Nonexistence of proper p-biharmonic maps and Liouville type theorems I: case of \(p\ge 2\)

  • Yingbo Han
  • Yong LuoEmail author


Let \(u: (M, g)\rightarrow (N, h)\) be a map between Riemannian manifolds (Mg) and (Nh). The p-bienergy of u is defined by \(E_p(u)=\int _M|\tau (u)|^pd\nu _g\), where \(\tau (u)\) is the tension field of u and \(p>1\). Critical points of \(E_p(\cdot )\) are called p-biharmonic maps. In this paper we will prove nonexistence result of proper p-biharmonic maps when \(p\ge 2\). In particular when \(M=\mathbb {R}^m\), we get Liouville type results under proper integral conditions , which extend the related results of Baird, Fardoun and Ouakkas (2010).


p-biharmonic maps Nonpositive curvature Rigidity 

Mathematics Subject Classification

53C24 53C43 



Yingbo Han was supported by NSF of China (No.11971415) and Nanhu Scholars Program for Young Scholars of XYNU and the Universities Young Teachers Program of Henan Province (2016GGJS-096). Yong Luo was supported by the NSF of China (No.11501421). Both authors would like to thank the anonymous reviewer for the suggestions which make this paper more readable.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Human participants or animals

This article does not contain any studies with human participants or animals performed by any of the authors.

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Informed consent was obtained from all individual participants included in the study.


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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXinyang Normal UniversityXinyangChina
  2. 2.School of Mathematics and statisticsWuhan UniversityWuhanChina

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