Some classifications of biharmonic hypersurfaces with constant scalar curvature

Shun Maeta and Ye-Lin Ou

Vol. 306 (2020), No. 1, 281–290

DOI: 10.2140/pjm.2020.306.281

Abstract

We give some classifications of biharmonic hypersurfaces with constant scalar curvature. These include biharmonic Einstein hypersurfaces in space forms, compact biharmonic hypersurfaces with constant scalar curvature in a sphere, and some complete biharmonic hypersurfaces of constant scalar curvature in space forms and in a nonpositively curved Einstein space. Our results provide additional cases (Theorem 2.3 and Proposition 2.8) that support the conjecture that a biharmonic submanifold in $S^{m+1}$ has constant mean curvature, and two more cases that support Chen’s conjecture on biharmonic hypersurfaces (Corollaries 2.2 and 2.7).

Keywords

biharmonic hypersurfaces, Einstein manifolds, constant scalar curvature, constant mean curvature

Mathematical Subject Classification 2010

Primary: 58E20

Secondary: 53C12

Milestones

Received: 4 November 2018

Revised: 8 September 2019

Accepted: 12 December 2019

Published: 14 June 2020