Yau's conjecture

{{Short description|Mathematical conjecture}}

{{distinguish|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.{{cite encyclopedia|last1=Yau|first1=Shing Tung|zbl=0479.53001|title=Problem section|encyclopedia=Seminar on Differential Geometry|pages=669–706|series=Annals of Mathematics Studies|volume=102|publisher=Princeton University Press|location=Princeton, NJ|year=1982|editor-last1=Yau|editor-first1=Shing-Tung|doi=10.1515/9781400881918-035|isbn=978-1-4008-8191-8|mr=0645762|author-link1=Shing-Tung Yau}}

The conjecture was resolved by Kei Irie, Fernando Codá Marques and André Neves in the generic case,{{cite journal |last1=Irie |first1=Kei |last2=Marques |first2=Fernando C. |author-link2=Fernando Codá Marques |last3=Neves |first3=André |author-link3=André Neves |year=2018|title=Density of minimal hypersurfaces for generic metrics |journal=Annals of Mathematics|volume=187|issue=3|pages=963–972|doi=10.4007/annals.2018.187.3.8|doi-access=free|arxiv=1710.10752 }} and by Antoine Song in full generality.{{cite journal |last1=Song |first1=Antoine |author-link1=Antoine Song|title=Existence of infinitely many minimal hypersurfaces in closed manifolds |year=2023|pages=859–895|journal=Annals of Mathematics|volume=197|issue=3|doi=10.4007/annals.2023.197.3.1|doi-access=free|arxiv=1806.08816}}