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Article

Chen’s Biharmonic Conjecture and Submanifolds with Parallel Normalized Mean Curvature Vector

Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, MI 48824–1027, USA
Mathematics 2019, 7(8), 710; https://doi.org/10.3390/math7080710
Received: 30 June 2019 / Revised: 30 July 2019 / Accepted: 1 August 2019 / Published: 6 August 2019
(This article belongs to the Special Issue Complex and Contact Manifolds)
The well known Chen’s conjecture on biharmonic submanifolds in Euclidean spaces states that every biharmonic submanifold in a Euclidean space is a minimal one. For hypersurfaces, we know from Chen and Jiang that the conjecture is true for biharmonic surfaces in E 3 . Also, Hasanis and Vlachos proved that biharmonic hypersurfaces in E 4 ; and Dimitric proved that biharmonic hypersurfaces in E m with at most two distinct principal curvatures. Chen and Munteanu showed that the conjecture is true for δ ( 2 ) -ideal and δ ( 3 ) -ideal hypersurfaces in E m . Further, Fu proved that the conjecture is true for hypersurfaces with three distinct principal curvatures in E m with arbitrary m. In this article, we provide another solution to the conjecture, namely, we prove that biharmonic surfaces do not exist in any Euclidean space with parallel normalized mean curvature vectors. View Full-Text
Keywords: biharmonic submanifold; B.-Y. Chen’s conjecture; δ-invariant; submanifolds with parallel normalized mean curvature vector biharmonic submanifold; B.-Y. Chen’s conjecture; δ-invariant; submanifolds with parallel normalized mean curvature vector
MDPI and ACS Style

Chen, B.-Y. Chen’s Biharmonic Conjecture and Submanifolds with Parallel Normalized Mean Curvature Vector. Mathematics 2019, 7, 710. https://doi.org/10.3390/math7080710

AMA Style

Chen B-Y. Chen’s Biharmonic Conjecture and Submanifolds with Parallel Normalized Mean Curvature Vector. Mathematics. 2019; 7(8):710. https://doi.org/10.3390/math7080710

Chicago/Turabian Style

Chen, Bang-Yen. 2019. "Chen’s Biharmonic Conjecture and Submanifolds with Parallel Normalized Mean Curvature Vector" Mathematics 7, no. 8: 710. https://doi.org/10.3390/math7080710

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