Recent Developments in Chen’s Biharmonic Conjecture and Some Related Topics
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 (<i>x</i>) satisfies the b...
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2025-04-01
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| description | 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 (<i>x</i>) satisfies the biharmonic equation, i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mo>Δ</mo><mn>2</mn></msup><mi>x</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>. 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. |
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| spelling | doaj-art-1c25e7f8d5a84ffd98dcba0e78e88ce72025-08-20T02:59:11ZengMDPI AGMathematics2227-73902025-04-01139141710.3390/math13091417Recent Developments in Chen’s Biharmonic Conjecture and Some Related TopicsBang-Yen Chen0Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, MI 48824-1027, USAThe 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 (<i>x</i>) satisfies the biharmonic equation, i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mo>Δ</mo><mn>2</mn></msup><mi>x</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>. 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.https://www.mdpi.com/2227-7390/13/9/1417biharmonic submanifoldbiconservative submanifoldsChen’s conjecturegeneralized Chen conjecturesBMO conjectures<i>L<sub>k</sub></i> conjecture |
| spellingShingle | Bang-Yen Chen Recent Developments in Chen’s Biharmonic Conjecture and Some Related Topics Mathematics biharmonic submanifold biconservative submanifolds Chen’s conjecture generalized Chen conjectures BMO conjectures <i>L<sub>k</sub></i> conjecture |
| title | Recent Developments in Chen’s Biharmonic Conjecture and Some Related Topics |
| title_full | Recent Developments in Chen’s Biharmonic Conjecture and Some Related Topics |
| title_fullStr | Recent Developments in Chen’s Biharmonic Conjecture and Some Related Topics |
| title_full_unstemmed | Recent Developments in Chen’s Biharmonic Conjecture and Some Related Topics |
| title_short | Recent Developments in Chen’s Biharmonic Conjecture and Some Related Topics |
| title_sort | recent developments in chen s biharmonic conjecture and some related topics |
| topic | biharmonic submanifold biconservative submanifolds Chen’s conjecture generalized Chen conjectures BMO conjectures <i>L<sub>k</sub></i> conjecture |
| url | https://www.mdpi.com/2227-7390/13/9/1417 |
| work_keys_str_mv | AT bangyenchen recentdevelopmentsinchensbiharmonicconjectureandsomerelatedtopics |