Skip to main content

Exponentially harmonic maps between Finsler manifolds

Abstract

Exponentially harmonic maps and harmonic maps are different. In this paper, we derive the first and second variations of the exponential energy of a smooth map between Finsler manifolds. We show that a non-constant exponentially harmonic map f from a unit m-sphere \(S^m\) (\(m\ge 3\)) into a Finsler manifold is stable in case \(|df|^2\ge m- 2\), and is unstable in case \(|df|^2< m-2\).

This is a preview of subscription content, access via your institution.

References

  1. Bao, D., Chern, S.S., Shen, Z.: An introduction to Riemann-Finsler Geometry. Springer, New York (2000)

    Book  MATH  Google Scholar 

  2. Bao, D., Chern, S.S.: A note on Gauss-Bonnet theorem for Finsler spaces. Ann. Math. 143, 943–957 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  3. Chiang, Y.J.: Exponential harmonic maps, exponential stress energy and stability. Commun. Contemp. Math. 18(06), 1–14 (2016)

    MathSciNet  Article  Google Scholar 

  4. Chiang, Y.J.: Exponential harmonic maps and their properties. Math. Nachr. 228(7–8), 1970–1980 (2015)

    Article  MATH  Google Scholar 

  5. Chiang, Y.J.: Developments of harmonic maps, wave maps and Yang-Mills fields into biharmonic maps, biwave maps and bi-Yang-Mills fields. Frontier in Mathematics, Birkhäuser, Springer, Basel, xxi+399 (2013)

  6. Chiang, Y.J., Pan, H.: On exponentially harmonic maps. Acta Math. Sinica 58(1), 131–140 (2015). (Chinese)

    MathSciNet  MATH  Google Scholar 

  7. Chiang, Y.J., Wolak, R.: Transversal wave maps and transversal exponential wave maps. J. Geom. 104(3), 443–459 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  8. Chiang, Y.J., Yang, Y.H.: Exponential wave maps. J. Geom. Phys. 57(12), 2521–2532 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  9. Cheung, L.-F., Leung, P.-F.: The second variation formula for exponentially harmonic maps. Bull. Aust. Math. Soc. 59, 509–514 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  10. Duc, D.M., Eells, J.: Regularity of exponentially harmonic functions. Int. J. Math. 2(1), 395–398 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  11. Eells, J., Lemaire, L.: Some properties of exponentially harmonic maps. In: Partial Differential Equations, Part 1, 2 (Warsaw, 1990), Banach Center Publ. 27, pp. 129–136. Polish Acad. Sci., Warsaw (1992)

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

    Article  MATH  Google Scholar 

  13. He, Q., Shen, Y.B.: Some results on harmonic maps for Finsler manifolds. Int. J. Math. 16, 1017–1031 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  14. Hong, M.C.: On the conformal equivalence of harmonic maps and exponentially harmonic maps. Bull. Lond. Math. Soc. 24, 488–492 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  15. Hong, J.Q., Yang, Y.: Some results on exponentially harmonic maps. Chin. Ann. Math. Ser. A 14(6), 686–691 (1993). (Chinese)

    MathSciNet  MATH  Google Scholar 

  16. Kanfon, A.D., Füzfa, A., Lambert, D.: Some examples of exponentially harmonic maps. J. Phys. A Math. Gen. 35, 7629–7639 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  17. Mo, X.H.: Harmonic maps from Finsler manifolds. Illinois J. Math. 45, 1331–1345 (2001)

    MathSciNet  MATH  Google Scholar 

  18. Omori, T.: On Eells–Sampson’s existence theorem for harmonic maps via exponentially harmonic maps. Nagoya Math. J. 201, 133–146 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  19. Omori, T.: On Sacks–Uhlenbeck’s existence theorem for harmonic maps via exponentially harmonic maps. Int. J. Math. 23(10), 1–6 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  20. Shen, Y.B., Zhang, Y.: The second variation of harmonic maps between Finsler manifolds. Sci. China Ser. A 47, 35–51 (2004)

    MathSciNet  Article  Google Scholar 

  21. Shen, Z.: Differential Geometry of Spray and Finsler Spaces. Kluwer Acad. Publ, Dordrecht (2001)

    Book  MATH  Google Scholar 

  22. Smith, R.T.: Harmonic maps of spheres. Am. J. Math. 97, 229–236 (1975)

    Article  Google Scholar 

  23. Wei, S.W., Shen, Y.B.: The stability of harmonic maps on Finsler manifolds. Houst. J. Math. 34, 1049–1056 (2008)

    MathSciNet  MATH  Google Scholar 

  24. Zhang, Y., Wang, Y., Liu, J.: Negative exponentially harmonic maps. J. Beijing Normal Univ. (Nat. Sci.) 34(3), 324–329 (1998)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yuan-Jen Chiang.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chiang, YJ. Exponentially harmonic maps between Finsler manifolds. manuscripta math. 157, 101–119 (2018). https://doi.org/10.1007/s00229-017-0981-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-017-0981-0

Mathematics Subject Classification

  • 58E20
  • 53C60
  • 53B40
  • 58B20