Yau's conjecture
| Yau's conjecture | |
|---|---|
| Field | Differential geometry |
| Conjectured by | Shing-Tung Yau |
| Conjectured in | 1982 |
| First proof by | Antoine Song |
| First proof in | 2018 |
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 first resolved by Fernando Codá Marques and André Neves in the case of positive Ricci curvature. Then, for the case of generic metrics, there are independent solutions from Kei Irie, Fernando Codá Marques and André Neves[2], Otis Chodosh and Christos Mantoulidis[3], and also Xin Zhou[4]. Finally, Antoine Song proved the conjectures for all metrics.[5] The proofs above use min-max theory, which is a form of infinite dimensional Morse theory for the area functional. A proof for the generic metric case via gluing construction was also later found by Adrian Chun-Pong Chu and Daniel Stern[6], in which the minimal surfaces constructed have bounded area but unbounded genus.
References
[edit]- ↑ 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.
- ↑ Chodosh, Otis; Mantoulidis, Christos (2020-01-01). "Minimal surfaces and the Allen--Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates". Annals of Mathematics. 191 (1). arXiv:1803.02716. doi:10.4007/annals.2020.191.1.4. ISSN 0003-486X.
- ↑ Zhou, Xin (2020-11-01). "On the Multiplicity One Conjecture in min-max theory". Annals of Mathematics. 192 (3). doi:10.4007/annals.2020.192.3.3. ISSN 0003-486X.
- ↑ 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.
- ↑ Chu, Adrian Chun-Pong; Stern, Daniel (2025-09-23). "Minimal surface doublings and electrostatics for Schr\"odinger operators". arXiv.org. Retrieved 2026-04-16.
External links
[edit]- Carlos Matheus (November 5, 2017). "Yau's conjecture of abundance of minimal hypersurfaces is generically true (in low dimensions)".
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