Abstract
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 (N, h) a Riemannian manifold, under certain conditions we establish a Liouville type 2 theorem for exponentially harmonic maps \(f:(M, \rho ^2 g_0)\rightarrow N\), \(0< \rho \in C^\infty (M)\).
Similar content being viewed by others
References
Baird, P.: Stress-energy tensors and the Lichnerowicz Laplacian. J. Geom. Phys. 58(10), 1329–1342 (2008)
Cheung, L.-F., Leung, P.-F.: The second variation formula for exponentially harmonic maps. Bull. Austral. Math. Soc. 59(3), 509–514 (1999)
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)
Chiang, Y.-J.: Exponentially harmonic maps and their properties. Math. Nachr. 288(17–18), 1970–1980 (2015)
Chiang, Y.-J.: Exponential harmonic maps, exponential stress energy and stability. Commun. Contemp. Math. 18(6), 1550076 (2016)
Chiang, Y.-J.: Exponentially harmonic maps between Finsler manifolds. Manuscripta Math. 157(1–2), 101–119 (2018)
Chiang, Y.J., Pan, H.: Exponentially harmonic maps. Acta Math. Sinica (Chin. Ser.) 58(1), 131–140 (2015) (in Chinese)
Chiang, Y.-J., Wolak, R.A.: 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)
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)
Duc, D.M., Eells, J.: Regularity of exponentially harmonic functions. Internat. J. Math. 2(4), 395–4098 (1991)
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)
Eells Jr., J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Amer. J. Math. 86, 109–160 (1964)
El Soufi, A., Lejeune, A.: Indice de Morse des applications \(p\)-harmoniques. Ann. Inst. H. Poincaré Anal. Non Linéaire 13(2), 229–250 (1996)
Gordon, W.B.: Convex functions and harmonic maps. Proc. Amer. Math. Soc. 33(2), 433–437 (1972)
Hong, J.Q., Yang, Y.H.: Some results on exponentially harmonic maps. Chinese Ann. Math. Ser. A 14(6), 686–691 (1993). (in Chinese)
Hong, M.C.: On the conformal equivalence of harmonic maps and exponentially harmonic maps. Bull. Lond. Math. Soc. 24(5), 488–492 (1992)
Kanfon, A.D., Füzfa, A., Lambert, D.: Some examples of exponentially harmonic maps. J. Phys. A 35(35), 7629–7639 (2002)
Kawai, S.: \(p\)-harmonic maps and convex functions. Geom. Dedicata 74(3), 261–265 (1999)
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)
Liu, J.C.: Liouville-type theorems for exponentially harmonic maps. J. Lanzhou Univ. Nat. Sci. 41(6), 122–124 (2005). (in Chinese)
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. Internat. J. Math. 23(10), 1250105 (2012)
Sacks, J., Uhlenbeck, K.: The existence of minimal immersions of 2-spheres. Ann. Math. 113(1), 1–24 (1981)
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)
Author information
Authors and Affiliations
Corresponding author
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, Morse index and Liouville type theorems. European Journal of Mathematics 6, 1388–1402 (2020). https://doi.org/10.1007/s40879-019-00362-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40879-019-00362-3