream/handle/11858/00-1735-0000-0028-8738-7/callies.pdf?sequence=1

  6. Haydys, A.: Generalized Seiberg–Witten Equations and HyperKähler Geometry. PhD in mathematics, Georg–August–Universität, Göttingen (2006). https://ediss.uni-goettingen.de/bitstream/handle/11858/00-1735-0000-0006-B381-C/haydys.pdf?sequence=1

  7. Haydys, A., Walpuski, T.: A compactness theorem for the Seiberg–Witten equations with multiple spinors in dimension three. Geom. Funct. Anal. 25, 1799–1821 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hitchin, N.J.: Monopoles, Minimal Surfaces and Algebraic Curves. Presses De Luniversité De Montréal, Montreal (1987)

  9. Kolář, I., Michor, P.W., Slovák, J.: Natural Operations in Differential Geometry. Springer, Berlin (1993)

    Book  MATH  Google Scholar 

  10. Kronheimer, P.B.: Instantons and the geometry of the nilpotent variety. J. Differ. Geom. 32, 473–490 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kobak, P., Swann, A.: The HyperKähler Geometry associated to Wolf Spaces. Mathematics 3(3), 587–595 (2001)

    MATH  Google Scholar 

  12. Pidstrygach, V.Y.: HyperKähler manifolds and Seiberg–Witten equations. In: Proceedings of the Steklov Institute of Mathematics, pp. 249–262 (2004)

  13. Swann, A.: HyperKähler and quaterninoic Kähler geometry. Math. Ann. 289(3), 421–450 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  14. Schumacher, H.: Generalized Seiberg–Witten Equations: Swann Bundles and \(L^\infty \)–Estimates. Master’s thesis, Mathematisches Institut, Georg–August–Universität, Göttingen (2010). https://www.uni-math.gwdg.de/preprint/mg.2010.02.pdf

  15. Taubes, C.H.: Nonlinear Generalizations of a 3-Manifold’s Dirac Operator. Trends in Mathematical Physics. AMS/IP Studies in Advanced Mathematics, Knoxville, TN, vol. 13, Amer. Math. Soc. Providence, RI, 1999, 475–486 (1998)

  16. Taubes, C.H.: Compactness theorems for \(SL(2; {\mathbb{C}})\) generalizations of the 4-dimensional anti-self dual equations (2013). arXiv:1307.6447v5

  17. Taubes, C.H.: On the behavior of sequences of solutions to U(1) Seiberg–Witten systems in dimension 4 (2016). arXiv:1610.07163

  18. Thakre, V.: Generalised Seiberg–Witten equations and almost-Hermitian geometry. J. Geom. Phys. 134, 119–132 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wolf, J.A.: Complex homogeneous contact manifolds and quaternionic symmetric spaces. J. Math. Mech. 14, 1033–1047 (1965)

    MathSciNet  MATH  Google Scholar 

  20. Wehrheim, K.: Uhlenbeck Compactness. European Mathematical Society, Zurich (2004)

    Book  MATH  Google Scholar 

  21. Walpuski, T.: A compactness theorem for Fueter sections. Comment. Math. Helv. 92, 751–776 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  22. Walpuski, T., Zhang, B.: On the compactness problem for a family of generalized Seiberg–Witten equations in dimension three (2019). arXiv:1904.03749

Download references

Acknowledgements

This work is carried out at the University of Adelaide, the author acknowledges the comfortable environment provided by the School of Mathematics and the support from ARC Discovery project grant DP170101054 with Chief Investigators Mathai Varghese and David Baraglia. The author would like to thank Dr.Varun Thakre’s helpful comments. He also wants to thank the anonymous referee, whose comments and suggestions have greatly improved this paper.

Author information

Authors and Affiliations

  1. Yau Center, Southeast University, Nanjing, China

    Guanheng Chen

Authors
  1. Guanheng Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guanheng Chen.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, G. Local behaviour of sequences of three-dimensional generalised monopoles. manuscripta math. 171, 595–620 (2023). https://doi.org/10.1007/s00229-022-01395-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-022-01395-x

Mathematics Subject Classification

Search

Navigation