Vol. 306, No. 1, 2020

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Some classifications of biharmonic hypersurfaces with constant scalar curvature

Shun Maeta and Ye-Lin Ou

Vol. 306 (2020), No. 1, 281–290
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).

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Keywords
biharmonic hypersurfaces, Einstein manifolds, constant scalar curvature, constant mean curvature
Primary: 58E20
Secondary: 53C12
Milestones
Received: 4 November 2018
Revised: 8 September 2019
Accepted: 12 December 2019
Published: 14 June 2020
Authors
 Shun Maeta Department of Mathematics Shimane University Matsue Japan Ye-Lin Ou Department of Mathematics Texas A & M University Commerce, TX United States