Biharmonic properly immersed submanifolds in Euclidean spaces

Abstract

We consider a complete biharmonic immersed submanifold $M$ in an 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.