Geometriae Dedicata

, Volume 164, Issue 1, pp 351–355 | Cite as

Biharmonic properly immersed submanifolds in Euclidean spaces

Original Paper

Abstract

We consider a complete biharmonic immersed submanifold M in a Euclidean space \({\mathbb{E}^N}\). Assume that the immersion is proper, that is, the preimage of every compact set in \({\mathbb{E}^N}\) is also compact in M. Then, we prove that M is minimal. It is considered as an affirmative answer to the global version of Chen’s conjecture for biharmonic submanifolds.

Keywords

Biharmonic map Biharmonic submanifold Chen’s conjecture 

Mathematics Subject Classification (2010)

Primary: 58E20 Secondary: 53C43 53A07 

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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Division of Mathematics, GSISTohoku UniversitySendaiJapan

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