Biharmonic hypersurfaces with constant scalar curvature in space forms

Yu Fu and Min-Chun Hong

Vol. 294 (2018), No. 2, 329–350

DOI: 10.2140/pjm.2018.294.329

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

Milestones

Received: 9 June 2016

Revised: 7 February 2017

Accepted: 5 October 2017

Published: 20 February 2018