A Schwarz lemma and a Liouville theorem for generalized harmonic maps. (January 2022)

• Record Type: Journal Article
• Authors: Chen, Qun; Li, Kaipeng; Qiu, Hongbing
• Is Part Of: Nonlinear analysis. Volume 214(2022)
• Publication Date: 2022-01
• Subjects: 58E20; Schwarz lemma -- Liouville theorem -- f-harmonic maps -- V-harmonic maps -- Gradient estimates; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2021.112556
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: 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.