A Schwarz lemma and a Liouville theorem for generalized harmonic maps

When the sectional curvature of the target manifold is negative, we establish a Schwarz lemma for f-harmonic maps, if the dimension of the domain and the target is large, the result improves Theorem 3 in Chen and Zhao (2017) for the case of V=∇f. When the sectional curvature of the target is nonposi...

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Bibliographic Details
Published inNonlinear analysis Vol. 214; p. 112556
Main Authors Chen, Qun, Li, Kaipeng, Qiu, Hongbing
Format Journal Article
LanguageEnglish
Published Elsevier Ltd 01.01.2022
Subjects
[formula omitted]-harmonic maps
Gradient estimates
Liouville theorem
Schwarz lemma
Schwarz lemma
V-harmonic maps
f-harmonic maps
Gradient estimates
58E20
Liouville theorem
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Summary:When the sectional curvature of the target manifold is negative, we establish a Schwarz lemma for f-harmonic maps, if the dimension of the domain and the target is large, the result improves Theorem 3 in Chen and Zhao (2017) for the case of V=∇f. When the sectional curvature of the target is nonpositive, we obtain a Liouville theorem for the general V-harmonic maps, as a consequence, any V-harmonic function u, satisfying |u(x)|=o(r(x)), on a complete Riemannian manifold with nonnegative Bakry–Emery–Ricci curvature is a constant. We also give some applications on gradient Ricci solitons and gradient solitons with potential which are solutions to Ricci-harmonic flow.
ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2021.112556