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Proceedings of the American Mathematical Society

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$ V$-harmonic morphisms between Riemannian manifolds


Author: Guangwen Zhao
Journal: Proc. Amer. Math. Soc. 148 (2020), 1351-1361
MSC (2010): Primary 58E20, 53C43; Secondary 32Q60, 35B53
DOI: https://doi.org/10.1090/proc/14811
Published electronically: November 4, 2019
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Abstract: A $ V$-harmonic morphism $ u:M\to N$ between Riemannian manifolds is a smooth map which pulls back germs of harmonic functions on $ N$ to germs of $ V$-harmonic functions on $ M$, where $ V$ is a smooth vector field on $ M$. In this paper, we give some characterizations and examples of $ V$-harmonic morphisms. In addition, a dilation estimate and a Liouville-type theorem of $ V$-harmonic morphisms from noncompact complete manifolds are also established. As applications, we obtain the Liouville-type theorems for $ V$-harmonic morphisms from complete manifolds of nonnegative Bakry-Émery Ricci curvature, especially complete steady or shrinking Ricci solitons, to manifolds of dimension at least three or compact Riemann surface of genus at least two.


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Additional Information

Guangwen Zhao
Affiliation: School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China
Email: gwzhao@fudan.edu.cn

DOI: https://doi.org/10.1090/proc/14811
Keywords: $V$-harmonic map, $V$-harmonic morphism, horizontally weakly conformal map, dilation, Liouville-type theorem
Received by editor(s): May 26, 2019
Received by editor(s) in revised form: July 22, 2019
Published electronically: November 4, 2019
Communicated by: Guofang Wei
Article copyright: © Copyright 2019 American Mathematical Society