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

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, 2025, vol. 13, issue 9, 1-33

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.

Keywords: biharmonic submanifold; biconservative submanifolds; Chen’s conjecture; generalized Chen conjectures; BMO conjectures; L k conjecture; k -harmonic conjecture (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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