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Dirac-harmonic maps from index theory

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Abstract

We prove existence results for Dirac-harmonic maps using index theoretical tools. They are mainly interesting if the source manifold has dimension 1 or 2 modulo 8. Our solutions are uncoupled in the sense that the underlying map between the source and target manifolds is a harmonic map.

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References

  1. Ammann B., Dahl M., Humbert E.: Harmonic spinors and local deformations of the metric. Math. Res. Lett. 18, 927–936 (2011)

    MathSciNet  MATH  Google Scholar 

  2. Ammann B., Dahl M., Humbert E.: Surgery and harmonic spinors. Adv. Math. 220, 523–539 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ammann, B., Ginoux, N.: Examples of Dirac-harmonic maps after Jost-Mo-Zhu. http://www.mathematik.uni-regensburg.de/ginoux (2009)

  4. Bär C.: Metrics with harmonic spinors. Geom. Funct. Anal. 6, 899–942 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bär C.: Harmonic spinors for twisted Dirac operators. Math. Ann. 309(2), 225–246 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bär C., Dahl M.: Surgery and the Spectrum of the Dirac operator. J. Reine. Angew. Math. 552, 53–76 (2002)

    MathSciNet  MATH  Google Scholar 

  7. Bär C., Schmutz P.: Harmonic spinors on Riemann surfaces. Ann. Glob. Anal. Geom. 10, 263–273 (1992)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen Q., Jost J., Wang G.: Liouville theorems for Dirac-harmonic maps. J. Math. Phys. 48(13), 113517 (2007)

    Article  MathSciNet  Google Scholar 

  11. Chen Q., Jost J., Wang G.: Nonlinear Dirac equations on Riemann surfaces. Ann. Glob. Anal. Geom. 33(3), 253–270 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dahl M.: On the space of metrics with invertible Dirac operator. Comment. Math. Helv. 83(2), 451–469 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Deligne, P., Freed, D.: Supersolutions. In: 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, pp. 227–356. AMS, Providence (1999)

  14. Eells J., Sampson J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–160 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  15. Freed, D.: Five Lectures on Supersymmetry. American Mathematical Society, Providence (1999)

  16. Friedrich, T.: Dirac operators in Riemannian geometry. In: Graduate Studies in Mathematics, vol. 25. American Mathematical Society, Providence (2000)

  17. Ginoux N., Habib G.: The spectrum of the twisted Dirac operator on Kähler submanifolds of the complex projective space. Manuscr. Math. 137(1–2), 215–231 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hijazi, O.: Spectral properties of the Dirac operator and geometrical structures. In: Geometric Methods for Quantum Field Theory (Villa de Leyva, 1999), pp. 116–169. World Science Publication, River Edge (2001)

  19. Hitchin N.: Harmonic spinors. Adv. Math. 14, 1–55 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  20. Isobe T.: Nonlinear Dirac equations with critical nonlinearities on compact Spin manifolds. J. Funct. Anal. 260(1), 253–307 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Kosinski, A.A.: Differential manifolds. In: Pure and Applied Mathematics, vol. 138. Academic Press Inc., Boston (1993)

  23. Lawson, H.B., Michelsohn, M.-L.: Spin geometry. In: Princeton Mathematical Series, vol. 38. Princeton University Press, Princeton (1989)

  24. Lemaire L.: Applications harmoniques de surfaces riemanniennes. J. Differ. Geom. 13(1), 51–78 (1978)

    MathSciNet  MATH  Google Scholar 

  25. Milnor, J.W., Stasheff, J.D.: Characteristic classes. In: Annals of Mathematics Studies, vol. 76. Princeton University Press, Princeton (1974)

  26. Mo X.: Some rigidity results for Dirac-harmonic maps. Publ. Math. Debrecen 77(3-4), 427–442 (2010)

    MathSciNet  MATH  Google Scholar 

  27. Parker T.H.: Bubble tree convergence for harmonic maps. J. Differ. Geom. 44, 595–633 (1996)

    MATH  Google Scholar 

  28. Parker T.H.: What is a bubble tree?. Notices AMS 50, 666–667 (2003)

    Google Scholar 

  29. Sacks J., Uhlenbeck K.: The existence of minimal immersions of 2-spheres. Ann. Math. 2(113), 1–24 (1981)

    Article  MathSciNet  Google Scholar 

  30. Schoen R., Yau S.-T.: Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative Ricci curvature. Comment. Math. Helv. 51(3), 333–341 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  31. Seeger, L.: Harmonische Spinoren auf gerade-dimensionalen Sphären. PhD Thesis, University of Hamburg, Shaker, Aachen. ISBN:978-3-8265-8755-9 (2000)

  32. Steenrod, N.: The topology of fibre bundles. In: Princeton Mathematical Series, vol. 14. Princeton University Press, Princeton (1951)

  33. Wang, C., Xu, D.: Regularity of Dirac-Harmonic Maps. Int. Math. Res. Not.IMRN 2009, no. 20, pp. 3759–3792 (2009)

  34. Weidmann, J.: Linear operators in Hilbert spaces. In: Graduate Texts in Mathematics, vol. 68. Springer, New York (1980)

  35. Xin, Y.L.: Nonexistence and existence for harmonic maps in Riemannian manifolds. In: Proceedings of the 1981 Shanghai Symposium on Differential Geometry and Differential Equations (Shanghai/Hefei, 1981), pp. 529–538. Science Press, Beijing (1984)

  36. Xin, Y.L.: Geometry of harmonic maps. In: Progress in Nonlinear Differential Equations and Their Applications, vol. 23. Birkhäuser, Boston (1996)

  37. Yang L.: A structure theorem of Dirac-harmonic maps between spheres. Calc. Var. 35(4), 409–420 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. Zhao L.: Energy identities for Dirac-harmonic maps. Calc. Var. 28(1), 121–138 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhu M.: Dirac-harmonic maps from degenerating spin surfaces. I. The Neveu–Schwarz case. Calc. Var. 35(2), 169–189 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zhu M.: Regularity for weakly Dirac-harmonic maps to hypersurfaces. Ann. Glob. Anal. Geom. 35(4), 405–412 (2009)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Regensburg, 93040, Regensburg, Germany

    Bernd Ammann & Nicolas Ginoux

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  1. Bernd Ammann
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  2. Nicolas Ginoux
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Correspondence to Bernd Ammann.

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Communicated by J.Jost.

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Ammann, B., Ginoux, N. Dirac-harmonic maps from index theory. Calc. Var. 47, 739–762 (2013). https://doi.org/10.1007/s00526-012-0534-z

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