Analysis and Mathematical Physics

, Volume 9, Issue 4, pp 1729–1739 | Cite as

Exponentially harmonic maps between surfaces

  • Yuan-Jen ChiangEmail author


We show the maximum principle for exponential energy minimizing maps. We then estimate the distance of two image points of an exponentially harmonic map between surfaces. We also study the existence of an exponentially harmonic map between surfaces if the image is contained in a convex disc. We finally investigate the existence of an exponentially harmonic map \(f:M_1\rightarrow M_2\) between surfaces in case \(\pi _2 (M_2) = \emptyset \).


Exponential energy Exponential tension field Exponentially harmonic map 

Mathematics Subject Classification

58E20 58G11 35K05 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Bers, L.: Quasiconformal mappings and Teichmüller’s theorem. In: Analytic Functions, pp. 89–119. Princeton University Press, Princeton (1960)Google Scholar
  2. 2.
    Chiang, Y.J.: Exponentially harmonic maps between Finsler manifolds, Manuscripta Mathematica published online in October (2017)Google Scholar
  3. 3.
    Chiang, Y.J.: Equivariant exponentially harmonic maps between manifolds with metrics of signatures. Asian Eur. J. Math. (WSP) 10, 1–16 (2017)Google Scholar
  4. 4.
    Chiang, Y.J.: Exponentially harmonic maps, exponential stress energy and stability. Commun. Contemp. Math. (WSP) 18(6), 1–14 (2016)Google Scholar
  5. 5.
    Chiang, Y.J.: Exponentially harmonic maps and their properties. Math. Nachr. 228(7–8), 1970–1980 (2015)Google Scholar
  6. 6.
    Chiang, Y.J.: Developments of Harmonic Maps, Wave Maps and Yang-Mills Fields into Biharmonic Maps, Biwave Maps and Bi-Yang-Mills Fields. Frontiers in Mathematics, p. xxi+399. Birkhä user, Springer, Basel (2013)Google Scholar
  7. 7.
    Chiang, Y.J., Heng, P.: On exponentially harmonic maps. Acta Math. Sin. 58(1), 131–140 (2015)Google Scholar
  8. 8.
    Chiang, Y.J., Wolak, R.: Transversal wave maps and transversal exponential wave maps. J. Geom. 104(3), 443–459 (2013)Google Scholar
  9. 9.
    Chiang, Y.J., Yang, Y.H.: Exponential wave maps. J. Geom. Phys. 57(12), 2521–2532 (2007)Google Scholar
  10. 10.
    Cheung, L.-F., Leung, P.-F.: The second variation formula for exponentially harmonic maps. Bull. Aust. Math. Soc. 59, 509–514 (1999)Google Scholar
  11. 11.
    Duc, D.M., Eells, J.: Regularity of exponentially harmonic functions. Int. J. Math. 2(1), 395–398 (1991)Google Scholar
  12. 12.
    Eells, J., Lemaire, L.: Some properties of exponentially harmonic maps. In: Partial Differential Equations, Part 1, 2 (Warsaw, 1990), 129–136, Banach Center Pulb. 27, Polish Academy of Sciences, Warsaw (1992)Google Scholar
  13. 13.
    Hildebrandt, S.: On the Plateau problem for surfaces of constant mean curvature. Commun. Pure Appl. Math. 23, 97–114 (1970)Google Scholar
  14. 14.
    Hildebrandt, S.: Nonlinear elliptic systems and harmonic mappings. In: Beijing Symposium on Differential Geometry and Differential Geometry and Differential Equations. Science Press, Beijing, 1982, also in SFB 72, Vorlesungsreihe, no. 3, Bonn (1980)Google Scholar
  15. 15.
    Hildebrandt, S., Widman, K.-O.: Some regularity results for quasilinear elliptic systems of second order. Math. Z. 42, 67–86 (1975)Google Scholar
  16. 16.
    Hildebrandt, S., Kaul, H., Widman, K.-O.: An existence theorem for harmonic mappings of Riemannian manifolds. Acta Math. 138, 1–16 (1977)Google Scholar
  17. 17.
    Hong, M.C.: On the conformal equivalence of harmonic maps and exponentially harmonic maps. Bull. Lond. Math. Soc. 24, 488–492 (1992)Google Scholar
  18. 18.
    Hong, J.Q., Yang, Y.: Some results on exponentially harmonic maps. Chin. Ann. Math. Ser. A 14(6), 686–691 (1993)Google Scholar
  19. 19.
    Jost, J.: Existence proofs for harmonic mappings with the help of a maximum principle. Math. Z. 184, 489–496 (1983)Google Scholar
  20. 20.
    Jost, J.: Harmonic Maps Between Surfaces. Lecture Notes in Mathematics, vol. 1062. Springer, Berlin (1984)Google Scholar
  21. 21.
    Kanfon, A.D., Füzfa, A., Lambert, D.: Some examples of exponentially harmonic maps. J. Phys. A Math. Gen. 35, 7629–7639 (2002)Google Scholar
  22. 22.
    Lemaire, L.: Applications harmoniques de surfaces Riemanniennes. J. Differ. Geom. 4(8), 51–78 (1978)Google Scholar
  23. 23.
    Lemaire, L.: Boundary vale problems for harmonic and minimal maps of surfaces into manifolds. Ann. Sc. Norm. Sup. Pisa 8(4), 91–103 (1982)Google Scholar
  24. 24.
    Liu, J.: Nonexistence of stable exponentially harmonic maps from or into compact convex hypersurfaces in \({{\mathbb{R}}}^{m+1}\). Turk. J. Math. 32, 117–126 (2008)Google Scholar
  25. 25.
    Morrey, C.: Multiple Integrals in the Calculus of Variations, Grundlehren Math Wiss, vol. 130. Springer, Berlin (1966)Google Scholar
  26. 26.
    Omori, T.: On Eells–Sampson’s existence theorem for harmonic maps via exponentially harmonic maps. Nagoya Math. J. 201, 133–146 (2011)Google Scholar
  27. 27.
    Omori, T.: On Sacks–Uhlenbeck’s existence theorem for harmonic maps via exponentially harmonic maps. Int. J. Math. 23(10), 1–6 (2012)Google Scholar
  28. 28.
    Sacks, J., Uhlenbeck, K.: The existence of minimal immersions of 2-spheres. Ann. Math. 113, 1–24 (1981)Google Scholar
  29. 29.
    Zhang, Y., Wang, Y., Liu, J.: Negative exponential harmonic maps. J. Beijing Normal Univ. (Nat. Sci.) 34(3), 324–329 (1998)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Mary WashingtonFredericksburgUSA

Personalised recommendations