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

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  • Bang-Yen Chen

    (Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, MI 48824–1027, USA)

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.

Suggested Citation

  • Bang-Yen Chen, 2019. "Chen’s Biharmonic Conjecture and Submanifolds with Parallel Normalized Mean Curvature Vector," Mathematics, MDPI, vol. 7(8), pages 1-7, August.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:8:p:710-:d:255290
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    References listed on IDEAS

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    1. Yu Fu, 2015. "Biharmonic hypersurfaces with three distinct principal curvatures in spheres," Mathematische Nachrichten, Wiley Blackwell, vol. 288(7), pages 763-774, May.
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