Abstract
We note the existence of exponentially harmonic maps from complete Riemannian manifolds into arbitrary compact Riemannian manifolds. We also investigate the constancy of exponentially harmonic maps with finite exponential energy under some curvature assumptions.
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Burstall, F.E.: Harmonic maps of finite energy from noncompact manifolds. J. Lond. Math. Soc. (2) 30(2), 361–370 (1984). https://doi.org/10.1112/jlms/s2-30.2.361
Chang, S.-C., Chen, J.-T., Wei, S.W.: Liouville properties for p-harmonic maps with finite q-energy. Trans. Am. Math. Soc. 368(2), 787–825 (2016). https://doi.org/10.1090/tran/6351
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/Springer, Basel (2013)
Choi, G., Yun, G.: A theorem of Liouville type for p-harmonic morphisms. Geom. Dedic. 101, 55–59 (2003). https://doi.org/10.1023/A:1026343820908
Duc, D.M.: Variational problems of certain functionals. Int. J. Math. 6(4), 503–535 (1995). https://doi.org/10.1142/S0129167X95000195
Eells, J., Lemaire, L.: Some properties of exponentially harmonic maps, Partial differential equations, Part 1, 2 (Warsaw, : Banach Center Publ., 27, Part 1, vol. 2, Polish Acad. Sci. Inst. Math. Warsaw 1992, 129–136 (1990). https://doi.org/10.1142/1438
Hong, M.C.: The equator map and the negative exponential functional. Manuscr. Math. 75(1), 49–63 (1992)
Hong, J.Q., Yang, Y.H.: Some results on exponentially harmonic maps. Chin. Ann. Math. Ser. A 14(6), 686–691 (1993). (Chinese, with Chinese summary)
Kassi, M.: A Liouville theorem for F-harmonic maps with finite F-energy. Electron. J. Differ. Equ. 2006(15), 9 (2006)
Liu, J.: Liouville-type theorems for F-harmonic maps on non-compact manifolds. Kodai Math. J. 28(3), 483–493 (2005). https://doi.org/10.2996/kmj/1134397762
Naito, H.: On a local Hölder continuity for a minimizer of the exponential energy functional. Nagoya Math. J. 129, 97–113 (1993)
Nakauchi, N.: A Liouville type theorem for p-harmonic maps. Osaka J. Math. 35(2), 303–312 (1998)
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), 1250105 (2012). https://doi.org/10.1142/S0129167X12501054
Pigola, S., Veronelli, G.: On the homotopy class of maps with finite p-energy into non-positively curved manifolds. Geom. Dedic. 143, 109–116 (2009). https://doi.org/10.1007/s10711-009-9376-z
Pigola, S., Rigoli, M., Setti, A.G.: Vanishing theorems on Riemannian manifolds, and geometric applications. J. Funct. Anal. 229(2), 424–461 (2005). https://doi.org/10.1016/j.jfa.2005.05.007
Pigola, S., Rigoli, M., Setti, A.G.: Constancy of p-harmonic maps of finite q-energy into non-positively curved manifolds. Math. Z. 258(2), 347–362 (2008). https://doi.org/10.1007/s00209-007-0175-7
Saloff-Coste, L.: Uniformly elliptic operators on Riemannian manifolds. J. Differ. Geom. 36(2), 417–450 (1992)
Schoen, R., Yau, S.T.: Harmonic maps and the topology of stable hypersurfaces and manifolds with nonnegative Ricci curvature. Comment. Math. Helv. 51(3), 333–341 (1976). https://doi.org/10.1007/BF02568161
Takegoshi, K.: A maximum principle for P-harmonic maps with Lq finite energy. Proc. Am. Math. Soc. 126(12), 3749–3753 (1998). https://doi.org/10.1090/S0002-9939-98-04609-7
Takeuchi, H.: Stability and Liouville theorems of p-harmonic maps. Jpn. J. Math. (N.S.) 17(2), 317–332 (1991)
Wei, S.W.: The minima of the p-energy functional, Elliptic and parabolic methods in geometry, pp. 171–203. A K Peters, Minneapolis (1994) (1996)
Wei, S.W.: Representing homotopy groups and spaces of maps by p-harmonic maps. Indiana Univ. Math. J. 47(2), 625–670 (1998). https://doi.org/10.1512/iumj.1998.47.1179
Acknowledgements
The author would like to thank the referee for her or his careful reading of the manuscript and many helpful suggestions. he is partially supported by Grant-in-Aid for Young Scientists (B) (KAKENHI No. 15K17546) and also by Grant-in-Aid for Scientific Research on Innovative Areas (KAKENHI No. 17H06466) both from JSPS. This work is also supported by JST CREST Grant Number JPMJCR17J4.
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Omori, T. Exponentially harmonic maps of complete Riemannian manifolds. manuscripta math. 161, 205–212 (2020). https://doi.org/10.1007/s00229-018-1084-2
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DOI: https://doi.org/10.1007/s00229-018-1084-2