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 \(|df|^2< m-2\).
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References
Bao, D., Chern, S.S., Shen, Z.: An introduction to Riemann-Finsler Geometry. Springer, New York (2000)
Bao, D., Chern, S.S.: A note on Gauss-Bonnet theorem for Finsler spaces. Ann. Math. 143, 943–957 (1996)
Chiang, Y.J.: Exponential harmonic maps, exponential stress energy and stability. Commun. Contemp. Math. 18(06), 1–14 (2016)
Chiang, Y.J.: Exponential 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. Frontier in Mathematics, Birkhäuser, Springer, Basel, xxi+399 (2013)
Chiang, Y.J., Pan, H.: On exponentially harmonic maps. Acta Math. Sinica 58(1), 131–140 (2015). (Chinese)
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), Banach Center Publ. 27, pp. 129–136. Polish Acad. Sci., Warsaw (1992)
Eells, J., Sampson, J.H.: Harmonic maps of Riemannian manifolds. Am. J. Math. 86, 109–160 (1964)
He, Q., Shen, Y.B.: Some results on harmonic maps for Finsler manifolds. Int. J. Math. 16, 1017–1031 (2005)
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). (Chinese)
Kanfon, A.D., Füzfa, A., Lambert, D.: Some examples of exponentially harmonic maps. J. Phys. A Math. Gen. 35, 7629–7639 (2002)
Mo, X.H.: Harmonic maps from Finsler manifolds. Illinois J. Math. 45, 1331–1345 (2001)
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)
Shen, Y.B., Zhang, Y.: The second variation of harmonic maps between Finsler manifolds. Sci. China Ser. A 47, 35–51 (2004)
Shen, Z.: Differential Geometry of Spray and Finsler Spaces. Kluwer Acad. Publ, Dordrecht (2001)
Smith, R.T.: Harmonic maps of spheres. Am. J. Math. 97, 229–236 (1975)
Wei, S.W., Shen, Y.B.: The stability of harmonic maps on Finsler manifolds. Houst. J. Math. 34, 1049–1056 (2008)
Zhang, Y., Wang, Y., Liu, J.: Negative exponentially harmonic maps. J. Beijing Normal Univ. (Nat. Sci.) 34(3), 324–329 (1998)
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Chiang, YJ. Exponentially harmonic maps between Finsler manifolds. manuscripta math. 157, 101–119 (2018). https://doi.org/10.1007/s00229-017-0981-0
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DOI: https://doi.org/10.1007/s00229-017-0981-0