Skip to main content
SpringerLink
Log in

Dirac-geodesics and their heat flows

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

Dirac-geodesics are Dirac-harmonic maps from one dimensional domains. In this paper, we introduce the heat flow for Dirac-geodesics and establish its long-time existence and an asymptotic property of the global solution. We classify Dirac-geodesics on the standard 2-sphere \(S^2(1)\) and the hyperbolic plane \(\mathbb {H}^2\), and derive existence results on topological spheres and hyperbolic surfaces. These solutions constitute new examples of coupled Dirac-harmonic maps (in the sense that the map part is not simply a harmonic map).

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. Ammann, B., Ginoux, N.: Dirac-harmonic maps from index theory. Calc. Var. Partial Differ. Equ. 47(3–4), 739–762 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  2. Branding, V.: The evolution equations for Dirac-harmonic maps. Ph.D. thesis, Universitätsbibliothek (2012)

  3. Branding, V.: The evolution equations for regularized Dirac-geodesics. arXiv:1311.3581 (2013)

  4. Chen, Q., Jost, J., Li, J.Y., Wang, G.F.: Regularity theorems and energy identities for Dirac-harmonic maps. Math. Z. 251(1), 61–84 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chen, Q., Jost, J., Li, J.Y., Wang, G.F.: Dirac-harmonic maps. Math. Z. 254(2), 409–432 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chen, Q., Jost, J., Sun, L.L., Zhu, M.M.: Boundary value problmes for Dirac equations and the flow of Dirac-harmonic maps. MPI MIS 79, (2014)

  7. Chen, Q., Jost, J., Wang, G., Zhu, M.M.: The boundary value problem for Dirac-harmonic maps. J. Eur. Math. Soc. 15(3), 997–1031 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  8. Choi, K., Parker, T.H.: Convergence of the heat flow for closed geodesics (2013)

  9. Deligne, P., Etingof, P., Freed, D.S., Jeffrey, L.C., Kazhdan, D., Morgan, J.W., Morrison, D.R., Witten, E. (eds.): Quantum fields and strings: a course for mathematicians. vol. 1, 2, American Mathematical Society, Providence, RI; Institute for Advanced Study (IAS), Princeton, NJ, 1999, Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ (1996–1997)

  10. Friedrich, T.: Dirac operators in Riemannian geometry, Graduate Studies in Mathematics, vol. 25. American Mathematical Society, Providence, RI (2000, Translated from the 1997 German original by Andreas Nestke)

  11. Isobe, T.: On the existence of nonlinear Dirac-geodesics on compact manifolds. Calc. Var. Partial Differ. Equ. 43(1–2), 83–121 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  12. Jost, J.: Geometry and Physics. Springer-Verlag, Berlin (2009)

    Book  MATH  Google Scholar 

  13. Jost, J.: Riemannian Geometry and Geometric Analysis, 6th edn. Universitext, Springer, Heidelberg (2011)

    Book  MATH  Google Scholar 

  14. Jost, J., Mo, X.H., Zhu, M.M.: Some explicit constructions of Dirac-harmonic maps. J. Geom. Phys. 59(11), 1512–1527 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. Koh, D.: On the evolution equation for magnetic geodesics. Calc. Var. Partial Differ. Equ. 36(3), 453–480 (2009)

    Article  MATH  Google Scholar 

  16. Lawson, H.B., Michelsohn, M.-L.: Spin Geometry, Princeton Mathematical Series, vol. 38. Princeton University Press, Princeton, NJ (1989)

    Google Scholar 

  17. Lin, L.Z., Wang, L.: Existence of good sweepouts on closed manifolds. Proc. Am. Math. Soc. 138(11), 4081–4088 (2010)

    Article  MATH  Google Scholar 

  18. Ottarsson, S.K.: Closed geodesics on Riemannian manifolds via the heat flow. J. Geom. Phys. 2(1), 49–72 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  19. Rosenberg, H., Schneider, M.: Embedded constant-curvature curves on convex surfaces. Pacific J. Math. 253(1), 213–218 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  20. Schneider, M.: Closed magnetic geodesics on \(S^2\). J. Differ. Geom. 87(2), 343–388 (2011)

    MATH  Google Scholar 

  21. Schneider, M.: Closed magnetic geodesics on closed hyperbolic Riemann surfaces. Proc. Lond. Math. Soc. 105(2), 424–446 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  22. Sharp, B., Zhu, M.M.: Regularity at the free boundary for Dirac-harmonic maps from surfaces. arXiv:1306.4260 (2013)

  23. Topping, P.M.: Rigidity in the harmonic map heat flow. J. Differ. Geom. 45(3), 593–610 (1997)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Hubei, 430072, China

    Qun Chen & Linlin Sun

  2. Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103, Leipzig, Germany

    Jürgen Jost, Linlin Sun & Miaomiao Zhu

Authors
  1. Qun Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jürgen Jost
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Linlin Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Miaomiao Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jürgen Jost.

Additional information

Communicated by L. Ambrosio.

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013) / ERC Grant Agreement No. 267087. The research of QC is partially supported by NSFC of China. The research of LLS is partially supported by CSC of China. The authors thank the Max Planck Institute for Mathematics in the Sciences for good working conditions when this work was carried out.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, Q., Jost, J., Sun, L. et al. Dirac-geodesics and their heat flows. Calc. Var. 54, 2615–2635 (2015). https://doi.org/10.1007/s00526-015-0877-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-015-0877-3

Mathematics Subject Classification

Search

Navigation