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

Bang-Yen Chen
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Bang-Yen Chen: Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, MI 48824–1027, USA

Mathematics, 2019, vol. 7, issue 8, 1-7

Abstract: 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.

Keywords: biharmonic submanifold; B.-Y. Chen’s conjecture; ? -invariant; submanifolds with parallel normalized mean curvature vector (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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