Vol. 294, No. 2, 2018

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ISSN: 0030-8730
Biharmonic hypersurfaces with constant scalar curvature in space forms

Yu Fu and Min-Chun Hong

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

Let Mn be a biharmonic hypersurface with constant scalar curvature in a space form Mn+1(c). We show that Mn has constant mean curvature if c > 0 and Mn is minimal if c 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 En+1 or hyperbolic space 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
Authors
Yu Fu
School of Mathematics
Dongbei University of Finance and Economics
Dalian
China
Min-Chun Hong
Department of Mathematics
University of Queensland
Brisbane
Australia