In differential geometry, Yau's conjecture is a mathematical conjecture which states that any closed Riemannian 3-manifold has infinitely many smooth closed immersed minimal surfaces. It is named after Shing-Tung Yau, who posed it as the 88th entry in his 1982 list of open problems in differential geometry.
The conjecture was resolved by Kei Irie, Fernando Codá Marques and André Neves in the generic case, and by Antoine Song in full generality.
References
- Yau, Shing Tung (1982). "Problem section". In Yau, Shing-Tung (ed.). Seminar on Differential Geometry. Annals of Mathematics Studies. Vol. 102. Princeton, NJ: Princeton University Press. pp. 669–706. doi:10.1515/9781400881918-035. ISBN 978-1-4008-8191-8. MR 0645762. Zbl 0479.53001.
- Irie, Kei; Marques, Fernando C.; Neves, André (2018). "Density of minimal hypersurfaces for generic metrics". Annals of Mathematics. 187 (3): 963–972. arXiv:1710.10752. doi:10.4007/annals.2018.187.3.8.
- Song, Antoine (2023). "Existence of infinitely many minimal hypersurfaces in closed manifolds". Annals of Mathematics. 197 (3): 859–895. arXiv:1806.08816. doi:10.4007/annals.2023.197.3.1.
External links
- Carlos Matheus (November 5, 2017). "Yau's conjecture of abundance of minimal hypersurfaces is generically true (in low dimensions)".
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