Yau's conjecture

In differential geometry, Yau's conjecture from 1982, named after Shing-Tung Yau, is a mathematical conjecture which then states that a closed Riemannian three-manifold has an infinite number of smooth closed immersed minimal surfaces. It was the first problem in the Minimal submanifolds section in Yau's list of open problems.

The conjecture has recently been claimed by Kei Irie, Fernando Codá Marques and André Neves in the generic case,[1][2] and by Antoine Song in full generality.[3]

References

  1. ↑ Irie, Kei; Marques, Fernando Codá; Neves, André. "Density of minimal hypersurfaces for generic metrics". arXiv:1710.10752 [math.DG].
  2. ↑ Carlos Matheus. "Yau's conjecture of abundance of minimal hypersurfaces is generically true (in low dimensions)".
  3. ↑ Song, Antoine. "Existence of infinitely many minimal hypersurfaces in closed manifolds". arXiv:1806.08816 [math.DG].

Further reading

  • "Problem section" of Shing-Tung Yau's book Seminar on Differential Geometry.
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