## 1. Introduction

**Chen’s Conjecture.**Biharmonic submanifolds of Euclidean spaces are minimal.

**Theorem**

**1.**

## 2. Preliminaries

**Theorem**

**2.**

## 3. Proof of the Theorem 1

**Claim.**If M is a biharmonic surface in ${\mathbb{E}}^{m}$ with a parallel normalized mean curvature vector, then it has a flat normal connection in ${\mathbb{E}}^{m}$, i.e., ${\mathrm{R}}^{D}=0$.

## 4. Some Remarks

**Conjecture.**There do not exist biharmonic submanifolds in Euclidean spaces with a parallel normalized mean curvature vector.

## Funding

## Acknowledgments

## Conflicts of Interest

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