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Recent Developments in Chen’s Biharmonic Conjecture and Some Related Topics

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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 study of biharmonic submanifolds in Euclidean spaces was introduced in the middle of the 1980s by the author in his program studying finite-type submanifolds. He defined biharmonic submanifolds in Euclidean spaces as submanifolds whose position vector field ( x ) satisfies the biharmonic equation, i.e., Δ 2 x = 0 . A well-known conjecture proposed by the author in 1991 on biharmonic submanifolds states that every biharmonic submanifold of a Euclidean space is minimal, well known today as Chen’s biharmonic conjecture. On the other hand, independently, G.-Y. Jiang investigated biharmonic maps between Riemannian manifolds as the critical points of the bi-energy functional. In 2002, R. Caddeo, S. Montaldo, and C. Oniciuc pointed out that both definitions of biharmonicity of the author and G.-Y. Jiang coincide for the class of Euclidean submanifolds. Since then, the study of biharmonic submanifolds and biharmonic maps has attracted many researchers, and many interesting results have been achieved. A comprehensive survey of important results on this conjecture and on many related topics was presented by Y.-L. Ou and B.-Y. Chen in their 2020 book. The main purpose of this paper is to provide a detailed survey of recent developments in those subjects after the publication of Ou and Chen’s book.

Suggested Citation

  • Bang-Yen Chen, 2025. "Recent Developments in Chen’s Biharmonic Conjecture and Some Related Topics," Mathematics, MDPI, vol. 13(9), pages 1-33, April.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:9:p:1417-:d:1642634
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    References listed on IDEAS

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    1. 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.
    2. Hiba Bibi & Bang‐Yen Chen & Dorel Fetcu & Cezar Oniciuc, 2023. "Parallel mean curvature biconservative surfaces in complex space forms," Mathematische Nachrichten, Wiley Blackwell, vol. 296(8), pages 3192-3221, August.
    3. Yu Fu, 2015. "Biharmonic hypersurfaces with three distinct principal curvatures in spheres," Mathematische Nachrichten, Wiley Blackwell, vol. 288(7), pages 763-774, May.
    4. Nurettin Cenk Turgay & Abhitosh Upadhyay, 2019. "On biconservative hypersurfaces in 4‐dimensional Riemannian space forms," Mathematische Nachrichten, Wiley Blackwell, vol. 292(4), pages 905-921, April.
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    1. 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.

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