Abstract
The notion of exponentially harmonic maps was introduced by Eells and Lemaire (Proceedings of the Banach Center Semester on PDE, pp. 1990–1991, 1990). In this note, by using the maximum principle we get the gradient estimate of exponentially harmonic functions, and then derive a Liouville type theorem for bounded exponentially harmonic functions on a complete Riemannian manifold with nonnegative Ricci curvature and sectional curvature bounded below.
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References
Cheng S.Y.: Liouville theorem for harmonic maps. Proc. Symp. Pure Math. 36, 147–151 (1980)
Choi H.I.: On the Liouville theorem for harmonic maps. Proc. Am. Math. Soc. 85, 91–94 (1982)
Eells, J., Lemaire, L.: Some properties of exponentially harmonic maps. In: Proceedings of the Banach Center Semester on PDE, pp. 1990–1991 (1990)
Hong J., Yang Y.-H.: Some results on exponentially harmonic maps. Chin. Ann. Math. 6, 686–691 (1993)
Hong M.-C.: Liouville theorems for exponentially harmonic functions on Riemmannian manifolds. Manuscr. Math. 77(1), 41–46 (1992)
Li P., Yau S.T.: On the parabolic kernel of the Schrodinger operator. Acta Math. 156, 153–201 (1986)
Yau S.Y.: Harmonic functions on complete Riemannian manifolds. Commun. Pure Appl. Math. 28, 201–228 (1975)
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Qihua Ruan: Supported partially by NSF of Fujian Province of China (No. 2012J01015).
Yi-Hu Yang: Supported partially by NSF of China (No. 11171253).
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Wu, J., Ruan, Q. & Yang, YH. Gradient estimate for exponentially harmonic functions on complete Riemannian manifolds. manuscripta math. 143, 483–489 (2014). https://doi.org/10.1007/s00229-013-0633-y
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DOI: https://doi.org/10.1007/s00229-013-0633-y