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Exponentially harmonic maps, Morse index and Liouville type theorems


We obtain a result on the Morse index of an exponentially harmonic map from a Riemannian manifold into the unit n-sphere. Next, we prove a Liouville type 1 theorem for exponentially harmonic maps between two Riemannian manifolds. Finally, let \((M, g_0)\) be a complete Riemannian manifold with a pole \(x_0\) and (Nh) a Riemannian manifold, under certain conditions we establish a Liouville type 2 theorem for exponentially harmonic maps \(f:(M, \rho ^2 g_0)\rightarrow N\), \(0< \rho \in C^\infty (M)\).

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  1. Baird, P.: Stress-energy tensors and the Lichnerowicz Laplacian. J. Geom. Phys. 58(10), 1329–1342 (2008)

    MathSciNet  Article  Google Scholar 

  2. Cheung, L.-F., Leung, P.-F.: The second variation formula for exponentially harmonic maps. Bull. Austral. Math. Soc. 59(3), 509–514 (1999)

    MathSciNet  Article  Google Scholar 

  3. 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. Birkhäuser, Basel (2013)

    Book  Google Scholar 

  4. Chiang, Y.-J.: Exponentially harmonic maps and their properties. Math. Nachr. 288(17–18), 1970–1980 (2015)

    MathSciNet  Article  Google Scholar 

  5. Chiang, Y.-J.: Exponential harmonic maps, exponential stress energy and stability. Commun. Contemp. Math. 18(6), 1550076 (2016)

    MathSciNet  Article  Google Scholar 

  6. Chiang, Y.-J.: Exponentially harmonic maps between Finsler manifolds. Manuscripta Math. 157(1–2), 101–119 (2018)

    MathSciNet  Article  Google Scholar 

  7. Chiang, Y.J., Pan, H.: Exponentially harmonic maps. Acta Math. Sinica (Chin. Ser.) 58(1), 131–140 (2015) (in Chinese)

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  10. Dong, Y., Wei, S.W.: On vanishing theorems for vector bundle valued \(p\)-forms and their applications. Commun. Math. Phys. 304(2), 329–368 (2011)

    MathSciNet  Article  Google Scholar 

  11. Duc, D.M., Eells, J.: Regularity of exponentially harmonic functions. Internat. J. Math. 2(4), 395–4098 (1991)

    MathSciNet  Article  Google Scholar 

  12. Eells, J., Lemaire, L.: Some properties of exponentially harmonic maps. In: Bojarski, B., Zajączkowski, W., Ziemian, B. (eds.) Partial Differential Equations, Part 1, 2. Banach Center Publications, vol. 27, pp. 129–136. Polish Academy of Sciences, Warsaw (1992)

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

    MathSciNet  Article  Google Scholar 

  14. El Soufi, A., Lejeune, A.: Indice de Morse des applications \(p\)-harmoniques. Ann. Inst. H. Poincaré Anal. Non Linéaire 13(2), 229–250 (1996)

    MathSciNet  Article  Google Scholar 

  15. Gordon, W.B.: Convex functions and harmonic maps. Proc. Amer. Math. Soc. 33(2), 433–437 (1972)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  17. Hong, M.C.: On the conformal equivalence of harmonic maps and exponentially harmonic maps. Bull. Lond. Math. Soc. 24(5), 488–492 (1992)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  19. Kawai, S.: \(p\)-harmonic maps and convex functions. Geom. Dedicata 74(3), 261–265 (1999)

    MathSciNet  Article  Google Scholar 

  20. Liu, J.: Nonexistence of stable exponentially harmonic maps from or into compact convex hypersurfaces in \({\mathbb{R}}^{m+1}\). Turkish J. Math. 32(2), 117–126 (2008)

    MathSciNet  MATH  Google Scholar 

  21. Liu, J.C.: Liouville-type theorems for exponentially harmonic maps. J. Lanzhou Univ. Nat. Sci. 41(6), 122–124 (2005). (in Chinese)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  23. Omori, T.: On Sacks–Uhlenbeck’s existence theorem for harmonic maps via exponentially harmonic maps. Internat. J. Math. 23(10), 1250105 (2012)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  25. Zhang, Y.T., Wang, Y.N., Liu, J.Z.: Some results on negative exponential harmonic maps. Beijing Shifan Daxue Xuebao 34(3), 324–329 (1998) (in Chinese)

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Correspondence to Yuan-Jen Chiang.

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Chiang, YJ. Exponentially harmonic maps, Morse index and Liouville type theorems. European Journal of Mathematics 6, 1388–1402 (2020).

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  • Exponentially harmonic map
  • Morse index
  • Liouville type theorems

Mathematics Subject Classification

  • 58E20
  • 58G11
  • 35J20