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Korean Mathematical Society JKMS

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J. Korean Math. Soc. 2025; 62(1): 165-178

Online first article October 11, 2024      Printed January 1, 2025

https://doi.org/10.4134/JKMS.j230637

Copyright © The Korean Mathematical Society.

Local gradient estimate on Riemannian manifold with asymptotically non-negative Ricci curvature

Tian Chong; Hui Liu; Lingen Lu; Jingjing Zhang

Shanghai\ Polytechnic\ University,; Fudan\ University,; Fudan\ University,; Yunnan \ Normal \ University,

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In this paper, we study the asymptotic behavior of the energy densities of harmonic maps, exponentially harmonic functions and positive $p$-harmonic functions at infinity of a Riemannian manifold with asymptotically non-negative curvature. We prove that the energy densities of bounded harmonic maps, exponentially harmonic functions and positive $p$-harmonic functions all vanish at infinity.

Keywords: Asymptotically non-negative Ricci curvature, exponentially harmonic function, harmonic map, positive $p$-harmonic function, gradient estimate

MSC numbers: 53B20, 53C21, 53C43, 58E20

Supported by: Tian Chong is supported by the National Natural Science Foundation of China (NSFC) No.12001360. And Jingjing Zhang is supported by NSFC No.12261105.

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