Yau's conjecture

Not to be confused with Yau's conjecture on the first eigenvalue.

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.[1]

The conjecture was resolved by Kei Irie, Fernando Codá Marques and André Neves in the generic case,[2] and by Antoine Song in full generality.[3]

References

1. 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.
2. 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.
3. 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)".