#### Vol. 294, No. 2, 2018

 Recent Issues Vol. 297: 1 Vol. 296: 1  2 Vol. 295: 1  2 Vol. 294: 1  2 Vol. 293: 1  2 Vol. 292: 1  2 Vol. 291: 1  2 Vol. 290: 1  2 Online Archive Volume: Issue:
 The Journal Subscriptions Editorial Board Officers Special Issues Submission Guidelines Submission Form Contacts Author Index To Appear ISSN: 0030-8730 Other MSP Journals
Biharmonic hypersurfaces with constant scalar curvature in space forms

### Yu Fu and Min-Chun Hong

Vol. 294 (2018), No. 2, 329–350
##### Abstract

Let ${M}^{n}$ be a biharmonic hypersurface with constant scalar curvature in a space form ${\mathbb{M}}^{n+1}\left(c\right)$. We show that ${M}^{n}$ has constant mean curvature if $c>0$ and ${M}^{n}$ is minimal if $c\le 0$, provided that the number of distinct principal curvatures is no more than 6. This partially confirms Chen’s conjecture and the generalized Chen’s conjecture. As a consequence, we prove that there exist no proper biharmonic hypersurfaces with constant scalar curvature in Euclidean space ${\mathbb{E}}^{n+1}$ or hyperbolic space ${ℍ}^{n+1}$ for $n<7$.

##### Keywords
biharmonic maps, biharmonic submanifolds, Chen's conjecture, generalized Chen's conjecture
##### Mathematical Subject Classification 2010
Primary: 53C40, 53D12
Secondary: 53C42