Abstract
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 \(\pi _2 (M_2) = \emptyset \).
Similar content being viewed by others
References
Bers, L.: Quasiconformal mappings and Teichmüller’s theorem. In: Analytic Functions, pp. 89–119. Princeton University Press, Princeton (1960)
Chiang, Y.J.: Exponentially harmonic maps between Finsler manifolds, Manuscripta Mathematica published online in October (2017)
Chiang, Y.J.: Equivariant exponentially harmonic maps between manifolds with metrics of signatures. Asian Eur. J. Math. (WSP) 10, 1–16 (2017)
Chiang, Y.J.: Exponentially harmonic maps, exponential stress energy and stability. Commun. Contemp. Math. (WSP) 18(6), 1–14 (2016)
Chiang, Y.J.: Exponentially harmonic maps and their properties. Math. Nachr. 228(7–8), 1970–1980 (2015)
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)
Chiang, Y.J., Heng, P.: On exponentially harmonic maps. Acta Math. Sin. 58(1), 131–140 (2015)
Chiang, Y.J., Wolak, R.: Transversal wave maps and transversal exponential wave maps. J. Geom. 104(3), 443–459 (2013)
Chiang, Y.J., Yang, Y.H.: Exponential wave maps. J. Geom. Phys. 57(12), 2521–2532 (2007)
Cheung, L.-F., Leung, P.-F.: The second variation formula for exponentially harmonic maps. Bull. Aust. Math. Soc. 59, 509–514 (1999)
Duc, D.M., Eells, J.: Regularity of exponentially harmonic functions. Int. J. Math. 2(1), 395–398 (1991)
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)
Hildebrandt, S.: On the Plateau problem for surfaces of constant mean curvature. Commun. Pure Appl. Math. 23, 97–114 (1970)
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)
Hildebrandt, S., Widman, K.-O.: Some regularity results for quasilinear elliptic systems of second order. Math. Z. 42, 67–86 (1975)
Hildebrandt, S., Kaul, H., Widman, K.-O.: An existence theorem for harmonic mappings of Riemannian manifolds. Acta Math. 138, 1–16 (1977)
Hong, M.C.: On the conformal equivalence of harmonic maps and exponentially harmonic maps. Bull. Lond. Math. Soc. 24, 488–492 (1992)
Hong, J.Q., Yang, Y.: Some results on exponentially harmonic maps. Chin. Ann. Math. Ser. A 14(6), 686–691 (1993)
Jost, J.: Existence proofs for harmonic mappings with the help of a maximum principle. Math. Z. 184, 489–496 (1983)
Jost, J.: Harmonic Maps Between Surfaces. Lecture Notes in Mathematics, vol. 1062. Springer, Berlin (1984)
Kanfon, A.D., Füzfa, A., Lambert, D.: Some examples of exponentially harmonic maps. J. Phys. A Math. Gen. 35, 7629–7639 (2002)
Lemaire, L.: Applications harmoniques de surfaces Riemanniennes. J. Differ. Geom. 4(8), 51–78 (1978)
Lemaire, L.: Boundary vale problems for harmonic and minimal maps of surfaces into manifolds. Ann. Sc. Norm. Sup. Pisa 8(4), 91–103 (1982)
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)
Morrey, C.: Multiple Integrals in the Calculus of Variations, Grundlehren Math Wiss, vol. 130. Springer, Berlin (1966)
Omori, T.: On Eells–Sampson’s existence theorem for harmonic maps via exponentially harmonic maps. Nagoya Math. J. 201, 133–146 (2011)
Omori, T.: On Sacks–Uhlenbeck’s existence theorem for harmonic maps via exponentially harmonic maps. Int. J. Math. 23(10), 1–6 (2012)
Sacks, J., Uhlenbeck, K.: The existence of minimal immersions of 2-spheres. Ann. Math. 113, 1–24 (1981)
Zhang, Y., Wang, Y., Liu, J.: Negative exponential harmonic maps. J. Beijing Normal Univ. (Nat. Sci.) 34(3), 324–329 (1998)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Chiang, YJ. Exponentially harmonic maps between surfaces. Anal.Math.Phys. 9, 1729–1739 (2019). https://doi.org/10.1007/s13324-018-0270-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13324-018-0270-4