$V$-harmonic morphisms between Riemannian manifolds
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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.References
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Additional Information
- Guangwen Zhao
- Affiliation: School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China
- MR Author ID: 1238820
- Email: gwzhao@fudan.edu.cn
- 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
- © Copyright 2019 American Mathematical Society
- 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
- MathSciNet review: 4055960